Analytical Calculation of Magnetic Field and Analysis of Rotor Permeability Effects on Permanent Magnet Synchronous Motor with Fractional Slot Concentrated Winding

Abstract

Accurate calculation of the flux and the magnetic field distribution of fractional slot concentrated winding permanent magnet synchronous motor (FSCW PMSM) is the basis for motor performance analysis, and rapid calculation is key. In this paper, to solve the problem of difficult modeling and accuracy guarantee of the flux linkage differential method, a method is proposed to calculate the flux and the no-load back EMF by the slotless subdomain model. By introducing the leakage flux calculation link, the calculation accuracy is improved, the analytical method results are compared with the finite element method results, and the effectiveness of the proposed method is verified. On this basis, the nonlinear variations of the magnetic field and the no-load back EMF with rotor permeability are determined, and the influence mechanism of rotor length and rotor permeability on the main magnetic circuit is revealed. Finally, an experiment of the prototype is carried out, and the correctness and accuracy of the analytical method and the finite element method is verified by comparing with the experimental results.

1. Introduction

In recent years, FSCW PMSM has been extensively utilized in wind power generation, electric vehicles and low-speed high-torque direct drive field due to its short axial length, high torque density, and excellent magnetization and speed regulation capabilities. Consequently, they have become a significant focus of research [1,2].

Fast and accurate calculation of the magnetic field distribution and flux of the motor is crucial. Current methods used in motor magnetic fields research include the finite element method (FEM), the equivalent magnetic circuit method (EMC), and the analytical calculation method [3]. Among these, the analytical calculation method, particularly the one based on the sub-domain model and Fourier decomposition, stands out due to its clear physical concepts and short computation time, making it highly suitable for shortening the design cycle [4].

Recently, there has been a significant increase in research focused on the analytical calculation method to optimize various motor performances [5,6].

By the correct description of the magnetic field in the slot region, B. Prieto proposes the expressions for calculating the leakage inductance with different winding layers in [7]; the results are highly accurate even for large slot-opening-to-slot-pitch ratios. In references [8,9], the analytical model of a surface inset permanent magnet motor is established, and the effect of q-axis air-gap length and pole arc ratio on the cogging torque, back-electromotive force (EMF), saliency ratio, and output torque is also investigated. M. Hajdinjak has analyzed and predicted the flux density distribution in the air gap and the permanent magnets of the u-shaped slotless brushless motor; the proposed method can also be applied to v-type IPMs with modifications [10,11]. Additionally, vernier motor performance has been analyzed through analytical methods, identifying feasible combinations of PM and rotor slot numbers [12].

Different from the integral slot distributed winding (ISDW) PMSM, FSCW PMSM has a close number of slots and poles and have wider stator teeth, and the structural feature will cause larger leakage flux. To calculate the flux of a motor, an analytical model with slot domains must be built to integrate the vector magnetic potential in the stator slot [13,14]. However, due to the large number of periodic boundary conditions, and multidimensional vector variables, the modeling method of the slotted subdomain model is difficult to establish and the calculation results are prone to errors due to the size of the slot subdomain model [15,16].

Moreover, conventional analytical models often neglect the magnetoresistance of the rotor core to simplify calculations, despite the rotor material’s electromagnetic characteristics significantly affecting the motor’s magnetic circuit and performance [17,18]. In more industrial applications, the rotor is usually of solid structure due to the process technology and structural dimension, and the influence of rotor material properties and magnetic circuit length on motor performance must be fully considered [19]. Especially in the servo field, where high accuracy and response speed are required, analysis of the effect of rotor permeability is of even greater importance and practical application [20,21].

Thus, in this paper, an 8-pole/12-slot FSCW PMSM is taken as an example. Firstly, the subdomain model of the motor is established using an analytical method. Subsequently, according to the structural characteristics of the FSCW PMSM, the zigzag leakage coefficient of the motor is calculated, and the accurate and fast calculation of the flux and the no-load back-EMF using the slotless subdomain model is realized. The calculation results of the air-gap magnetic field are compared with the FEM results to verify the accuracy of the analytical models. On this basis, the effect of rotor permeability on the motor performance is analyzed by using the proposed method, and the influence mechanism of the rotor permeability and thickness is revealed. Finally, the correctness and accuracy of the calculation method are verified through experimental tests.

2. Analytical Subdomain Mode

In this section, the modeling of motor subdomains was described (include Appendix A). The following assumptions are made.

(1)

The end effect is neglected, and the materials are isotropic.

(2)

The magnetic vector potential has only the Z-axis component and is uniformly distributed along the axial direction.

(3)

The slots effect is equated by using the air gap coefficient. The expression of the equivalent air-gap length $g_{\delta }$ is as follows [22]:

$$g_{\delta }=\frac{2}{\pi }\left\{arctan\frac{1}{2}\left(\frac{b_0}{h_{PM}+g}\right)-\left(\frac{h_{PM}+g}{b_0}\right)·\right.\left.ln\left[1+\frac{1}{4}{\left(\frac{b_0}{h_{PM}+g}\right)}^2\right]\right\}$$

(1)

The inner radius of the equivalent stator can be determined as follows:

$$R_{g\delta }=R_r+g_{\delta }$$

(2)

where, $b_0$ is the slot notch width, $h_{PM}$ is the thickness of the PM, and $g$ is the air gap length. The motor subdomain model is shown in Figure 1.

