Abstract
When the magnetic bearing rotor is working, the iron loss concentration phenomenon may occur on the rotor journal, which will cause the temperature gradient on the rotor journal and thermal unbalance. In order to study the iron loss concentration phenomenon, a multi-physics simulation method considering the magnetic-thermal coupling is proposed. Using 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; using COMSOL for magnetic-thermal coupling in magnetic bearing to obtain the influence of iron loss concentration on temperature distribution. The simulation results show that: because of the initial disturbance, the control effect of the magnetic bearing makes the magnetic field distribution on the circumference of the rotor journal uneven, resulting in the phenomenon of iron loss concentration; control current and bias current have significant influence on iron loss distribution, while bias current and speed have a significant influence on iron loss value; iron loss concentration will result in a temperature gradient on the rotor surface. The magnetic-thermal coupling finite element analysis is verified by the temperature rise detection experiment. It provides a theoretical basis and reference for the loss and thermal unbalance of magnetic bearing.
Introduction
Magnetic bearing is a new type of high-performance bearing that can realize rotor suspension through electromagnetic force. Compared with traditional mechanical bearing, the magnetic bearing has advantages such as no mechanical contact, no friction, long service life, low energy consumption, no lubrication, and active control, etc. It has a wide range of application prospects in the field of high-speed rotating machinery [1–3]. Although the speed of the magnetic bearing can reach the height that the traditional bearing cannot reach, the loss of the magnetic bearing is also inevitable. Due to the magnetization characteristics of magnetic bearing core materials and high-frequency working characteristics, the high iron loss will be generated inside the magnetic bearing, which will be converted into heat and make the internal temperature of the magnetic bearing rise, then cause thermal unbalance and affect the stability of the rotor [4]. An earlier reported study points out that a certain type of magnetic suspension centrifugal compressor produces an iron loss concentration phenomenon under specific working conditions, resulting in a spiral increase of rotor vibration amplitude [5]. Therefore, it is necessary to thoroughly study the law of iron loss in magnetic bearings.
For the magnetic bearing system, iron loss is an important part of the total loss. In recent years, many scholars have done relevant researches on the iron loss of magnetic bearings. Desvaux et al. focused on a 2D iron loss analytical model considering the variations of magnetic flux density for fast computation of iron losses due to eddy currents in the permanent magnets and ferromagnetic parts [6]. Tian et al. presented a magnetic circuit model of eddy current loss for a magnetic thrust bearing based on the effective reluctance and the magnetic circuit theorem [7]. Considering the skin effect of eddy current, then they analyzed the magnetic flux, coil impedance and eddy current loss by introducing the effective reluctance into the flux path. Shelke analyzed the influence of pole pairs on the loss of radial magnetic bearing using theoretical and genetic algorithm approach for different rotor speeds [8]. It was found that there was a drastic decrease in the total loss from two-pole pair to four-pole pair, and later there was slow increase of total loss with the number of pole pairs, and the total loss was found to increase with speed. Ren et al. used a calculation method considering the coupling of electromagnetic and temperature to calculate the iron loss and the heat generated by the iron loss in the magnetic bearing, and found that the heating of the magnetic bearing was related to the iron loss and closely related to the stability of the magnetic bearing [9]. Zhang et al. used the field-circuit coupled simulation method and electromagnetic-thermal iteration calculation method to investigate the loss and thermal analysis of high-speed permanent magnet synchronous machines with amorphous alloy stator core and interior permanent magnet rotor [10]. Li et al. established a 3D equivalent thermal network model with hysteresis loss and copper loss as heat sources [11]. Then by applying the thermal network model, the influence of different speeds and loads on temperature rise is analyzed. Han et al. calculated the loss of high-speed permanent magnet motor through finite element method and carried out thermal analysis based on the loss, then they compared it with the experimental measurement data to verify the correctness of the loss calculation and thermal analysis results [12]. Cheng et al. developed a 3D finite element loss model of a permanent magnet motor, in which magnet axial segmentation was taken into account [13]. Then using 3D FEM to deduce the convective heat-transfer coefficient between external frame and ambient, and the coefficient of convection heat transfer in the end region. Zhang et al. proposed an iron loss prediction model and experimental separation method for radial AMB of turbomolecular pump considering temperature influence to calculate hysteresis coefficient, eddy-current coefficient, and the iron loss of steel laminations [14,15].
