Analysis of temperature field in innovative axial magnetic levitation bearings

Abstract

In this paper, innovative axial magnetic levitation bearings with high redundancy and low heating characteristics are designed for axial magnetic bearings, the design criteria for stacked redundant axial magnetic bearings are proposed, and a finite element model of innovative axial magnetic levitation on bearings is established to conduct the combined thermal and magnetic simulation analysis of innovative axial magnetic levitation bearings, thus analyzing temperature field and eddy current energy consumption distribution of the structure. A comparative analysis is also performed on temperature field and eddy current energy consumption distribution of classic axial magnetic bearings with the same volume. The study shows that, under the same volume and current conditions, the eddy current loss generated by the stacked six-ring structure is smaller than that generated by the traditional concentric two-ring structure, and the temperature rise of the stacked six-ring structure is smaller than that of the traditional concentric two-ring structure.

Keywords

stacked redundant axial magnetic bearing eddy current loss temperature field

Introduction

Magnetic bearings can use the controllable electromagnetic forces generated by electromagnet to make the rotor suspend without contact.^1–3 Compared with traditional bearings, magnetic bearings have certain advantages, such as no friction, no need for lubrication, and stiffness and damping can be adjusted.^4–6 Since the magnetic bearings are mostly used for high-speed rotation, they often suffer a prominent problem of eddy current loss under high-speed rotation. The heat caused by eddy current loss will cause thermal expansion of the fixed rotor of the magnetic bearing, leading to fixed rotor deformation, and affecting the positioning accuracy of the sensor and resulting in a significant decrease in the reliability of the entire maglev system. Therefore, it is necessary to improve the reliability of maglev system by reducing the energy consumption of maglev system.^7,8

Currently, research on the thermal performance of magnetic bearings in China and overseas countries mainly includes the following : Ao Wang et al investigated the pressure, temperature, and magnetic intensity distribution of magnetic fluids under different eccentricity conditions.⁹ Chaowu Jin et al used Ansys Maxwell for modeling and finite element analysis to obtain the causes of iron loss concentration in magnetic bearing and the influence of different parameters on the size and distribution of iron loss.¹⁰ Yincai Zou et al used finite element analysis (FEA) to calculate the flux density and distribution of a permanent magnet ring and rotor.¹¹ Shaojie Guo et al analyzed the coupling effect of the unbalanced magnetic pull (UMP), nonlinear ball bearing forces (NBB), and operating temperature on the motor rotordynamic behaviors.¹² Based on the established electromagnetic heat model, Wenjiao Yang et al studied the dynamic characteristics of linear high temperature superconducting magnetic suspension (maglev) bearings.¹³

Design of innovative axial magnetic levitation bearings

Each silicon steel sheet is placed within a range of $\frac{360^{\circ}}{n}$ , replacing rectangular silicon steel with trapezoidal silicon steel sheets, which can effectively utilize the magnetic pole area and ensure that adjacent coils won’t interfere with each other. The structure is shown in Figure 1.

Figure 1.

Structure diagram of innovative axial magnetic levitation bearings.

As shown in Figure 2, there is no interference between the coils on the premise that the length of the coil cavity is P. Figure 3 shows the structure diagram indicating the length of each part of a pair of silicon steel sheets. In the figure, the distance between the magnetic pole in the silicon steel sheet and the center of the stator disk is represented by Y. In the design, so as to make full use of the coil cavity area to increase the electromagnetic force, theoretically, the coil shall occupies the entire coil cavity and adjacent to line P.

Figure 2.

Dimensional diagram of the silicon steel sheet.

Figure 3.

Assembly drawing of the silicon steel sheet and coil.

When the stator volume is constant, in order to maximize the electromagnetic force generated by the magnetic levitation bearing $F$ , it is necessary to select appropriate X, Y, Z, and P values and $N$ value, which is the number of coil winding turns. The objective function shows that the axial magnetic bearing has the maximum electromagnetic force $F$ . Because the new structure is equal to 6 silicon steel sheets working in parallel, so it is only necessary to determine that a specific silicon steel sheet generates the maximum electromagnetic force.