Regions 1–4 are the equivalent stator, airgap, permanent magnet, and rotor, respectively. $R_{g\delta }$ is the inner radius of the equivalent stator, $R_m$ is the outer radius of the permanent magnet region, $R_r$ is the rotor radius, and $R_a$ is the inner radius of the rotor. $\theta$ is the angle in the circumferential direction.

2.1. Equivalent Model of Remanence

When using the analytical calculation method, the mathematical model of the PM region is usually modeled by simulating the volume current.

Considering that there is no macroscopic current in the PM, for the pre-magnetized permanent magnet, the remanence can be expressed as follows:

$$\overrightarrow{M}=M_r\overrightarrow{r}+M_{\theta }\overrightarrow{\theta }$$

(3)

The series expansions of $M_r$ and $M_{\theta }$ are as follows:

$$\left\{\begin{array}{c}{M}_r^{\prime }={\displaystyle \sum _{n=\mathrm{1,3},5\dots }^{np}}{M}_{rn}\cos \left(np\theta +np{\theta }_0\right)\\ M_{\theta }^{\prime }={\displaystyle \sum _{n=\mathrm{1,3},5\dots }^{np}}{M}_{\theta n}\sin \left(np\theta +np{\theta }_0\right)\end{array}\right.$$

(4)

where, $\overrightarrow{M}$ is the remanence vector, $M_r$ and $M_{\theta }$ are the radial and tangential components of remanence, respectively, $n$ is the harmonic order, $p$ is pole-pairs, and $\theta$ is the angle in the circumferential direction.

2.2. Vector Partial Differential Equations

When analyzing the PMSM performance, the conductivity of the motor core can be ignored. The flux equation of each subdomain can be represented as follows:

$$\left\{\begin{array}{l}\frac{{\partial }^2{A}_s}{\partial r^2}+\frac{1}{r}\cdot \frac{\partial A_s}{\partial r}+\frac{1}{r^2}\cdot \frac{{\partial }^2{A}_s}{\partial \theta }=0\\ \frac{{\partial }^2{A}_{gap}}{\partial r^2}+\frac{1}{r}\cdot \frac{\partial A_{gap}}{\partial r}+\frac{1}{r^2}\cdot \frac{{\partial }^2{A}_{gap}}{\partial \theta }=0\\ \frac{{\partial }^2{A}_{pm}}{\partial r^2}+\frac{1}{r}\cdot \frac{\partial A_{pm}}{\partial r}+\frac{1}{r^2}\cdot \frac{{\partial }^2{A}_{pm}}{\partial \theta }=\frac{{\mu }_0}{r}\left(M_{\theta }^{\prime }-\frac{\partial M_r^{\prime }}{\partial \theta }\right)\\ \frac{{\partial }^2{A}_r}{\partial r^2}+\frac{1}{r}\cdot \frac{\partial A_r}{\partial r}+\frac{1}{r^2}\cdot \frac{{\partial }^2{A}_r}{\partial \theta }=0\end{array}\right.$$

(5)

where, $A_s$ is the stator region vector magnetic potentials, $A_{gap}$ is the vector air gap region magnetic potentials, $A_{pm}$ is the permanent magnet vector magnetic potentials, and $A_r$ is the vector magnetic potentials of the rotor region.

2.3. Laplace’s Solution Equation and Quasi-Poisson’s Equation

Based on the Green formula, the equation can be further obtained:

The general solution of the equations can be obtained as follows:

In the passive region,

$$A_{s,gap,r}={\sum }_n\left[\left(A_n^{s,gap,r}{\left(\frac{r}{R_1}\right)}^{np}+B_n^{s,gap,r}{\left(\frac{r}{R_2}\right)}^{-np}\right)cos\left(np\theta \right) \right.+\left. (C_n^{s,gap,r}{\left(\frac{r}{R_1}\right)}^{np}+D_n^{s,gap,r}{\left(\frac{r}{R_2}\right)}^{-np})sin(np\theta )\right]$$

(6)

where $R_1$ and $R_2$ are the solution region outer radius and inner radius, respectively. $A_n^{s,gap,r}$, $B_n^{s,gap,r}$, $C_n^{s,gap,r}$, and $D_n^{s,gap,r}$ are the corresponding undetermined coefficients for each region.

In the active region,

$$\begin{gathered} A_{pm}=\sum _n\left[\begin{array}{c}\left(A_n^{\prime }{\left(\frac{r}{R_m}\right)}^{np}+B_n^{\prime }{\left(\frac{r}{R_r}\right)}^{-np}\right)cos\left(np\theta \right)\\ +(C_n^{\prime }{\left(\frac{r}{R_m}\right)}^{np}+D_n^{\prime }{\left(\frac{r}{R_r}\right)}^{-np})\sin (np\theta )\end{array}\right] \\ +\sum _n\left[{\Gamma }_s(r)+{\Gamma }_c(r)\right]\sin (np\theta ) \end{gathered}$$

(7)

$${\Gamma }_s(r)=\left\{\begin{array}{c}\frac{(np){\mu }_0}{1-{(np)}^2}{rM}_{rn} np\ne 1\\ \frac{{\mu }_0}{2}rlnr M_{rn} np=1\end{array}\right.$$

(8)

$${\Gamma }_c\left(r\right)=\left\{\begin{array}{c}\frac{{\mu }_0}{1-{\left(np\right)}^2}{rM}_{\theta n} np\ne 1\\ \frac{{\mu }_0}{2}rlnr M_{\theta n} np=1\end{array}\right.$$

(9)

where $A_n^{\prime }$, $B_n^{\prime }$, $C_n^{\prime }$, and $D_n^{\prime }$ are the undetermined coefficients.

