Analysis of high speed permanent magnet synchronous motor rotor supported by air foil bearings

Abstract

The electromagnetism and rotor dynamics have been drawing great research interest due to their significance in the design of the high speed permanent magnet synchronous motor. However, it is challenging but essential to study the rotor supported by air foil bearings with cylindrical permanent magnet. Here the approach is reported to analyze a high speed PMSM rotor at the rating of 28.5 kW and 96,000 rpm. Through the analytic method and the finite element simulation, the expression for the magnetic flux density is obtained. Based on the rotor-bearing system discrete model, the critical speed is calculated by Campbell diagram. It comes to the conclusion that the radius and length of cylindrical permanent magnet is reciprocally related to the magnetic flux density and critical speed values. Finally, results of the PMSM experimental device operating at 96,000 rpm are represented by rotor frequency spectrum and orbit diagram.

Keywords

High speed motor cylindrical permanent magnet magnetic field rotor critical speed air foil bearing

1. Introduction

The permanent magnet synchronous motor (PMSM) supported by air foil bearings are now gaining more and more popularity in the rotating machinery industry. In the high speed PMSM, the excitation magnetic field of the rotor permanent magnet interacts with the current density of the stator, and the electromagnetic torque is generated to drive the rotor rotation. The high speed PMSM has many advantages: firstly, the efficiency is high when the high speed PMSM works in the synchronous rotation [1]; secondly, the structure with a coaxial compressor or turbine can reduce the system complexity to be more compact [2]; thirdly, it has high specific energy and low power loss [3].

To analyze the rotor excitation magnetic field in theory, several researchers have worked on the analytic expression for different kinds of the permanent magnet [4–7]. The structure of the rotor and stator is different, so solution procedures are not suitable for the below-mentioned cases. Many other scholars have studied the rotor dynamics research of the high speed rotor supported by air foil bearings [2,8–10]. These references introduced the analysis of air foil bearing and rotor dynamic characteristics, including unbalance vibration response and critical speed. But effects of the permanent magnet on rotor dynamics have not been involved. References [1–3,11,12] present experimental realization in different levels of the power and rotation speed, which can be used as the reference for the PMSM performance experimental device. In this paper, it is necessary to obtain the analytic prediction solution of the excitation magnetic field in the bipolar cylindrical permanent magnet which is magnetized in parallel. Then the influence of permanent magnet dimensions on the magnetic flux density and critical speed can be taken into account. Therefore, this paper carried out further research on the electromagnetism, rotor-bearing system dynamics and performance experiment in the high speed PMSM.

This paper introduced the research organized as follows: first, through the analytic and simulation method, the magnetic flux density of the cylindrical permanent magnet is calculated. Then, the discrete model of the rotor-bearing system is built to analyze the critical speed. Finally, the experiment is carried out to validate the performance of the high speed PMSM. Through the reasonable and effective analysis of the electromagnetism and rotor dynamics, results obtained in this paper have significance to the theoretical study, design and manufacture of the high speed PMSM rotor.

2. Analytic and simulation analysis

The analytic and simulation analysis is based on a high speed PMSM driven by 28.5 kW at the rated speed of 96,000 rpm. The PMSM rotor has a nonferromagnetic sleeve and a bipolar cylindrical permanent magnet magnetized in parallel. The material of the sleeve and the permanent magnet is nickel-based superalloy and Sm₂Co₁₇ alloy, respectively.

2.1. Rotor electromagnetic analysis

2.1.1. Magnetic analytic expression of cylindrical permanent magnet

The magnetic field is directly related to the electromagnetic energy transfer between the rotor permanent magnet and stator. So the magnetic field analysis based on the permanent magnet is the important foundation for the high speed PMSM performance analysis and the characteristic calculation.

Figure 1 shows the excitation magnetic field model for the bipolar cylindrical permanent magnet which is magnetized in parallel. R ₁, R ₂ and R ₃ represent the outer radius of the rotor sleeve, the radius of the permanent magnet and the inner radius of the stator, respectively. The regions of the permanent magnet and the gap are defined as P and G, respectively. The gap is regarded as nonmagnetic material.

Fig. 1.

Analytic model of excitation magnetic field.