Based on the above objective function and constraints are as follows:

Q + X + P + X + P \sin \frac{360 °}{n} - \sqrt{95^{2} - {(\frac{Y}{2} + P \cos \frac{360 °}{n})}^{2}} < 0

\frac{Y}{2} - X \frac{360 °}{2 n} + \frac{P}{\cos \frac{360 °}{2 n}} - (X + P + Q) \tan \frac{360 °}{2 n} < 0

(1)

X + \frac{Z}{2} < 48

30 - Q < 0

MATLAB is used to solve the optimization problem of the maximum electromagnetic force under a certain volume. The objective function is to maximize the electromagnetic force of the axial magnetic levitation bearing. During the design process, the stacked structure is equivalent to a piece of silicon steel sheet working in parallel, and there is no coupling between adjacent silicon steel sheets, so it is only necessary to determine the electromagnetic force generated by a piece of silicon steel sheet as the maximum. The variation curve of the bearing capacity of the magnetic suspension bearing with the area of the magnetic pole and the area of the coil cavity under a certain volume can be obtained, as shown in the Figure 4. The initial conditions and final optimization results are as shown in Tables 1 and 2.

Figure 4.

Electromagnetic force curve.

Table 1.

Structural design condition.

Name	Sign	Value
Air gap dimension	$x_{0}$	1 (mm)
Rotor core inner diameter (mm)	$r_{n}$	30
Outer diameter of the stator (mm)	$r_{w}$	95
Height of stator	$h$	48 (mm)
Maximum density of magnetic flux density (T)	$B_{s}$	1.8 (T)
Ampere density	$J$	5 (A/mm²)
Maximum current	$I_{m}$	6 (A)
Coil cavity filling rate	$λ$	$0.7$

Table 2.

Dimension parameters of silicon steel sheet.

X	Y	Z	P	Q	N
10.60	40.48	74.81	19.22	33.11	337

Analysis on eddy current loss and temperature field of stacked structure and traditional concentric structure

Losses in stacked redundant axial magnetic bearings are mainly categorized as copper losses and iron losses. The iron loss of magnetic levitation bearing is mainly composed of eddy current loss and hysteresis loss, which can be calculated according to the following empirical formula:

P_{F e} = P_{w} + P_{c}

P_{w} = σ_{w} {(\frac{f}{100} B_{m})}^{2} V_{f e} r_{f e}

(2)

P_{c} = σ_{c} {(\frac{f}{100} \frac{B_{m}}{10^{4}})}^{β} V_{f e} r_{f e}

Formula $σ_{w}$ indicates eddy current loss coefficient; when the core material is silicon steel sheet, $σ_{w}$ = 0.4; when the core material is electrical pure iron, $σ_{w}$ = 2; ƒ indicates the frequency of the high-frequency current at the output of the switching amplifier; $σ_{c}$ indicates the hysteresis loss coefficient; silicon steel sheet takes $σ_{c}$ = 0.2; $B_{m}$ is the amplitude of the alternating magnetic induction; $V_{f e}$ refers to the volume of the iron core; $r_{f e}$ is the specific gravity of the core; the silicon steel sheet is 7.65 × 103; electrotechnical pure iron is taken as 7.87 × 103; when $B_{m}$ < 1, $β$ = 1.6.

The analysis shows that the hysteresis loss is at least 4 orders of magnitude smaller than the eddy current loss. Therefore, the hysteresis loss is ignored when calculating the iron loss of the magnetic levitation bearing.

During the operation of the innovative axial magnetic levitation bearings, the copper loss caused by coil heating and the eddy current loss generated by the iron core in the alternating magnetic field are the main heating sources of temperature field variation in the system. In order to accurately analyze the distribution of temperature variation of the stacked structure in steady state, a joint analysis and study of these two heating sources shall be conducted during the simulation analysis.

The magnetic-thermal coupling joint simulation is used to analyze variation in temperature of the innovative axial magnetic levitation bearings. Firstly, the eddy current loss of the stacked structure under a certain volume and current is analyzed, and the analysis results are used to analyze the distribution of temperature variation of each component of the structure. The eddy current power loss of each component material at static suspension state is shown in Table 3.

Table 3.

Summary of eddy current loss of various components of the stacked six-ring structure in stable suspension state.

Component	Silicon steel sheet	Reinforcement ring	Stator disc	Thrust disc
Eddy current loss (W)	57.88	0.014	0.0014	27.36

For the traditional concentric two-ring structure, the different ways of energization will result in different eddy current losses of each component under the same current, which will in turn lead to different temperature field distributions of each component. The eddy current losses and temperature field distributions of the traditional concentric two-ring structure under different current directions are analyzed separately as below.

The calculated eddy current losses of various components of the traditional concentric two-ring structure under the condition of the same current size but opposite current directions are as shown in Tables 4 and 5.

Table 4.

Eddy current losses of various components of traditional concentric two-ring structure in stable suspension with opposite current directions.