2.4. Interfaces Conditions

The conditions for the boundaries and intersections of the different subdomains can be defined as follows:

• The permeability of the stator is infinite, the flux enters the stator core vertically, and the tangential component of the magnetic field intensity is 0.

  $$\frac{1}{{\mu }_0}{\left.\frac{\partial A_{gap}}{\partial r}\right|}_{r=R_{g\delta }}=0$$

  (10)
• The magnetic field is continuous, and the tangential component of the flux density is continuous.

  $$\left\{\begin{array}{c}{\left.A_{gap}\right|}_{r=R_m}={\left.A_{pm}\right|}_{r=R_m}\\ \frac{1}{{\mu }_0}{\left.\frac{\partial A_{gap}}{\partial r}\right|}_{r=R_m}=\frac{1}{{\mu }_r}{\left.\frac{\partial A_{pm}}{\partial r}\right|}_{r=R_m}\end{array}\right.$$

  (11)

  $$\left\{\begin{array}{c}{\left.A_{pm}\right|}_{r=R_r}={\left.A_r\right|}_{r=R_r}\\ \frac{1}{{\mu }_r}{\left.\frac{\partial A_{pm}}{\partial r}\right|}_{r=R_r}=\frac{1}{{\mu }_{Fe}}{\left.\frac{\partial A_r}{\partial r}\right|}_{r=R_r}\end{array}\right.$$

  (12)
• There is no flux leakage at the inner boundary of the rotor, and the normal component of the magnetic flux density is zero.

  $$\frac{1}{r}{\left.\frac{\partial A_r}{\partial \theta }\right|}_{r=R_a}=0$$

  (13)

2.5. Matrix Equation and Solution

According to the conditions for the boundaries and intersections of the different subdomains, the differential equations can be established. Construct the matrix equations in terms of the n-th components of the magnetic field; the undetermined coefficients can be further obtained.

When the PM is magnetized in the radial direction,

$$\left\{\begin{array}{c}{M}_{rn}=2\frac{B_r}{{\mu }_0}{\alpha }_p\frac{sin\frac{n\pi {\alpha }_p}{2}}{\frac{n\pi {\alpha }_p}{2}}\\ M_{\theta n}=0\end{array}\right.$$

(14)

where, $B_r$ is the remanence of permanent magnet, ${\alpha }_p$ is pole-arc to pole-pitch ratio, and ${\mu }_0$ is the permeability of vacuum.

The form of the general solution in Equations (6) and (7) can be simplified as follows:

$$\left\{\begin{array}{c}{A}_s={\displaystyle \sum _n}(C_{1n}{\left(\frac{r}{R_o}\right)}^{np}+D_{1n}{\left(\frac{r}{R_g}\right)}^{-np})sin(np\theta )\\ A_{gap}={\displaystyle \sum _n}(C_{2n}{\left(\frac{r}{R_g}\right)}^{np}+D_{2n}{\left(\frac{r}{R_m}\right)}^{-np})sin(np\theta )\\ A_{pm}={\displaystyle \sum _n}\left[\left({C_{3n}\left(\frac{r}{R_m}\right)}^{np}+D_{3n}{\left(\frac{r}{R_r}\right)}^{-np}\right)+{\Gamma }_s(r)\right]\sin (np\theta )\\ A_r={\displaystyle \sum _n}(C_{4n}{\left(\frac{r}{R_r}\right)}^{np}+D_{4n}{\left(\frac{r}{R_a}\right)}^{-np})sin(np\theta )\end{array} \right.$$

(15)

By applying the boundary condition (28), at $r=R_{g\delta }$,

$$C_{2n}-D_{2n}{\left(\frac{R_m}{R_{g\delta }}\right)}^{np}=0$$

(16)

The above equations can be written in the matrix format as follows

$$K_{11}{C}_2+K_{12}{D}_2=0$$

(17)

$$K_{11}=I_N, K_{12}={-G}_1$$

(18)

where $I_N$ is the $n\times n$ identity matrix, $C_2$ and $D_2$ are the matrices composed of the undetermined coefficients with different harmonics, and $G_1$ is the matrix composed of the constant.

$$C_2=diag{\left[C_{2(p)} C_{2(3p)} C_{2(np)}\right]}_{n\times n}$$

(19)

$$G_1=diag{\left[{\left(\frac{R_m}{R_{g\delta }}\right)}^{np}\right]}_{n\times n}$$

(20)

By applying the boundary condition (31), at $r=R_a$,

$$C_{4n}{\left(\frac{R_a}{R_r}\right)}^{np}+D_{4n}=0$$

(21)