In the excitation magnetic field model, B is permanent magnet magnetic flux density; M _R is rotor permanent magnet magnetization; H is magnetic field strength of the permanent magnet. They are in the same direction and the constitutive relation is written as $\begin{eqnarray}\displaystyle B={\mu}_{0}{\mu}_{\text{r}}H+{\mu}_{0}M_{R} & & \displaystyle\end{eqnarray}$ (1) where 𝜇₀ is the vacuum permeability, and 𝜇_r is the relative permeability of permanent magnet [13].

The analytic expression of the excitation magnetic field is defined in the polar coordinate system, which have polar coordinate parameters r and θ. For a cylindrical magnet that is magnetized in parallel, the magnetization vector M _R can be expressed by $\begin{eqnarray}\displaystyle \mathbf{M}_{\text{R}}=\mathbf{M}_{\text{Rr}}+\mathbf{M}_{\text{R}{\theta}}=M_{\text{Rr}}\mathbf{e}_{\text{r}}+M_{\text{R}{\theta}}\mathbf{e}_{{\theta}}=M_{\text{R}}\cos {\theta}\mathbf{e}_{\text{r}}+M_{\text{R}}\sin {\theta}\mathbf{e}_{{\theta}}. & & \displaystyle\end{eqnarray}$ (2) Then the magnetization component M _Rr and M _Rθ can be expressed as a Fourier expansion: $\begin{eqnarray}\displaystyle M_{\text{Rr}}+M_{\text{R}{\theta}}=\mathop{\sum }_{m=1,3,5,\ldots }^{\infty }\sqrt{M_{\text{r}m}^{2}+M_{{\theta}m}^{2}}\sin \left(m{\theta}+\arctan \frac{M_{\text{r}m}}{M_{{\theta}m}}\right) & & \displaystyle\end{eqnarray}$ (3) where $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}ll@{}}\begin{array}{@{}l@{}}\displaystyle M_{\text{r}m}=M_{\text{R}}\left\{\frac{\sin \left[(m+1)\frac{{\pi}}{2}\right]}{(m+1)\frac{{\pi}}{2}}+\frac{\sin \left[(m-1)\frac{{\pi}}{2}\right]}{(m-1)\frac{{\pi}}{2}}\right\},\\[20.0pt] \displaystyle M_{{\theta}m}=M_{\text{R}}\left\{\frac{\sin \left[(m+1)\frac{{\pi}}{2}\right]}{(m+1)\frac{{\pi}}{2}}-\frac{\sin \left[(m-1)\frac{{\pi}}{2}\right]}{(m-1)\frac{{\pi}}{2}}\right\}\end{array} & m\neq 1\\[40.0pt] M_{\text{r1}}=M_{\text{R}},\quad M_{{\theta}1}=-M_{\text{R}} & m=1.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$

In the permanent magnet excitation magnetic field, the magnetic vector potential A is expressed by $\begin{eqnarray}\displaystyle \mathbf{B}={\nabla}\times \mathbf{A}. & & \displaystyle\end{eqnarray}$ (4) Substituting Eq. (2) into Eq. (1), and Eq. (1) can be calculated by convolution as: $\begin{eqnarray}\displaystyle {\nabla}\times \mathbf{B}={\nabla}\times ({\nabla}\times \mathbf{A})={\mu}_{0}{\mu}_{\text{r}}({\nabla}\times \mathbf{H})+{\nabla}\times ({\mu}_{0}\mathbf{M}_{\text{R}}). & & \displaystyle\end{eqnarray}$ (5)

In the static or steady-state magnetic field, the eddy current can be ignored inside the permanent magnet, i.e., 𝛻 × H = 0. And through Eqs (2) and (3), 𝛻 × M _R = 0. The magnetic vector potential has only the axial component, which can be expressed as A = A (r, θ)e _z. So Eq. (5) can be simplified and the governing equation for the the rotor region (P and G) can be written as $\begin{eqnarray}\displaystyle {\nabla}^{2}A_{\text{P}}(r,{\theta})={\nabla}^{2}A_{\text{G}}(r,{\theta})=0 & & \displaystyle\end{eqnarray}$ (6) where A _P, A _G are magnetic vector potential in region P, G, respectively, and 𝛻² is the Laplacian on the axial component of the magnetic vector potential [14].