Component	Stator disc	Thrust disc
Eddy current loss (W)	95.39	22.49

Table 5.

Eddy current losses of various components of traditional concentric two-ring structure in stable suspension with the same current direction.

Component	Stator disc	Thrust disc
Eddy current loss	392.77 $W$	64.57 $W$

The stator of the common axial magnetic suspension bearings is made of electrical pure iron, which will result in a large amount of eddy current loss during operation, reducing the reliability of the entire system. In order to reduce the eddy current loss, the stator structure is therefore changed to a stacked redundant axial magnetic bearing form. From the previous analysis, it can be seen that under the same volume and current, the total eddy current loss of the traditional concentric multi-ring structure is 117.884, including 95.394 generated by the stator. For the stacked structure, the total eddy current loss is only 37.03, and the part generated by the stator is 9.67. It can be seen from the intuitive data that when the stator adopts a stacked structure, the eddy current loss is significantly reduced compared to the traditional concentric two-ring structure. This result can also be reflected in the eddy current density contour map of the stator; please see Figures 5 and 6.

Figure 5.

Eddy current density contour map of innovative axial magnetic levitation bearings.

Figure 6.

Eddy current density contour map of traditional concentric two-ring structure.

From Figures 5 and 6, it can be seen that the eddy current density of the stacked six-ring structure appears more concentrated on the silicon steel sheet, while the eddy current density of the traditional concentric two-ring structure is more evenly distributed. For the stacked six-ring structure, due to the concentrated eddy current density on the silicon steel sheet, there are more eddy current losses and higher temperatures on the silicon steel sheet. For the traditional concentric two-ring structure, due to the uniform and larger eddy current density distribution, the overall loss of the stator disc is greater and the temperature distribution is more uniform. However, since the eddy current loss value of the stator in the stacked six-ring structure is much smaller than that in the traditional concentric multi-ring structure, the stator temperature of the traditional concentric two-ring structure is higher than that of the stacked six-ring structure.

Based on the calculated heat generation rates of the various components of the innovative axial magnetic levitation on bearings and the traditional concentric redundant axial magnetic bearing, the convective heat transfer coefficients between each component and the surrounding air, as well as the surrounding air temperature, are set, and the temperature results that need to be viewed are determined for simulation calculation. The simulation results are as shown in Table 6. The results of the temperature field simulation of each component of the laminated six-ring structure are shown in Figure 7.

Table 6.

Temperature results of various components of innovative axial magnetic levitation bearings and traditional concentric two-ring structure.

	Rotor	Coil	Stator
	Rotor	Coil	Reinforcement ring	Silicon steel sheet	Base
Stacked structure	68.6°C	38.3°C	37.8°C	37.91°C	37.46°C
Concentric structure (same direction)	140.12°C	46.46°C		271.83°C
Concentric structure (opposite direction)	63.2°C	46.46°C		84.67°C

Figure 7.

Temperature field distribution diagram of each component of the laminated six-ring structure.

For the stacked six-ring structure, the temperature variation range of the stator structure is between 37.33°C and 38.46°C, and the highest temperature appears on the thrust disc, which is 67°C. For the traditional concentric two-ring structure with opposite current directions, the highest temperature appears on the stator disc, with a temperature variation range of 78.36°C–90.98°C, and the highest temperature of the thrust disc is 64.2°C. Therefore, for the stacked six-ring structure, the highest temperature of the stator disc is much lower than that of the traditional concentric two-ring structure, while the difference in the highest temperature on the thrust disc between the two structures is not significant.

In addition, further comparison of the heating situation of the coils in the two structures under the same current conditions shows that the average temperature of the coil in the stacked six-ring structure is 38.2°C, while the average temperature of the coil in the traditional concentric two-ring structure is 46.5°C. Therefore, under a certain current, the temperature rise of the coil in the stacked six-ring structure is lower than that in the traditional concentric two-ring structure, and as the current increases, the safety of the stacked six-ring structure coil is higher than that of the traditional concentric two-ring structure.

When the rotor (thrust disk) rotates at a high speed, it will affect the stator and rotor temperatures of the magnetic levitation system to a certain extent. Under the action of the same current, assuming that the rotor rotational speed is 1000

r p m

, 10,000

r p m

, and 20,000

r p m

, the eddy current loss values of the stacked redundant axial magnetic levitation bearings are shown in Table 7.

Table 7.

Summary of eddy current loss of laminated six-ring stator at different speeds.