Construct the similar matrix equations as follows:

$$K_{65}{C}_4+K_{66}{D}_4=0$$

(22)

$$K_{65}=G_3,K_{66}=I_N$$

(23)

$$G_3=diag{\left[{\left(\frac{R_a}{R_r}\right)}^{np}\right]}_{n\times n}$$

(24)

By applying the boundary condition (11), at $r=R_m$, for $np\ne 1$,

$$\left\{\begin{array}{c}{C}_{2n}{\left(\frac{R_m}{R_g}\right)}^{np}+D_{2n}-C_{3n}-D_{3n}{\left(\frac{R_r}{R_m}\right)}^{np}=\frac{(np){\mu }_0}{1-{(np)}^2}{R_{m}M}_{rn}\\ {\mu }_r\left(C_{2n}{\left(\frac{R_m}{R_g}\right)}^{np}-D_{2n}\right)-{\mu }_0\left(C_{3n}-D_{3n}{\left(\frac{R_m}{R_r}\right)}^{np}\right)\\ =\frac{{{\mu }_0}^2}{1-{(np)}^2}{M}_{rn}\end{array} \right.$$

(25)

For $np=1$

$$\left\{\begin{array}{c}{C}_{21}\left(\frac{R_m}{R_g}\right)+D_{21}-C_{31}-D_{31}\left(\frac{R_r}{R_m}\right)=\frac{{\mu }_0}{2}{R}_{m}ln{R}_m{M}_{r1}\\ {\mu }_r\left(C_{21}\left(\frac{R_m}{R_g}\right)-D_{21}\right)-{\mu }_0\left(C_{31}-D_{31}\left(\frac{R_m}{R_r}\right)\right)\\ ={\mu }_0\frac{{\mu }_0}{2}\left(1+ln{R}_m\right)M_{r1}\end{array} \right.$$

(26)

It also can be obtained that,

$$\left\{\begin{array}{c}{K}_{21}{C}_2+K_{22}{D}_2+K_{23}{C}_3+K_{24}{D}_3=Y_1\\ K_{31}{C}_2+K_{32}{D}_2+K_{33}{C}_3+K_{34}{D}_3=Y_2\end{array}\right.$$

(27)

$$Y_1=\frac{{\mu }_{0}K}{I_K-K^2}{R}_m{M}_r^{\prime }$$

(28)

$$Y_2=\frac{{{\mu }_0}^2}{I_K-K^2}{R}_m{M}_r^{\prime }$$

(29)

$$K_{21}=G_1, K_{22}=I_N, K_{23}=-I_N, K_{24}={-G}_2$$

(30)

$$K_{31}={{\mu }_{r}G}_1, K_{32}={-\mu }_r{I}_N, K_{33}=-{{\mu }_{0}I}_N, K_{34}={{\mu }_{0}G}_2$$

(31)

where $M_r$ is the matrix composed of the residual magnetism harmonics, $G_2$ is the matrix composed of constant.

$$G_2=diag{\left[{\left(\frac{R_a}{R_r}\right)}^{np}\right]}_{n\times n}$$

(32)

$$K=diag{\left[1,3\cdots N\right]}_{n\times n}$$

(33)

By applying the boundary condition (12), at $r=R_r$, for $np\ne 1$,

$$\left\{\begin{array}{c}{C}_{3n}{\left(\frac{R_m}{R_g}\right)}^{np}+D_{3n}-C_{4n}-D_{4n}{\left(\frac{R_r}{R_m}\right)}^{np}\\ =\frac{(np){\mu }_0}{1-{(np)}^2}{R_{r}M}_{rn}\\ {\mu }_r\left(C_{3n}{\left(\frac{R_r}{R_m}\right)}^{np}-D_{3n}\right)-{\mu }_{Fe}\left(C_{4n}-D_{4n}{\left(\frac{R_r}{R_a}\right)}^{np}\right)\\ ={\mu }_r\frac{{\mu }_0}{1-{(np)}^2}{M}_{rn}\end{array} \right.$$

(34)

For $np=1$

$$\left\{\begin{array}{c}{C}_{31}\left(\frac{R_m}{R_g}\right)+D_{31}-C_{41}-D_{41}\left(\frac{R_r}{R_m}\right)=-\frac{{\mu }_0}{2}{R}_{r}ln{R}_r{M}_{r1}\\ {\mu }_r\left(C_{31}\left(\frac{R_r}{R_m}\right)-D_{31}\right)-{\mu }_{Fe}\left(C_{41}-D_{41}\left(\frac{R_r}{R_a}\right)\right)=\\ -{\mu }_r\frac{{\mu }_0}{2}\left(1+ln{R}_r\right)M_{r1}\end{array} \right.$$

(35)

$$\left\{\begin{array}{c}{K}_{43}{C}_3+K_{44}{D}_3+K_{45}{C}_4+K_{46}{D}_4=Y_3\\ K_{53}{C}_3+K_{54}{D}_3+K_{55}{C}_4+K_{56}{D}_4=Y_4\end{array}\right.$$