The separated variable method can be used when the periodic equation is assumed via A(r, θ) = A(r, θ + 2π). Then the general solution can be given by $\begin{eqnarray}\displaystyle A(r,{\theta})={\alpha}_{0}+{\beta}_{0}\ln (r)+\mathop{\sum }_{m=1}^{\infty }({\alpha}_{m}r^{m}+{\beta}_{m}r^{-m})({\eta}_{m}\sin m{\theta}+{\lambda}_{m}\cos m{\theta}) & & \displaystyle\end{eqnarray}$ (7) where 𝛼₀, 𝛽₀, 𝛼_m, 𝛽_m, 𝜂_m, 𝜆_m are coefficients determined by boundary conditions. In Eq. (7), 𝛼₀ + 𝛽₀ ln(r) indicates the vector and scalar magnetic potential; other parts indicate the influence of boundary conditions [15].

In the radial and circumference magnetic flux density B _r and B _θ, there are $\begin{eqnarray}\displaystyle \text{} & B_{\text{r}}(r,{\theta})=B_{\text{r}}(r,-{\theta})=-B_{\text{r}}(r,{\pi}-{\theta}) & \displaystyle\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & B_{{\theta}}(r,{\theta})=-B_{{\theta}}(r,-{\theta})=B_{{\theta}}(r,{\pi}-{\theta}). & \displaystyle\end{eqnarray}$ (9) According to Eqs (8) and (9), Eq. (7) can be solved and 𝛼₀ = 𝛽₀ = 𝜆_m = 0, 𝜂_m = 1.

For the excitation magnetic field in Fig. 1, there are the following boundary conditions: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}ll@{}}\displaystyle -\frac{1}{{\mu}_{0}}\frac{\partial A_{\text{S}}}{\partial r}=0 & (r=R_{3})\\[10.0pt] \displaystyle \frac{1}{r}\frac{\partial A_{\text{P}}}{\partial {\theta}}=\frac{1}{r}\frac{\partial A_{\text{G}}}{\partial {\theta}},\quad -\frac{1}{{\mu}_{0}{\mu}_{r}}\frac{\partial A_{\text{P}}}{\partial r}=\frac{1}{{\mu}_{0}}\left(-\frac{\partial A_{\text{G}}}{\partial r}-{\mu}_{0}M_{\text{R}{\theta}}\right) & (r=R_{2}).\end{array}\right. & & \displaystyle\end{eqnarray}$ (10)

Substituting Eqs (3) and (10) into Eq. (7), all of undetermined coefficients can be solved. Thus, based on Eqs (4) and (7), the analytic expression of the magnetic flux density can be written as $\begin{eqnarray}\displaystyle B_{\text{r}}(r,{\theta})=k\frac{(R_{3}^{2}+r^{2})}{r^{2}}\cos {\theta},\quad B_{{\theta}}(r,{\theta})=k\frac{(R_{3}^{2}-r^{2})}{r^{2}}\sin {\theta} & & \displaystyle\end{eqnarray}$ (11) where $k=-\mu_{0} M_{\mathrm{R}} R_{2}^{2} /[ {(\mu_{r}-1)R_{2}^{2} -(\mu_{r} +1)R_{3}^{2} } ]$ , which is constant determined by PMSM.

2.1.2. Finite element simulation and analytic solution

According to the finite element method (FEM), a finite element model of the stator and rotor is constructed in ANSYS Maxwell software and all parameters conform to the high speed PMSM driven by 28.5 kW. In order to improve the calculation accuracy, the element density in the gap region is largely raised as shown in Fig. 2. Totally, there are 24,882 elements in the finite element model. Through the electromagnetic simulation, Fig. 2 shows the distribution nephogram of the magnetic flux density in the stator and rotor. In the gap region, it can be seen that with a decrease in the distance away from the permanent magnet outer circle, the value of the magnetic flux density increases.

Fig. 2.

Magnetic flux density distribution nephogram (28.5 kW PMSM).

The simulation result can be compared with the analytic solution to verify Eq. (11). In the middle of gap region, the radius is 15.55 mm as shown in Fig. 2. The magnetic flux density results (in radius r = 15.55 mm) of the analytic solution and simulation is illustrated in Fig. 3. It can be seen that curves comprise analytic and simulation results of the radial component B _r and the circumference component B _θ. The distribution of the magnetic flux density is close to the sine function. The comparative analysis of analytic values and FEM simulation results reveals their close fit for both radial and circumference component. This implies that the analytic expression is suitable for the magnetic flux density of the cylindrical permanent magnet parallel magnetized. So a feasibility calculation method can be applied for the magnetic field of the PMSM rotor.