However, the above studies only calculated the total loss value, did not take into account the possibility of iron loss concentration in the magnetic bearing on the circumference of the rotor journal. Since the electromagnetic loss generates heat, the iron loss concentration will make the temperature at the rotor journal asymmetrical. Then the temperature gradient at the journal will cause different expansions on both sides of the rotor, resulting in thermal bending deformation and reducing the performance of the magnetic bearing system. Therefore, it is necessary to study the iron loss concentration of magnetic bearing.
In this paper, the magnetic-thermal coupling simulation method is to obtain the iron loss through the electromagnetic field simulation by ANSYS Maxwell software and load it in COMSOL software as a heat source for temperature field simulation. The law of iron loss distribution can be obtained from the electromagnetic field simulation, and the influence of iron loss concentration on temperature distribution can be further obtained from the temperature field simulation.
The rest of this article is arranged as follows: The second section introduces the classification of loss in magnetic bearings and theoretically analyzes the reasons for iron loss concentration of magnetic bearing; in the third section, based on electromagnetic simulation, the reason for the different distribution of magnetic bearing iron loss is analyzed; in the fourth section, the influence of different parameters on iron loss is studied; in the fifth section, a magnetic-thermal coupling simulation analysis is carried out to study the influence of iron loss concentration on temperature distribution; in the sixth section, the experiment is conducted to verify the correctness of the magnetic-thermal coupling simulation; the seventh section is conclusions.
Analysis on the causes of iron loss concentration of magnetic bearing
Analysis of iron loss of magnetic bearing
As shown in Fig. 1, according to the classification of components, the energy loss of the magnetic bearing system can be divided into electronic component loss, stator loss, rotor loss and switching power amplifier loss. The electronic component loss mainly includes the loss of controllers, sensors, and others; the switching power amplifier loss mainly refers to the switching loss of the power amplifier; the stator loss mainly includes the internal windings copper loss and stator iron loss; the rotor loss mainly includes the internal iron loss and the air friction loss on the rotor surface. Iron loss can be divided into hysteresis loss, eddy current loss and extra loss. Each part has its generation mechanism and corresponding expression. Because this paper studies the iron loss concentration on the rotor journal, it mainly studies the iron loss inside the rotor.
Schematic diagram of the composition of losses in magnetic bearing.
Figure 2 shows the formation process of iron loss concentration in the magnetic bearing. In the magnetic bearing system, there is no iron loss concentration on the rotor journal at the initial time. But the rotor is affected by the initial disturbance, the rotation center will deviate from the balance position during rotation. When the displacement sensor detects the displacement of the rotor deviates from the equilibrium position, the displacement signal will be transmitted to the controller and converted into a control voltage signal, then the power amplifier converts it into control current signal. Due to the existence of control current, two pairs of magnetic poles separated by 180° will generate magnetic fields with different magnetic flux densities, which will cause uneven iron loss in the rotor. At this time, the highest iron loss point and the lowest iron loss point will appear on the rotor journal, resulting in iron loss concentration.
The formation process of iron loss concentration.
This paper takes the cantilever rotor as an example to analyze the iron loss characteristics of the magnetic bearing. The structures of stator and rotor are shown in Fig. 3. PID (Proportional-Integral-Differential) control law is chosen as the control method of magnetic bearing.
Structure diagrams of stator and rotor.
ANSYS Maxwell is a professional commercial software for electromagnetic field simulation. In this section, ANSYS Maxwell is used to calculate and analyze the magnetic field distribution and iron loss of magnetic bearing. Both the stator and rotor in the bearing adopt a laminated structure. The purpose of this paper is to study the iron loss concentration phenomenon on the journal of the magnetic bearing rotor. According to the structural characteristics, the two-dimensional model is selected for electromagnetic simulation. Therefore, using two-dimensional model to simulate the iron loss distribution can not only save the calculation cost, but also fully meet the engineering requirements in calculation accuracy. Both the magnetic bearing stator and rotor are made of 20WTG1500 silicon steel. The loss coefficients obtained in the above section are imported into ANSYS Maxwell to set the material properties. The structural diagram of the simulation model is shown in Fig. 4, and its structural parameters are shown in Table 1. The mesh division of the simulation model is shown in Fig. 5. Because the rotor iron loss is mainly concentrated at the rotor journal, it is necessary to encrypt the mesh in this area, which can make the finite element results more accurate.
