Effect of AMB winding configurations on static and dynamic performances and power consumptions of the AMB-rotor system

Abstract

Active magnetic bearings (AMBs) bring extraordinary benefits such as free of contact, elimination of lubrication, active control of rotor position, and a built-in monitoring system. In the design of AMBs, the bearing structure is of significance since it has an important impact on bearing performances. However, the effect of winding configurations of AMBs is still obscure. In this paper, the system-level implications of two radial AMB winding configurations termed the Ortho and the Cross types are investigated, including the bearing characteristics, power consumptions and rotor dynamic behaviors. The simulation results demonstrate that the Cross type contributes to larger load capacity in the direction of gravity. In condition of a heavier gravity load (−200 N), the Cross-type winding configuration saves power consumption with a percentage of 17.2% at steady state (20,000 rpm) and 12.7% considering unbalance mass during the run-up process. However, the rotor vibrations of the Cross type in case of external loads and unbalance mass are larger than the Ortho type. The proposed results of this paper provide some useful information for the AMB winding configuration design.

Keywords

Active magnetic bearing winding configuration power consumption rotor dynamics

1. Introduction

Active magnetic bearings (AMBs) use adjustable magnetic force to levitate the rotor, thus there is no mechanical contact between the rotor and the stator [1,2]. High speed, high accuracy, no abrasion and free of lubrication, makes AMBs an ideal supporting element in industry such as ultra-clean devices, vacuum equipment, turbo machinery, and high-speed electrical spindles [3–5]. Compared with conventional bearings, AMBs possess unique advantages, including remote monitoring, active controlling, vibration isolation or elimination, semi-permanent life and reduced maintenance [6,7]. However, major concerns in the industrial implementation of AMBs are the large power consumptions and limited load capacity [8–10], which are largely affected by mechanical design, including the winding configuration. Therefore, it is crucial to evaluate the comprehensive effects of coil windings in the AMB design.

Different magnetic pole counts and the corresponding winding schemes have been reported in the literature. Chen and Hsu [11] studied a three-pole AMB’s optimal design considering bias current, pole orientation angle, number of coil turns, and pole face area. Simulations show that only two power amplifiers are required and the copper losses are reduced not only at the steady state, but during the transient as well. Han [12] investigated a permanent magnet bias radial-axial magnetic bearing with four poles. The results manifest that the maximum power loss of the bearing set with two proposed bearings is reduced by 15.6% compared with the original bearing set which consists of two radial eight-pole AMBs and one pair of axial AMBs. The weight and the volume are reduced by 13% and 16.3%, respectively. Ji et al. [13] proposed a six-pole hybrid magnetic bearing and designed the bias flux density based on maximization of load capacity in different directions. Experimental results indicate that the total power consumption for a 2.2 kg rotor is only 13.35% of a traditional eight-pole AMB. Kasarda et al. [14] demonstrated the rotor losses with homopolar and heteropolar eight-pole AMB designs, which mainly differ in flux paths. Test results show that homopolar bearing has significantly lower power losses than the heteropolar bearing. However, the homopolar bearing occupies a longer axial space and limits the critical speed of the rigid rotor [15]. Ahrens et al. [16] presented an analytical model of an eight-pole heteropolar magnetic bearing with which magnetic fields, forces and losses are calculated. Analytical results reveal that the pole sequence NSNS has smaller losses than NNSS.

Furthermore, there are some other reports concerning the winding designs of AMBs. Lijesh and Hirani [17] provided winding constraint equations concerning wire diameter, number of turns and dimensions of the pole. This numerical optimization method also takes the power consumption as a design constraint and realizes maximal levitation force under given conditions. Štumberger et al. [18] optimized a radial AMB by differential evolution algorithm and two-dimensional (2D) finite element analysis. Maximum bearing force at an unchanged mass is obtained and the values of the force, position stiffness and current gain are determined. Xu et al. [19] optimized the flux density distribution of a Lorentz-type magnetic bearing by finite element simulation. The results demonstrate that the uniform degree of magnetic flux density distribution reaches 0.978 and the average value of the flux density is 0.482 T. Baumgartner et al. [20] presented the Lorentz-force-based AMBs, in which the AMB windings are placed in the inner diameter of the motor winding to obtain an external flux density generated by the permanent magnets on the rotor. Compared with conventional reluctance-force-based AMB, the overall system is more compact and the prototype reaches speeds of 505 kr/min.