Fig. 3.

Results of finite element simulation and analytic solution.

2.1.3. Case study and discussions

According to Eq. (11), it can be suggested that the magnetic flux density of the permanent magnet can be determined by the permanent magnet radius R ₂ when the stator inner radius R ₃ is a given value. So it is necessary to analyze the relationship between the magnetic flux density and the permanent magnet radius. The radial and circumferential maximum amplitude of the magnetic flux density can be written as $\begin{eqnarray}\displaystyle B_{\text{rmax}}(R_{2})=\left|\frac{{\mu}_{0}M_{\text{R}}(R_{3}^{2}R_{2}^{-2}+1)}{{\mu}_{r}-1-({\mu}_{r}+1)(R_{3}/R_{2})^{2}}\right|,\quad B_{{\theta}\text{max}}(R_{2})=\left|\frac{{\mu}_{0}M_{\text{R}}(R_{3}^{2}R_{2}^{-2}-1)}{{\mu}_{r}-1-({\mu}_{r}+1)(R_{3}/R_{2})^{2}}\right| & & \displaystyle\end{eqnarray}$ (12) where 𝜇₀ = 4π × 10⁻⁷ H/m, 𝜇_r = 1.03, M _R = 8.59 × 10⁵ A/m, R ₃ = 17.35 mm, which are based on parameters of the high speed PMSM driven by 28.5 kW.

From the calculation results, the relationship between R ₂ and B _rmax, B _θmax is shown as Fig. 4. It indicates that with an increase in R ₂, the radial maximum amplitude B _rmax increases and the circumferential maximum amplitude B _θmax decreases. Thus, permanent magnet radius affects the maximum amplitude of magnetic flux density, and the numerical relationship is expressed by Eq. (12).

Fig. 4.

Relationship between radius and maximum amplitude of magnetic flux density.

Above results covered only the magnetic flux density generated by permanent magnets. While, the torque comes from the interaction between the magnetic field of rotor permanent magnets and the current surface density of stator currents in the air gap. Mechanical vibrations in a PMSM can be caused by torque ripples due to many factors. So the proposed approach with the magnetic field calculation may be prospectively extended and used for mechanical vibration in the follow-up studies.

2.2. Rotor dynamics analysis

2.2.1. Rotor discrete model and critical speed

The structure of the rotor supported by radial air foil bearings (I and II) is shown in Fig. 5. The rotor is double-span combined and connected by a shaft coupling. The permanent magnet diameter D is 31.8 mm and the length L is 70 mm.

Fig. 5.

Discrete model of the rotor-bearing system (mm).

The dynamics equation of the rotor-bearing system is given by: $\begin{eqnarray}\displaystyle [\mathbf{M}]\{\ddot{\mathbf{X}}\}+([\mathbf{C}]+[\mathbf{C}_{\text{cor}}])\{\dot{\mathbf{X}}\}+[\mathbf{K}]\{\mathbf{X}\}=\{\mathbf{F}\} & & \displaystyle\end{eqnarray}$ (13) where [M] is mass matrix of the rotor and the thrust disc, [C] is gyroscope damping matrix of rotor and the damping matrix of air foil bearings, [C _cor] is Coriolis matrix, [K] is stiffness matrix of the rotor and bearings, $\{\mathbf{F}\}$ is force vector acting on the system, $\{\mathbf{X}\}$ is displacement vector.

FEM is a commonly used method in rotor dynamics analysis. A finite element model of the rotor-bearing system is constructed via ANSYS as shown in Fig. 6. Shafts can be modeled using Timoshenko beam element, and the thrust disc is treated as a rigid body. Air foil Bearings can be simplified as COMBIN14 element and equivalent to the form of stiffness and damping in the linear range. In the finite element model, the bearing supported stiffness is 1 × 10⁶ N/m and Coriolis force is added to allow for the effect of spin softening.

Fig. 6.

Finite element model.

By solving the eigenvalue of Eq. (12), damp algorithm is used to determine eigenvalues under the different speed. The Campbell diagram of the rotor whirling frequency is obtained through the imaginary part of eigenvalues. As shown in Fig. 7, the horizontal axis represents the rotor speed range which is from 0 to 250,000 rpm, and the vertical axis stands for the rotor whirling frequency calculated through FEM. The rotor forward whirl and backward whirl are shown by different symbol lines. The crossing points of whirl lines and a spin line (slope = 1) represent the critical speed.