Rotational speed	1000	10,000	20,000
Eddy current loss	80.05 $W$	80.89 W	81.28 W

From Table 7, it can be seen that the rotational speed has little effect on the eddy current loss of the laminated six-ring stator. Since the value of eddy current loss does not change much, it also has little effect on the temperature field distribution of the entire laminated six-ring stator structure.

The average temperature of the stator of laminated six-ring structure under different rotating speeds is shown in the Table 8.

Table 8.

Average stator temperature of laminated six-ring structure at different speeds.

Rotational speed	1000 $r p m$	10000 $r p m$	20000 $r p m$
Average stator temperature	71.6°C	72.1°C	72.3°C

For the laminated redundant axial magnetic suspension bearing, the copper loss and eddy current loss in the structure are changed due to failure.

Assume that the current in normal operation is 1

A

. When the laminated six-ring structure has a first-order failure, the required compensation current is 1.15

A

;when the laminated six-ring structure has a second-order failure, the required compensation current is 1.25

A

; when the laminated six-ring structure has a third-order failure, the required compensation current is 1.5

A

. When the normal operating current is 1

A

, the heat generation rate of the coil and fixed rotor corresponding to normal operation and failure of levels 1, 2, and 3 is shown in Table 9.

Table 9.

Heat generation rates of the components of the stacked six-ring structure in normal operation and under different failure scenarios.

Working conditions	Coil heat generation rate	Stator heat generation rate	Rotor heat generation rate
Proper functioning	$0.65 \times 10^{4}$	$0.33 \times 10^{4}$	$1.2 \times 10^{4}$
Level 1 failure	$0.86 \times 10^{4}$	$0.714 \times 10^{3}$	$1.65 \times 10^{4}$
Level 2 failure	$1.02 \times 10^{4}$	$0.584 \times 10^{3}$	$1.475 \times 10^{4}$
Level 3 failure	$1.463 \times 10^{4}$	$2.381 \times 10^{3}$	$1.21 \times 10^{4}$

When the laminated six-ring structure has different failures, the temperature distribution of the fixed rotor under the action of compensation current is shown in Figure 8.

Figure 8.

Temperature distribution of a fixed rotor with a laminated six-ring structure under the action of compensation current. (a) Stator and rotor temperature distribution during normal operation. (b) The temperature distribution of the rotor is determined when the first stage failure occurs. (c) The temperature distribution of the rotor is determined when the second stage fails. (d) The temperature distribution of the rotor is determined when the third stage fails.

As can be seen from Figure 9, when the coil of laminated six-ring structure fails, the value of the compensation current needs to be increased in the reconstruction process. However, with the increase of the current, the temperature distribution of both the rotor and the stator does not fluctuate to a large extent, which means that the temperature field distribution of laminated six-ring structure does not increase greatly with the increase of the compensation current during reconstruction. The reason why the average temperature of the laminated six-ring structure changes less during failure is that the heat dissipation area of the stator structure does not change during failure, and the compensation current through the reconstruction is not large.

Figure 9.

Curve of overall average temperature variation of stator-rotor under different cases of laminated structure.

Experimental study on innovative axial magnetic levitation bearings

In the previous section, the thermal performance of the redundant axial magnetic bearing with stacked six-ring structure was simulated and analyzed. In order to verify the correctness of the simulation, in this section, a specific experimental setup is used to verify the simulated thermal performance of the innovative axial magnetic levitation bearings.

The mechanical component of the experimental apparatus is depicted in Figure 10, and the control component of the experimental apparatus is depicted in Figure 11.

Figure 10.

Mechanical part of the experimental setup.

Figure 11.

Control part of the experimental setup.

Based on the above experimental setup, temperature variation measurement is carried out after the innovative axial magnetic levitation bearings entering stable suspension state. In temperature variation experiment, an infrared thermal imager is used to measure temperature variation distribution of the coils, stator, and rotor of the innovative axial magnetic levitation bearings.

Measurement method: After the innovative axial magnetic levitation bearings entering stable suspension for 5 minutes, the first time of measurement is carried out: measuring the temperatures of the coil, stator, and rotor. After that, the measurements are taken every 5 minutes until the temperatures of various parts of the bearing remain basically unchanged for a period of time, indicating that the temperatures of various parts of the innovative axial magnetic levitation bearings have reached thermal equilibrium with the surroundings. Figure 12 shows the temperature distribution at different times when the innovative axial magnetic levitation bearings is in normal suspension state.

Figure 12.

Temperature distribution at different times during normal suspension state.