Schematic diagram of the simulation model.
Main structural parameters
Schematic diagram of mesh division.
Due to uneven materials, machining errors or other reasons, there is inevitably unbalance mass in the rotor, which will produce unbalance force and cause the rotor to be disturbed. As the rotor rotates, the magnetic bearing generates synchronous electromagnetic force to suppress the unbalance vibration caused by the initial disturbance (such as the unbalance mass). The magnitude of the electromagnetic force is determined by the magnetic flux density passing through the air gap between the stator and rotor. Therefore, the electromagnetic force increases with the increase of the magnetic flux density. Through simulation analysis, the distribution of magnetic lines of force and electromagnetic force in the magnetic bearing system is shown in Fig. 6. It proves the relationship between electromagnetic force and magnetic flux density. The red dot represents the equivalent position of the unbalance mass.
Distribution of magnetic induction lines and electromagnetic force.
ANSYS Maxwell is used to calculate and analyze the magnetic flux density distribution and iron loss of magnetic bearing. When the electromagnetic field simulation software is used to calculate the iron loss of the magnetic bearing, the software does not directly use the loss curve. Instead, it conducts the loss experiment on silicon steel through the autotest system of magnetic materials (MATS-2010SA) and calculates the loss coefficients of the soft magnetic material.
The measurement principle of MATS-2010SA is the power meter method. The circuit schematic diagram is shown in Fig. 7. The variable frequency power supply provides sinusoidal excitation current for the primary winding N 1, so that the alternating magnetic flux is induced in the cross-section of the test sample.
Schematic diagram of power meter method.
The detection power in the circuit is P m , which contains the power consumed by instruments in the secondary circuit. Because the secondary voltage is sinusoidal, the approximate value is equal to (1.111|U 2|)2∕R i .
Therefore, the total loss P
c
can be calculated as follows:
The specific total loss P
s
is calculated by:
Then, the specific loss of each magnetic flux density is measured and fitted by the iron loss separation model. The loss coefficient can be obtained by data fitting. Figure 8 shows the loss measurement and fitting results, and the loss coefficient can be obtained by the curve. It can be seen from Fig. 8 that the fitting errors are all less than 5%, meeting the accuracy requirements.
Measurement and fitting curve of iron loss.
After that, Bertotti’s iron loss separation theory is used to divide the iron loss into hysteresis loss P
h
, eddy current loss P
c
and extra loss P
e
according to the generation mechanism [16], as follows:
Assuming 0° direction of electromagnetic force as the initial rotation position of rotor, through the post-processing function of the ANSYS Maxwell field calculator, the magnetic flux density curve of the rotor journal surface at that time is obtained, as shown in Fig. 9. The x-coordinate represents the angular position on the surface of the rotor journal in a rotating coordinate system. It can be seen from Fig. 9 that the magnetic field distribution on the surface of the rotor at this moment is uneven.
Distribution of magnetic flux density on the rotor surface when the electromagnetic force direction is 0°.
Superposition diagram of magnetic flux density on the surface of rotor journal under rotating condition.
In order to fully express the variation range of the magnetic flux density on the rotor surface during rotor rotation, as the rotor rotates 360° for one cycle, the rotor rotation of 11.25° is taken as one step. Each step is simulated to obtain a magnetic flux density curve, then 32 steps are taken to superpose and obtain the magnetic flux density curve on the rotor journal surface. The 32 steps have enough accuracy to represent the rotor rotation. The result is shown in Fig. 10. The 0° position of each curve is set to be in the same direction as the electromagnetic force, i.e., they represent the same point on the surface of the rotor journal. The function of the electromagnetic force is to suppress the disturbance force induced by the equivalent residual unbalance, which is opposite to the direction of the disturbance force. Therefore, each of the 0° positions of the curve is consistent with the direction of the electromagnetic force, while the equivalent residual unbalance is located at the 180° position. Obviously, during the movement of the rotor, the amplitude of the magnetic flux density varies everywhere on the journal surface. The amplitude of the magnetic flux density varies at the 0° position is greater than that at the 180° position.