Fig. 7.

Campbell diagram of the rotor.

The Campbell diagram shows that due to the gyroscopic effect, with an increase in the rotor speed, the backward whirling frequency reduces and the forward whirling frequency increases. The forward whirling frequency need to be considered in the rotor critical speed. Therefore, the critical speed of forward whirling from 1st to 5th are 26,182 rpm, 28,693 rpm, 35,307 rpm, 84,225 rpm and 203,250 rpm, respectively. Whirling orbit plots of the vibration mode are shown in Fig. 8. Operating of the rotor supported by air foil bearings is not affected by the critical speed of parallel and conical whirling, but the critical speed of the bending whirling should be avoided [2]. As shown in Fig. 8, the whirling orbits from 1st to 4th are parallel and conical whirling within the rated speed range (0–96,000 rpm). The rated speed is far away from the bending whirling critical speed, so the performance of rotor dynamics has sufficient safety margin. According to the bending whirling orbits, the bending area mainly appears in the part of the rotor sleeve and the permanent magnet.

Fig. 8.

Whirling orbit plots from mode 1 to mode 5.

2.2.2. Case study and discussions

From the above analysis results, it can be seen that the bending whirling primarily appears in the rotor part including the permanent magnet. High speed rotor dynamics can be influenced by the complex interaction between the inertial properties and the critical speed [16]. So different dimension parameters of the permanent magnet affect the dynamics performance of the whole rotor-bearing system.

The applied FEM made it possible to construct rotor-bearing models with various diameter and length of the permanent magnet. For each of the following cases, diameter D is chosen from 21.8 mm to 41.8 mm, and length L is chosen from 70 mm to 140 mm. Influence of the permanent magnet on the critical speed is calculated through the finite element simulation. As shown in Fig. 9, the respective curves of the forward whirling critical speed corresponding to different modes are calculated in the spin velocity ranging from 0 to 250,000 rpm.

Figure 9 properly reflects the change trend of the forward whirling critical speed along with different diameters and lengths. With the same length (L = 70 mm) and an increase in the diameter, values of the critical speed is reduced gradually; meanwhile, with the same diameter (D = 31.8 mm) and an increase in the length, values of the critical speed increases gradually. In conclusion, the case results obtained show that the variation of the forward whirling critical speed can be found through different diameters and lengths of the permanent magnet.

Fig. 9.

Influence of diameter and length on the forward whirling critical speed.

Based on two sections of the case study, from the comprehensive analysis of calculation results it can be seen that different radius of the cylindrical permanent magnet affects the magnetic flux density; meanwhile, different diameter and length affect the rotor critical speed. Given this, dimension parameters of the permanent magnet is reciprocally related to the magnetic flux density and critical speed.

3. Experiment results

In order to identify the performance in electromagnetism and rotor dynamics, a PMSM experimental device has already been made and illustrated in Fig. 10. In the experimental setup, there are a stator and a rotor-bearing system, including two sets of radial air foil bearings a shaft coupling, a thrust disc and shaft sleeves as show in Fig. 10(b–d). The bipolar cylindrical permanent magnet is press-fit in the rotor sleeve. The stator with the Wye distributed winding is assembled in the PMSM as shown in Fig. 10(a). Two high accuracy displacement sensors are arranged in the horizontal and vertical direction to measure the rotor orbit. The experiment process and data acquisition system are shown in Fig. 10(e,f).

When the high speed PMSM experiment was carried out at the speed of 96,000 rpm, all performances of the rotor-bearing system were verified successfully. As shown in Fig. 11(a), the rotor frequency spectrum indicates the horizontal vibration values at 96,000 rpm, and the maximum amplitude is 2.337 μm corresponding to the frequency of 1,600 Hz. Figure 11(b) shows the rotor orbit diagram obtained from two displacement sensors in the horizontal and vertical position at 96,000 rpm. The vertical displacement Y and the horizontal displacement X are both within the limits of 11 μm. The experimental results obtained at the rated speed suggest that the rotor supported by air foil bearings has the high stability and little vibration. It is anticipated that above results are useful for the design and analysis of the rotor electromagnetism and dynamics.

Fig. 10.

PMSM experimental device.

Fig. 11.

Rotor experimental results obtained at 96,000 rpm.