(36)

$$Y_3=-\frac{{\mu }_{0}K}{I_K-K^2}{R}_r{M}_r$$

(37)

$$Y_4=\frac{{\mu }_0{\mu }_r}{I_K-K^2}{R}_r{M}_r$$

(38)

$$K_{43}=G_2,{ K}_{44}=I_N,{ K}_{45}=-I_N,{ K}_{46}={-G}_3$$

(39)

$$K_{53}={{\mu }_{Fe}G}_2, K_{54}={-\mu }_{Fe}{I}_N, K_{55}=-{{\mu }_{r}I}_N, K_{56}={{\mu }_{0}G}_2$$

(40)

The final matrix equation is shown in Equation (41).

$$\left[\begin{array}{c}{K}_{11} K_{12} 0 0 0 0\\ K_{21} K_{22} K_{23} K_{24} 0 0\\ K_{31} K_{32} K_{33} K_{34} 0 0\\ 0 0 K_{43} K_{44} K_{45} K_{46}\\ 0 0 K_{53} K_{54} K_{55} K_{56}\\ 0 0 0 0 K_{56} K_{66}\end{array}\right]\left[\begin{array}{c}{C}_2\\ D_2\\ C_3\\ D_3\\ C_4\\ D_4\end{array}\right]=\left[\begin{array}{c}0\\ Y_1\\ Y_2\\ Y_3\\ Y_4\\ 0\end{array}\right]$$

(41)

By solving matrix equations, the undetermined coefficient can be obtained, and the electromagnetic performance of the motor can be calculated.

3. Leakage Flux Calculation

Due to the structural characteristics of FSCW PMSM, there are various types of leakage fluxes in the motor, which will have an important impact on the calculation of the average flux and the back-EMF when using the analytic method. Figure 2 shows the main leakages in the FSCW PMSM.

Where ${\Phi }_{\sigma 1}$ is the leakage flux between the PM and the rotor, and ${\Phi }_{\sigma 2}$ is the leakage between permanent magnets. ${\Phi }_{\sigma 31}$, ${\Phi }_{\sigma 32}$, and ${\Phi }_{\sigma 33}$ are the three forms of the zigzag leakages, of which, ${\Phi }_{\sigma 33}$ is the main component.

Based on the concept of zigzag leakage flux, in this paper, the average leakage flux coefficient of the motor is calculated according to the periodical variation of the leakage flux in a single stator tooth.

The zigzag leakage flux is related to the area of the permanent magnets facing the stator teeth. As shown in Figure 3a, assumed that when the axis of the i-th stator tooth coincides with the q-axis of the magnetic pole, $x=0$. At this position, the zigzag leakage flux maximum and the area of the PM facing the stator tooth is $∆x=\left(b_{\mathrm{s}}+b_{\mathrm{m}}-\tau \right)/2$.

Where, $b_{\mathrm{s}}$, $b_{\mathrm{m}}$, and $\tau$ are the stator tooth pitch, the width of PM, and the pole pitch, respectively.

As the rotor continues moving, $∆x$ gradually decreases, and when $x=\left(b_{\mathrm{s}}+b_{\mathrm{m}}-\tau \right)/2$, the leakage flux from the N pole will no longer enter the tooth top of the i-th stator tooth, and the zigzag leakage flux of the i-th stator tooth is 0 (as shown in Figure 3b). When the i-th tooth moves to the next permanent magnet region, and after $x\ge \left(b_s+b_m-\tau \right)/2$, the zigzag leakage flux of the i-th tooth will increase (as shown in Figure 3c).

With the increase in $x$, the zigzag leakage flux of the i-th stator tooth varies linearly and periodically. The variation shows symmetry in the region of a pair of poles and is shown in Figure 4.

The zigzag leakage flux of the i-th stator tooth ${\mathsf{\Phi }}_{\mathrm{L}\mathrm{t}\mathrm{i}}$ can be expressed as follows

$${\mathsf{\Phi }}_{\mathrm{L}\mathrm{t}\mathrm{i}}=\left\{\begin{array}{cc}0& [\xi ,\frac{\pi D_{\mathrm{g}}}{2p}-\xi ]\\ -\frac{2p{\mathsf{\Phi }}_{\mathrm{m}}}{{\alpha }_{\mathrm{p}}\pi D_{\mathrm{g}}}x+\frac{{\mathsf{\Phi }}_{\mathrm{m}}}{{\alpha }_{\mathrm{p}}}\left(\frac{p}{z}+\frac{{\alpha }_{\mathrm{p}}}{2}-\frac{1}{2}\right)& other\end{array}\right.$$

(42)

$${\mathsf{\Phi }}_{\mathrm{L}\mathrm{t}\mathrm{i}}=\left\{\begin{array}{cc}0& [\xi ,\frac{\pi D_{\mathrm{g}}}{2p}-\xi ]\\ -\frac{2p{\mathsf{\Phi }}_{\mathrm{m}}}{{\alpha }_{\mathrm{p}}\pi D_{\mathrm{g}}}x+\frac{{\mathsf{\Phi }}_{\mathrm{m}}}{{\alpha }_{\mathrm{p}}}\left(\frac{p}{z}+\frac{{\alpha }_{\mathrm{p}}}{2}-\frac{1}{2}\right)& other\end{array}\right.$$

(43)

$$\xi =\frac{\pi D_{\mathrm{g}}}{2}\left(\frac{1}{z}+\frac{{\alpha }_{\mathrm{p}}}{2p}-\frac{1}{2p}\right)$$