From temperature variation experimental data, it can be seen that during the normal suspension of the innovative axial magnetic levitation bearings, when thermal equilibrium is reached, the highest temperature appears in the coil, followed by the silicon steel sheets, the reinforcement ring, and the stator base, and the lowest temperature appears in the rotor. During the data collection process, the overall temperature of the innovative axial magnetic levitation bearings rises rapidly in the initial stage, and as time goes on, the temperatures of various parts gradually stabilize until reaching the final thermal equilibrium state.

In order to compare and analyze temperature variation distribution between simulation and actual experiments, temperature simulation is performed on the innovative axial magnetic levitation bearings based on the actual experimental setup size and the current applied on the bearing during normal suspension state. The comparison curve of experimental data and simulation data is as shown in Figure 13.

Figure 13.

Comparison of temperature curves between experimental and simulation results for different components of the stacked structure.

By comparing the simulation results with the experimental test results, it can be found that the highest temperature of the coil in the experiment after reaching thermal equilibrium is 54.1°C, while the simulation result is 51.5°C, and the difference between the experimental and simulation temperatures is small; the highest temperature of the stacked silicon steel sheets after reaching thermal equilibrium is 43.8°C, while the simulation result is 51.2°C, and the experimental result is about 15% lower than the simulation temperature; the highest temperature of the rotor after reaching thermal equilibrium is 28.6°C, while the simulation result is 37.1°C, and the experimental result is about 23% lower than the simulation temperature.

Conclusions

(1) Under the same volume and current conditions, the eddy current losses generated by the stacked six-ring structure are smaller than those generated by the traditional concentric two-ring structure.

(2) The temperature rise of the stator and rotor in the stacked six-ring structure is smaller than that in the traditional concentric two-ring structure.

(3) During the normal suspension state of the innovative axial magnetic levitation bearings, when reaching the thermal equilibrium state, the highest temperature appears in the coil, and the lowest temperature appears in the rotor.

Statements and declarations

Footnotes

Conflicting interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge key scientific and technological projects of Henan province (Grant: 232102220091) and key research projects of Henan higher schools (23B430015).

References

Wang

Liu

Jingbo

, et al. Analytical modeling of air gap magnetic fields and bearing force of a novel hybrid magnetic thrust bearing. IEEE Trans Magn. 2021; 57(10).

Dokyu

SeungJoon

JeongIk

. Instability study of magnetic journal bearing under S-CO2 condition. Appl Sci 2021; 11(8): 3491.

Xiaojun

Ming

Tianming

. Design and optimization of a radial high-temperature superconducting magnetic bearing. IEEE Trans Appl Supercond 2019; 29(2): 1–5.

InJun

MinKi

, et al. Design for reducing bearing force ripple and torque ripple of integrated magnetic bearing motor through Halbach array. Energies 2023; 16(3): 1249.

Xiao

Cheng

. Analysis of eddy current field for new type of thrust bearings based on single-objective genetic algorithm. Int J Appl Electromagn Mech 2020; 64(1-4): 263–270.

Satoshi

Masaya

Changan

. Development of an axial-flux self-bearing motor using two permanent magnet attractive type passive magnetic bearings. Int J Appl Electromagn Mech 2020; 64(1-4): 827–833.

Chen

Zhang

Luo

, et al. Response analysis of random base excitation on a magnetic bearing rotor system. MATEC Web Conf 2019; 293: 04004.

Zhu

Faller

. An integrated electric motor driven compressor supported by magnetic bearings. In: Electric Machines and Drives Conference (IEMDC), San Diego, CA, USA, 12–15 May 2019, 2019: 160–164.

Jiabao

Huaibiao

, et al. Structural design and lubrication properties under different eccentricity of magnetic fluid bearings. Appl Sci 2022; 12(14): 7051.

10.

Jin

Dong

, et al. Iron loss analysis of magnetic bearing system considering magnetic-thermal coupling. Int J Appl Electromagn Mech 2022; 70(1): 1–20.

11.

Zou

Xing

Shang

, et al. Calculation and measurement of the magnetic field of Nd ₂ Fe ₁₄ B magnets for high-temperature superconducting magnetic bearing rotor. J Supercond Nov Magnetism 2019; 33(4): 1–10.

12.

Shaojie

Changqing

. Experimental and numerical research studies on multifield coupling rotordynamic behaviors of motor. Shock Vib 2021; 2021(2): 1–23.

13.

Yang

Quéval

, et al. A 3-D strong-coupled electromagnetic-thermal model for HTS bulk and its uses to study the dynamic characteristics of a linear HTS maglev bearing. IEEE Trans Appl Supercond. 2020; 30(6).