4. Conclusion

In this paper, several analysis methods have been applied to deal with the high speed PMSM rotor at the rating of 28.5 kW and 96,000 rpm. Firstly, the analytic expression for the magnetic flux density generated by parallely magnetized cylindrical permanent magnet is given. The results obtained via analytic solution and FEM simulation are found to be approximate. Secondly, the rotor-bearing system discrete model is built and rotor dynamics analysis is carried out. The bending whirling critical speed is higher than the rated speed as proven in the Campbell diagram. Thirdly, based on results of the case study, dimension parameters of the permanent magnet are reciprocally related to the magnetic flux density and rotor critical speed. This result is instrumental in the integration design of the magnetic field and mechanical structure. Lastly, a PMSM experimental device has been constructed to identify the performance in electromagnetism and rotor dynamics. Through the realization of PMSM operation experiment at 96,000 rpm, rotor frequency spectrum and orbit diagram are both obtained. As a result, the performances of the rotor-bearing system are verified well.

Footnotes

Acknowledgements

The work is supported by Chinese National Natural Science Foundation (51805407), Natural Science Foundation in Shaanxi Province of China (Grant No. 2018JQ5148) and Key Research and Development Project in Shaanxi Province of China (Grant No. 2017ZDXM-GY-054).

References

Hong

Woo

Jeong

, Development of an ultra high speed permanent magnet synchronous motor, International Journal of Precision Engineering & Manufacturing 14 (2013), 493–499.

Hong

Woo

Lee

, Ultra high speed motor supported by air foil bearings for air blower cooling fuel cells, IEEE Transactions on Magnetics 48 (2012), 871–874.

Salehi

Heshmat

Walton

J.F.

, Operation of a mesoscopic gas turbine simulator at speeds in excess of 700,000 rpm on foil bearings, Journal of Engineering for Gas Turbines & Power 129 (2007), 87–94.

Selvaggi

J.P.

Salon

Kwon

O.M.

, Computation of the three-dimensional magnetic field from solid permanent-magnet bipolar cylinders by employing toroidal harmonics, IEEE Transactions on Magnetics 43 (2007), 3833–3839.

L.J.

Zhu

Z.Q.

Staton

, Analytical prediction of electromagnetic performance of surface-mounted PM machines based on subdomain model accounting for tooth-tips, Iet Electric Power Applications 5 (2011), 597–609.

Rahideh

and Korakianitis

, Analytical magnetic field distribution of slotless brushless permanent magnet motors, Electric Power Applications Iet 6 (2012), 628–638.

Lubin

Mezani

and Rezzoug

, Improved analytical model for surface-mounted pm motors considering slotting effects and armature reaction, Progr. Electromagn. Res. B 25 (2010), 293–314.

Larsen

J.S.

, Experimental and theoretical analysis of a rigid rotor supported by air foil bearings, Mécanique & Industries 16 (2015), 1–13.

Zhang

Sun

Liu

, Dynamic characteristics of self-acting gas bearing–flexible rotor coupling system based on the forecasting orbit method, Nonlinear Dynamics 69 (2012), 341–355.

10.

Feng

Geng

and Yu

, Rotordynamics analysis of a quill-shaft coupling-rotor-bearing system, Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science 229 (2015), 1385–1398.

11.

Sim

Lee

Song

, Identification of the dynamic performance of a gas foil journal bearing operating at high temperatures, Journal of Mechanical Science & Technology 28 (2014), 43–51.

12.

Lim

Bang

Choi

, Secondary flow stabilization of 100 kw-class micro gas turbine, Journal of Mechanical Science & Technology 31 (2017), 1753–1761.

13.

Xiao

Cheng

, Magnetic properties and eddy current losses in Fe-based soft magnetic composites, International Journal of Applied Electromagnetics & Mechanics 52 (2016), 1433–1441.

14.

Geng

Yang

, Analytical solution of air-gap field in motors with parallel magnetized permanent magnets, Proceedings of 2017 IEEE International Conference on Mechatronics and Automation 1 (2017), 984–989.

15.

Tang

and Liang

, Analysis and Computation of Electromagnetic Field of Motor, China Machine Press, Beijing, 2010.

16.

Xia

Sun

Zhang

, Impact of synchronous generator transient on gas turbine rotor, International Journal of Applied Electromagnetics & Mechanics 52 (2016), 281–288.