(44)

The average zigzag leakage flux of each stator tooth or each pole can be obtained according to the leakage flux variation in one cycle.

$${\mathsf{\Phi }}_{Lti-avg}=\frac{{{\displaystyle \int }}_c^{ }{\mathsf{\Phi }}_{Lti}}{\tau }=\frac{{\mathsf{\Phi }}_{\mathrm{m}}p}{{\alpha }_p}\left(\frac{p}{z}+\frac{{\alpha }_{\mathrm{p}}}{2}-\frac{1}{2}\right)\left(\frac{1}{z}+\frac{{\alpha }_{\mathrm{p}}}{2p}-\frac{1}{2p}\right)$$

(45)

$${\mathsf{\Phi }}_{avg}={\mathsf{\Phi }}_{Lti-avg}\frac{Z}{2p}$$

(46)

where ${\mathsf{\Phi }}_{Lti-avg}$ and ${\mathsf{\Phi }}_{avg}$ are the average zigzag leakage flux of a single tooth, and the average zigzag leakage flux of a pole, respectively. Therefore, the zigzag leakage coefficient of the FSCW PMSM is as follows

$${\sigma }_{Lt}=\frac{{\mathsf{\Phi }}_m}{{\mathsf{\Phi }}_m-{\mathsf{\Phi }}_{avg}}$$

(47)

4. Verification of Finite Element Method

In this section, the two-dimensional finite element model is established as shown in Figure 5 and the results of the analytical method were verified. The main parameters of the motor are shown in

Table 1. Main parameters of the motor.

Parameter	Quantity
Br (T)	1.21
Stator inner Diameter/mm	42.2
Rotor out Diameter/mm	34.85
Slot width angle (°)	12.28
Winding turns	62
PM Thickness/mm	2.775
Slot opening degree (°)	0.54
Length/mm	50.8
Number of Slots	12
Number of Poles	8
Air-gap Length/mm	0.9
Rated speed (rpm)	3000
Pole-arc coefficient	0.97
Stator out Diameter/mm	72
${\mu }_r$	1.08
Magnetization	Radial

.

4.1. Calculation of Air Gap Magnetic Density

By solving the above analytical model, the flux density can be obtained, and the radial component of the air gap magnetic density obtained by the analytical methods can be expressed as follows:

$$\begin{gathered} {\mathrm{B}}_{\mathrm{r}}=\frac{1}{\mathrm{r}}\frac{\partial {\mathrm{A}}_{\mathrm{g}\mathrm{a}\mathrm{p}}}{\partial \mathsf{\theta }}=-{\sum }_{\mathrm{n}}\left[\left({\mathrm{C}}_2\frac{\mathrm{n}\mathrm{p}}{{\mathrm{R}}_{\mathrm{g}}}{\left(\frac{\mathrm{r}}{{\mathrm{R}}_{\mathrm{g}}}\right)}^{\mathrm{n}\mathrm{p}-1}+{\mathrm{D}}_2\frac{\mathrm{n}\mathrm{p}}{{\mathrm{R}}_{\mathrm{m}}}{\left(\frac{\mathrm{r}}{{\mathrm{R}}_{\mathrm{m}}}\right)}^{-\mathrm{n}\mathrm{p}-1}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{n}\mathrm{p}\mathsf{\theta }\right)\right]+{\sum }_n\left[\left(C_2\frac{np}{R_g}{\left(\frac{r}{R_g}\right)}^{np-1}+ \\ {D}_2\frac{np}{R_m}{\left(\frac{r}{R_m}\right)}^{-np-1}\right)cos\left(np\theta \right)\right] \end{gathered}$$

(48)

$$\begin{gathered} B_t=\frac{\partial A_{gap}}{\partial r}=-{\sum }_n\left[\left(C_2\frac{np}{R_g}{\left(\frac{r}{R_g}\right)}^{np-1}-D_2\frac{np}{R_m}{\left(\frac{r}{R_m}\right)}^{-np-1}\right)cos\left(np\theta \right)\right]-{\sum }_n\left[\left(C_2\frac{np}{R_g}{\left(\frac{r}{R_g}\right)}^{np-1}- \\ {D}_2\frac{np}{R_m}{\left(\frac{r}{R_m}\right)}^{-np-1}\right)cos\left(np\theta \right)\right] \end{gathered}$$

(49)

Figure 6 shows the comparison of the radial flux density of the air gap calculated by the analytical method and the finite element method, at different moments.

As shown in Figure 6, the results obtained by using the two methods are highly consistent. But the flux density cannot be accurately predicted.

This is because the FSCW PMSMs are often equipped with the segmented stator and the semi-closed slots as the characteristics of the concentrated winding form, and the stator tooth tips are always thin, and the tooth tip area more prone to saturation. In addition, the zigzag leakage flux will aggravate the saturation of the stator tooth tip, which causes an increase in the calculation error of the magnetic field of this region. Figure 7 shows the magnetic density.

As can be seen from Figure 7, the magnetic flux density of different tooth tips shows significant differences. When the stator teeth axis is facing the d-axis of the rotor, the maximum flux density of the tooth tips on both sides of the teeth is 2.02 T. When the stator teeth axis is facing the q-axis of the rotor, the maximum flux density of the tooth tips on both sides of the teeth is 2.06 T.