Superposition diagram of rotor loss cloud diagram.
In the process of rotation, the silicon steel laminated parts of the rotor will produce iron loss under the changing magnetic flux. According to Bertotti’s iron loss separation theory, the iron loss is proportional to the magnitude of flux density. That is, the greater the change amplitude of magnetic flux density, the greater the iron loss of the rotor core will be. Considering the calculation cost, in a rotation period (T) of the rotor, take one step per T/32 interval to solve the iron loss cloud diagram. It can be seen from Fig. 11 that the iron loss at each point on the rotor is also distributed differently. The different color distributions represent the uneven distribution of iron loss. The comparison between Fig. 10 and Fig. 11 shows that the iron loss on the rotor is distributed differently, and the distribution law is related to magnetic flux density. The highest iron loss will be formed at the position of 0° and the lowest iron loss will be formed at the position of 180°, resulting in the iron loss concentration. Therefore, the iron loss distribution around the rotor journal surface can be expressed by magnetic flux density distribution.
When the structural parameters of the magnetic suspension rotor are fixed, the size and position distribution of iron loss are related to the magnetic flux density and frequency of the rotor’s magnetic field. Since there is a proportional relationship between frequency and speed, the speed is used instead of frequency for parameter analysis. The magnetic flux density is mainly determined by the winding coil current, which is divided into control current and bias current. To obtain the law affecting the rotor iron loss, this section uses simulation to qualitatively analyze the influence of control current, bias current and speed on rotor iron loss.
Influence of control current on rotor iron loss
This section mainly discusses the influence of control current on the iron loss of the rotor, which is divided into the influence of iron loss distribution and size. The following parameters are set: the speed is 30000 r/min, the bias current is 2 A and the current frequency is 500 Hz. According to the finite element simulation model established above, the magnetic flux density distribution curve on the surface of the rotor journal can be obtained when the control current is 0.236 A, 0.472 A, 0.708 A, 0.944 A, and 1.18 A, respectively, as shown in Fig. 12(a)∼(e). Besides, the relationship between the control current and the average iron loss in a period of rotor operation is shown in Fig. 12(f).
Influence of different control currents on magnetic flux density and iron loss.
It can be seen from Fig. 12 that with the increase of the control current, the difference in magnetic flux density on the circumference of the rotor journal becomes more and more obvious. From the above analysis of the distribution law between magnetic flux density and iron loss, it can also be deduced that the difference in the location distribution of iron loss is also more and more obvious. And with the increase of the control current, the average iron loss of the rotor in one rotation period also increases, but the increment is highly small.
This section mainly discusses the influence of bias current on the iron loss of the rotor. The following parameters are set: the speed is 30000 r/min, the control current is 1.18 A and the current frequency is 500 Hz. The magnetic flux density distribution curve on the surface of the rotor journal can be obtained when the bias current is 1.5 A, 2.0 A, 2.5 A, 3.0 A, and 3.5 A, respectively, as shown in Fig. 13(a)∼(e). The relationship between the bias current and the average iron loss in a period of rotor operation is shown in Fig. 13(f).
Influence of different bias currents on magnetic flux density and iron loss.
It can be seen from Fig. 13 that with the increase of the bias current, the difference in magnetic flux density on the circumference of the rotor journal becomes smaller and smaller, therefore it can be deduced that the difference in the position distribution of iron loss becomes more and more inconspicuous. As the bias current increases, the average iron loss of the rotor in one rotation period increases significantly.