There are two common winding schemes for the eight-pole AMB with NNSS configuration [21–24], termed as “Ortho” and “Cross” types in this paper. However, how the winding configurations influence the load characteristics, rotating losses and dynamic characteristics are obscure. Accordingly, this paper evaluates the comprehensive effect of two winding configurations. The magnetic fields of the two winding configurations are demonstrated by 2D finite element analysis. In addition, the corresponding dynamic characteristics and power consumptions of a 3D AMB-rotor system are studied via Simulink.

2. Virtual 3D AMB-rotor system

The virtual 3-dimensional (3D) AMB-rotor system for this study is established by Simulink. As demonstrated in Fig. 1, the virtual shaft-bearing system mainly includes three modules: magnetic bearings, the permanent magnet synchronous motor (PMSM) and the shaft. The magnetic bearing system mainly consists of the sensor module, the controller module and the actuators. The PMSM located between the front AMB and the rear AMB is comprised of the rotor, the stator and the permanent magnets. The shaft, supported by one axial and two radial magnetic bearings, is 0.484 m long and weighs around 8.1 kg. The specific details of radial AMBs, axial AMB and PMSM are listed in Table 1 and 2, respectively. The control algorithm is based on classical PID.

Fig. 1.

3-D model of the AMB-rotor system in Simulink.

Table 1

Parameters of the radial magnetic bearing

Parameters	Value	Parameters	Value
Number of poles	8	Area of pole A	3.648 × 10⁻⁵ m²
Nominal air gap x ₀	7 × 10⁻⁴ m	Bias current I ₀	1.2 A
Shaft radius r	1.75 × 10⁻² m	Bearing inner radius r _i	2.625 × 10⁻² m
Bearing outer radius r _o	3.34 × 10⁻² m	Coil turns N	240

Table 2

Parameters of the 3D axial magnetic bearing and PMSM

Parameters of the axial AMB	Value	Parameters of the PMSM	Value
Nominal air gap h _a	7 × 10⁻⁴ m	Rated speed w	12 krpm
Coil turns N _a	60	Radius of rotor r _r	3.5 × 10⁻² m
Area of pole A _a	7.21 × 10⁻⁴ m²	Width of rotor L _r	0.108 m
Inner radius of the pole r _ip	2.04 × 10⁻² m	Width of magnet/stator L _m	8.64 × 10⁻² m
Outer radius of the pole r _op	2.54 × 10⁻² m	Outer radius of magnet r _om	3.68 × 10⁻² m
Inner radius of axial stator r _ia	3.22 × 10⁻² m	Stator air-gap radius r _as	3.82 × 10⁻² m
Outer radius of axial stator r _oa	3.56 × 10⁻² m	Stator outer radius r _os	6.87 × 10⁻² m
Width of coil slot L _c	3.4 × 10⁻³ m	Phase number	3
Width of axial stator L _s	6.8 × 10⁻³ m	Number of pole pairs	4

The working principle of the virtual shaft-bearing system is as follows: the displacement sensors detect the journal position and transfer the signals to the controller, which outputs the control signals to excite the magnetic coils. In this way, the shaft is maintained at the equilibrium position rather than adhered to the bearing in case of any perturbance.

To verify the method by establishing virtual AMB-rotor system via Simulink, the simulation data was compared with experimental results. The parameters of the AMB test rig are from our previous work [1,25]. An unbalance mass of 0.002 kg and an eccentric distance of 0.01 m are assumed to be closer to the actual situation. The control currents were recorded under varied horizontal (Fig. 2a) and vertical (Fig. 2b) loads. It shows that the simulation data basically agree with experimental results, and the variation tendencies of control currents are the same: the control currents in X∕Y direction increase with the horizontal/vertical loads. As demonstrated in Fig. 3, the rotor displacements had similar case with the control currents. Considering that the AMB test rig is a complicated mechatronic device with many components, and some of them cannot be totally the same in the simulation, e.g. the PMSM. In view of the unknown machining error (which causes rotor unbalance mass) and some nonlinear factors from electronic devices, the virtual AMB-rotor system is an ideal mechatronic system and therefore inevitably exits some errors, yet the changing trends of control currents and rotor displacements are the same with experimental results, indicating the reliability of the conclusions in this paper.

Fig. 2.

Verification of control currents of the AMB-rotor system in Simulink: (a) control currents in X direction under horizontal loads; (b) control currents in Y direction under vertical loads.

Fig. 3.