4.2. Calculation of No-Load Back-EMF

The calculation method of the no-load back-EMF is as follows:

$$E_0=4.44fN{K}_{dp}{B}_1{K}_c\tau L_{ef}$$

(50)

where, $f$ is the frequency, $N$ is the effective number of turns per phase, $K_{dp}$ is the winding coefficient, $B_1$ is the fundamental component for the magnetic density, $L_{ef}$ is the effective length of the motor core, $K_c$ indicates the effect of the polar arc coefficients and waveform coefficients, and is expressed as follows:

$$K_c=\frac{{\alpha }_{pi}}{K_f}=\frac{B_{av}}{B_{\delta }}\frac{B_{\delta }}{B_1}=\frac{B_{av}}{B_1}=\frac{\sum _{i=\mathrm{1,35}\dots }\frac{1}{T}{\int }_T^i{B}_i}{B_1}$$

(51)

According to the analysis of the magnetic leakage in Section 2, the formula for calculating the no-load back-EMF of the FSCW PMSM can be revised as follows.

$$E=4.44fN{K}_{dp}{\mathsf{\Phi }}^{\prime }=4.44fN{K}_{dp}{B}_1{K}_c\tau L_{ef}\frac{1}{{\sigma }_{Lt}}$$

(52)

Table 2. Comparison of the no-load back-EMF calculation results by using different methods.

	Finite Element Method	Analytical Method Not Considering Flux Leakage	Analytical Method Considering Flux Leakage
E0 (V)	98.62	114.67	97.17
Error		16.27%	1.47%

shows the no-load back-EMF obtained by different methods.

Compared with the FEM result, the results obtained by the analytical method before and after considering the flux leakage differ by 16.27%, 1.47%, respectively.

In the case that flux leakage is not considered, the analytical method result is significantly larger than the FEM result. By introducing the flux leakage coefficient, the error has greatly reduced, and the effectiveness of the proposed method is verified.

5. The Influence of Rotor Permeability

As the important part of the motor main magnetic circuit, the electromagnetic properties of the rotor material are bound to have a significant impact on the performance of the motor. Especially for small motors, the motor rotor often adopts a solid structure, the determination of the rotor material is even more important, and the analysis of the rotor permeability is of great importance.

For the motor magnetic circuit, the effect of the material permeability can be determined by using the magnetic Ohm’s law. Neglecting the leakage flux circuit reluctance and the stator core reluctance, assuming that the permeability of permanent magnets and air is the same, it can be obtained that:

$$\mathsf{\Phi }=-\frac{{\mathrm{F}}_{\mathrm{m}}}{{\mathrm{R}}_{\mathrm{m}}}=-\frac{F_m}{\frac{1}{{\mu }_r{\mu }_0}\frac{L_1}{S_1}+\frac{1}{{\mu }_0}\frac{L_2}{S_2}}$$

(53)

$$L_2=g+h_{PM}$$

(54)

$L_1$ is the length of the magnetic circuit of the rotor region, $L_2$ is the total length of the magnetic circuit of the permanent magnet region and the air gap region, $S_1$ is the equivalent area of the magnetic circuit of the rotor region, and $S_2$ is the equivalent area of the magnetic circuit of the permanent magnet region and the air gap region.

$$\mathsf{\Phi }=-\frac{F_m{\mu }_0{S}_1{S}_2}{\left(g+h\right)S_1}·\frac{\left(g+h\right)S_1·{\mu }_r}{L_1{S}_2+\left(g+h\right)S_1·{\mu }_r}$$

(55)

As the rotor permeability gradually increases, the second term in the equation is closer to 1, and the effect of the change in rotor permeability on the main magnetic flux will be reduced, and when ${\mu }_r\gg \frac{\left(g+h\right)S_1}{L_1{S}_2}$, the effect of the change in rotor permeability can be neglected.

5.1. Influence on Air Gap Flux Density

Figure 8 shows the comparison of the analytical method and the FEM. Since the amplitudes of high-order harmonics of air gap flux density are less than 0.08 T, the effects of high-order harmonics can be ignored and will not be discussed here.

For the fundamental component, third harmonic and fifth harmonic, the maximum error of the results obtained by the two methods are only 1.09%, 1.37% and 3.38%, respectively. Combined with Formula (84), the Total Harmonics Distortion (THD) of air gap flux density is obtained. Figure 8d shows the variations of the THD of the air gap flux density, and the maximum relative error of the two methods is 2.81%. The analytical method has high accuracy to the FEM for the magnetic field prediction.

$$THD=\sqrt{{\displaystyle \sum _{i=2}^n}{\left(\frac{B_n}{B_1}\right)}^2}$$

(56)

where $B_n$ is the amplitude of the n-th harmonic and $B_1$ is the amplitude of the fundamental wave of the air gap flux density.

As the rotor permeability increases, the magnetic field strength in the motor gradually increases, saturation of the stator core is aggravated, and the result obtained by the analytical calculation is slightly higher. When the relative permeability of the rotor is greater than 100, the air gap flux density is less affected by the change of permeability, the saturation status of the motor core changes less, and the relative error of the two calculation methods tends to be constant. And with the growth of rotor permeability, the harmonics of air gap flux density show different nonlinear growth, so the THD also changes significantly.