This section mainly discusses the influence of speed on the iron loss of the magnetic rotor. The following parameters are set: the bias current is 2 A, the control current is 0.472 A and the current frequency is 500 Hz. According to the finite element simulation model established above, the magnetic flux density distribution curve on the surface of the rotor journal can be obtained when the speed is 6000 r/min, 18000 r/min, 30000 r/min, 42000 r/min, and 54000 r/min, respectively, as shown in Fig. 14(a)∼(e). The relationship between the speed and the average iron loss in a period of rotor operation is shown in Fig. 14(f).
Influence of different speeds on magnetic flux density and iron loss.
It can be seen from Fig. 12 that with the increase of the speed, the difference in magnetic flux density on the circumference of the rotor journal does not change, therefore it can be deduced that the difference in the position distribution of iron loss also changes little. As the speed increases, the average iron loss of the rotor in one rotation period increases observably, which is due to the significant influence of frequency on iron loss size.
From the above simulation results, the influence of control current and bias current on the uneven distribution of iron loss is greater than that of speed, and the influence of bias current and speed on iron loss size is greater than that of control current.
Finite element model and parameter setting
In this paper, the multi-field coupling analysis software COMSOL is used for the magnetic and thermal coupling analysis. The simulation model is shown in Fig. 15. In the multi-field coupling analysis, the unidirectional coupling method is adopted, i.e., the rotor iron loss is used as the heat source to calculate the temperature rise. Establish the above model in COMSOL, then define the material parameters of each part. The specific parameters are shown in Table 2. To simplify the analysis, the heat source is only defined at rotor assembly 1 in the temperature field. Besides, air with a constant temperature of 25 °C outside the rotor is set to simulate the real environment.
The structure diagram of the cantilever magnetic suspension rotor.
Main parts parameters
The mesh division of the finite element model is an essential part. The best mesh division is not only to meet the accuracy of finite element solution, but also to have a relatively small number of meshes to reduce the calculation cost and improve the calculation efficiency. According to the different structural characteristics of the rotor, the tetrahedral, triangular, edge element and vertex element are mixed to mesh the model.
Histogram of mesh quality results.
Cloud image of mesh quality.
The quality of mesh has a great influence on the finite element results, so it is necessary to detect the mesh quality. In this section, the skewness is selected to detect the mesh quality, and the results are shown in Fig. 16 and Fig. 17. Figure 16 is the histogram of mesh quality detection results, and the statistical results show that most mesh elements are to the right, tending to good. Figure 17 shows the cloud image of mesh quality. In the range of 0–1, the more the color tends to 1, the better the mesh quality is.
During the operation of the rotor, the silicon steel laminations produce iron loss and generate heat. Due to the thermal conductivity of the material itself, the heat is conducted inside the solid. Part of the heat is transferred to the surface of the rotor assembly to convective heat transfer with the air gap; the other part of the heat is transferred to the end face of the rotor or the surface of the core shaft to exchange heat with the surrounding air. At the same time, convective heat transfer and radiation heat transfer are included, so the composite heat transfer coefficient is used for equivalent [17].
Temperature cloud diagram of magnetic suspension rotor.
As time goes by, heat generation and dissipation gradually reach balance and the temperature of each part tends to be stable. As shown in Fig. 18(a), the region with the highest temperature at the rotor journal is the silicon steel lamination. The temperature distribution of the surface is shown in Fig. 18(b). There is an obvious temperature difference of 3 K between the highest temperature and the lowest temperature. It can be seen that the highest and lowest temperatures point corresponds to the 0° and 180° positions respectively in the previous section. Although the temperature difference of 3 K seems small, it can produce large thermal bending at the rotor journal, which may affect the stability of the rotor rotation. It is confirmed that the iron loss concentration in the magnetic bearing system affects the temperature distribution. Since iron loss is not easy to measure in practice, the equivalent iron loss distribution can be obtained by measuring the temperature distribution in the qualitative analysis experiments.
The difference between a stationary rotor and a rotating rotor lies in the amount of loss and heat dissipation, but the way of loss generation and heat dissipation is the same. Based on the simulation above, it can be concluded that the current has a great influence on iron loss distribution. The analogy principle is adopted to verify the temperature rise of the magnetic bearing system during the rotation, which uses the temperature rise experiment results of the magnetic bearing rotor under different current conditions to verify the simulation results under the same working conditions when the rotor is stationary. The simulation results indirectly prove the correctness of the magnetic-thermal coupling analysis method.
Experimental device and schematic.
When the rotor is stationary, 4 groups of different current are designed for experiments. The input DC bias current is 1.7 A, and the input AC control current amplitude are 0.25 A, 0.5 A, 0.75 A and 1 A respectively. First, conduct 4 sets of magnetic-thermal coupling simulations under different current conditions to get the simulation results. Then set up an experimental system to conduct experiments, and compare the simulation results with the experimental results. The experimental device is shown in Fig. 19(a).
Figure 19(b) shows the schematic diagram of the experimental system. First, the excitation voltage signal is generated by the signal generator, which is transformed into an alternating current signal with adjustable amplitude through the power amplifier in the control box of the magnetic levitation system. Then, the current passes through the transfer box and flows into a set of coils at the position of the PT100 temperature sensor 1. The current signal can be observed by the current probe and oscilloscope. The alternating current produces an alternating magnetic field. And under the alternating magnetic field, the magnetic bearing system will produce eddy current loss, hysteresis loss and extra loss. The heat generated by loss will heat the rotor assembly, causing a temperature gradient inside the rotor. The temperature on the surface of the rotor assembly is collected by the PT100 temperature sensor, then the temperature inspection instrument can visualize the collected temperature data.
Comparison of simulation results and experimental results.
On the one hand, since the measurement time of each group is 2 hours, the change of ambient temperature has a great influence on the experimental data during the temperature collection process. Therefore, keeping the ambient temperature constant is crucial to the authenticity of the experimental measurement data. Before the experiment, the indoor temperature should be maintained at a certain constant temperature through the adjustment of the air conditioner, and the room temperature should be kept as constant as possible during the experiment. On the other hand, due to the manufacturing deviation between PT100 temperature sensors, a pre-experiment is required, and then two PT100 temperature sensors with the same performance are selected and attached to the measured surface of the rotor assembly. Finally, it is necessary to carry out multiple experiments for each current condition to avoid the contingency of experimental data.
Among them, the rotor journal is a cylindrical regular symmetrical object. In order to study the temperature gradient on the journal, it is necessary to select two positions with an interval of 180° for comparative analysis. “Position 1 and Position 3” or “Position 2 and Position 4” can be selected for experiments. In this paper, the group of coils in position 1 are selected to conduct current, and position 3 is used as the treatment group.
In the experiment, the highest precision PT100 temperature sensor in industrial practice is used, and its accuracy is ±0.1 °C. Therefore, the experimental data has volatility. In data recording, the average value of temperature data is taken as the experimental data to reduce the experimental errors. After the experimental measurements, the simulation results were compared with the experimental results, as shown in Fig. 20. It shows that the simulation results at position 1 and position 3 are in good agreement with the experimental results, verifying that the magnetic-thermal coupling simulation method can accurately analyze the temperature difference distribution inside the rotor, thus confirming the iron loss concentration in the magnetic bearing.
In this paper, a multi-physical field simulation method considering magnetic-thermal coupling is proposed, which reveals the causes of iron loss concentration. The iron loss concentration is caused by the existence of the initial disturbance and the control effect of the magnetic bearing makes the magnetic field distribution on the circumference of the rotor journal uneven, which causes the uneven distribution of iron losses on the rotor journal.
The influence of different parameters on the iron loss is: the control current has a significant influence on the position distribution of iron loss, but little influence on loss; the bias current not only has a remarkable impact on the position distribution of iron loss, but also has a significant impact on loss; speed has little effect on the distribution of iron loss, but has a conspicuous effect on loss.
The magnetic-thermal coupling analysis and experimental test results prove the existence of iron loss concentration in the magnetic bearing and the iron loss concentration will cause the temperature gradient on the rotor journal.
The simulation and experimental research carried out in this paper provide a theoretical reference for the loss problem, temperature distribution and system stability of the magnetic bearing.
Footnotes
Acknowledgement
This work was supported by the National Natural Science Foundation of China (51875275), the Six Talent Peaks Project in Jiangsu Province (JNHB-041) and the Key R&D Program of Jiangsu Province (BE2019122).