When the rotor relative permeability is 100, compared with the case that the rotor relative permeability is 1, the fundamental component, the third harmonic, and the fifth harmonic of the air gap magnetic density increase by 128.7%, 17.7%, and 5.7%, respectively, and the THD decrease by 49.7%. However, when the rotor relative permeability is 4000, compared to the case of rotor relative permeability of 100, the fundamental, the third harmonic, and the fifth harmonic of the air gap magnetic density increase by 1.8%, 0.4%, and 0.8%, respectively, and the THD only decrease by 1.6%.

5.2. Influence on On-Load Back EMF

The no-load back-EMF of the motor with different rotor relative permeability was calculated, and the results are shown in Figure 9.

As can be seen from Figure 9, similar to the change in air gap flux density, the no-load back-EMF also shows an increasing trend as the rotor material permeability increases, but the change rate gradually decreases. When the rotor relative permeability increases from 1 to 100, the no-load back-EMF increases by 52%, while when the rotor relative permeability increases from 100 to 4000, the change rate of the back-EMF is less than 2%.

As the rotor permeability increases, the relative error between the FEM and the analytical method gradually increases (0.78% to 3.74%), but eventually reaches a constant.

This is because as the rotor permeability increases, the magnetic field strength gradually increases and the stator tooth is severely saturated, and the effect of this phenomenon on the flux leakage cannot be accurately considered by using the analytical method. When the relative permeability of the rotor core is greater than 100, the magnetic field strength hardly changes, and the calculation error no longer changes.

To clearly illustrate the effect of rotor size and permeability on motor no-load performance, the effects of rotor thickness and rotor material permeability on the air gap flux density fundamental component and the no-load back-EMF are investigated. The results are shown in Figure 10.

As can be seen from the figure, when the permeability of the rotor material is low, the motor air gap magnetic density and the back-EMF show an increasing trend with the increase in the rotor thickness. However, when the rotor permeability is greater than 100, the change of rotor thickness does not affect the magnetic field and magnetic flux of the motor.

6. Experimental Test and Data Comparison

To verify the accuracy of the calculations, the prototype is tested. The performance changes of the motor under different loads are tested. The power analyzer is used to record the waveform and related parameters of the motor under different loads. The experimental setup included a prototype, a power tester, and a magnetic control test bench.

During the load test, the motor operated at approximately 75 °C. Figure 11 displays the test platform and equipment, while

Table 3. Comparison of calculated data and experimental data.

Operation	Parameter	FEM Data	Experimental Data	Relative Error
No-load	EMF (V)	98.62	100.46	1.87%
0.95 kW	Current (A)	3.19	3.33	4.20%
	Power efficiency	98.84%	98.01%	0.83%
1.04 kW	Current (A)	3.45	3.31	4.23%

lists the collected data.

It can be seen from the table that the test results are in good agreement with the calculation results, and the maximum error does not exceed 5%. To validate the model’s accuracy, the no-load back-EMF tested was compared with the results obtained using the FEM, as shown in Figure 12.

The back-EMF results obtained from the FEM closely match the experimental results in terms of RMS value and waveform. This confirms the accuracy of both the FEM model and the analytical calculation results.

7. Conclusions

In this paper, by considering the zigzag leakage flux of the FSCW PMSM, a method for calculating the magnetic flux based on the slotless subdomain analytical model is proposed. Taking the 0.95 kW, 3000 r/min surface-mounted PMSM as an example, the effect of rotor core permeability on the motor no-load characteristics was investigated by using the described method, and the conclusions are as follows:

(1)

The radial air gap flux density obtained by the analytical method and the FEM are highly consistent. But the no-load back-EMF results calculated by the two methods differ by 16.27%. By introducing the leakage coefficient, the error between the analytical method and the FEM is reduced to 1.55%, and the error between the analytical method and the experimental test is 3.35%. This method of calculating the back-EMF for the FSCW PMSM is simple and feasible, although a fully accurate prediction requires a combination of other factors, such as the nonlinearity of the stator material B-H curve.

(2)

As the rotor permeability increases, the air gap flux density increases mainly in the fundamental component, the THD of air gap flux density gradually decreases, and the back-EMF gradually increases. When the rotor relative permeability increases from 1 to 100, the fundamental component of the air gap flux density and no-load back-EMF has increased by 128.7% and 52%, respectively, while the permeability of the rotor core, greater than a threshold value, that changes in rotor permeability or thickness has little effect on motor performance. This critical permeability may be only a few tenths of the permeability of silicon steel. When the rotor relative permeability increases from 100 to 4000, the fundamental component of the air gap flux density and no-load back-EMF have increased by 1.8% and 1.9%, respectively.

Despite the success of the proposed method, future research should investigate the impact of various stator material properties, particularly the nonlinearity of the B-H curve, on the accuracy of the proposed analytical method. Additionally, developing enhanced models that incorporate temperature effects and thermal properties of motor components is crucial to improve the accuracy of flux and back-EMF calculations under varying operational conditions. Addressing these aspects is essential as they may potentially influence the precision of motor performance analysis, warranting further research.