Verification of rotor displacements of the AMB-rotor system in Simulink under horizontal loads: (a) X displacement; (b) Y displacement.

3. Description and FE analysis of the two winding configurations

This section presents two winding configurations for radial AMBs, and provides associated 2D FE analysis of the AMBs.

3.1. Description of the two winding configurations

As shown in Fig. 4, there are two current excitation methods for eight-pole AMBs with the paired-poles winding configuration. One is the Ortho type with the excitation currents exerted on vertical and horizontal directions respectively. The other is the Cross type with two pairs of excitation currents inclined to the vertical at ±45° angles.

Fig. 4.

Schematic of the two winding configurations: (a) Ortho type; (b) Cross type.

For the Ortho type (Fig. 4a), the vertical bearing load is generated by two vertical pairs of coils. The resultant reluctance forces on the journal can be linearized as [26] $\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}f_{x}^{o}=k_{x}^{o}x+k_{ix}^{o}i_{x}\\[3.0pt] f_{y}^{o}=k_{y}^{o}y+k_{iy}^{o}i_{y}\end{array}\right\}\quad \text{where}∼\left\{\begin{array}{@{}l@{}}k_{k}^{o}={\mu}_{0}AN^{2}I_{0k}^{2}/h^{3}\\[3.0pt] k_{ik}^{o}={\mu}_{0}AN^{2}I_{0k}/h^{2}\end{array}\right.\quad (k=x,y) & & \displaystyle\end{eqnarray}$ (1) where μ₀ denotes the vacuum permeability, A the cross-section area, N the coil turns, h the nominal air gap, $k_{k}^{o}$ the displacement stiffness, and $k_{ik}^{o}$ the current stiffness.

However, for the Cross type (Fig. 4b), the coil currents have identical component forces in vertical and horizontal directions. The resultant forces are listed as follows, where ϵ and δ are the rotor displacements at 45 and 135 degrees respectively. $\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}f_{x}^{c}=\sin 45^{\circ }(k_{{\varepsilon}}^{c}{\varepsilon}+k_{i}^{c}i_{{\varepsilon}}+k_{{\delta}}^{c}{\delta}+k_{i}^{c}i_{{\delta}})\\[2.0pt] f_{y}^{c}=\cos 45^{\circ }(k_{{\varepsilon}}^{c}{\varepsilon}+k_{i}^{c}i_{{\varepsilon}}+k_{{\delta}}^{c}{\delta}+k_{i}^{c}i_{{\delta}})\end{array}\right\}\quad \text{where}∼\left\{\begin{array}{@{}l@{}}k_{k}^{c}={\mu}_{0}AN^{2}I_{0k}^{2}/h^{3}\\[2.0pt] k_{i}^{c}={\mu}_{0}AN^{2}I_{0k}/h^{2}\end{array}\right.\quad (k={\varepsilon},{\delta}). & & \displaystyle\end{eqnarray}$ (2) Based on the different winding configurations, the force directions and load capacities are different even with identical coil currents.

3.2. FE analysis of the AMB with different winding configurations

Since the magnetic circuit design of AMB is significant for system design to produce enough magnetic force [27], the FE analysis of the AMB is conducted in this section. The 2D static magnetic field analysis is presented to observe the distributions of magnetic flux and flux density of the proposed winding schemes.

The rotor is made of silicon steel 50W350, and the stator is made of No. 10 steel. Figure 5 demonstrates the distributions of the excitation currents for the Ortho and the Cross types. The key parameters of the AMB model used for the FE analysis are listed in Table 1.

Fig. 5.

Pole arrangements and excitation current distributions: (a) Ortho type; (b) Cross type.

Figure 6 demonstrates the magnetic flux distribution of the two winding types. For both winding types, the magnetic fluxes are symmetrically distributed and concentrated on four pairs of magnetic poles and their corresponding parts of the rotor. And magnetic flux leakage exists in both structures, but the leakage value is very small. For the Ortho type, the magnetic flux loops in the Y and the X directions are independent of each other. Nevertheless, for the Cross type, the magnetic flux loops are distributed at the 45- and the 135-degree angles, therefore coupling in the X and the Y directions.

Fig. 6.

Flux distributions of two distinct winding configurations: (a) Ortho type; (b) Cross type.

Figure 7 shows the flux density distributions when the rotor is at the equilibrium position. For both winding types, local magnetic fields are formed by each pair of magnetic poles and the corresponding rotor. And there is almost no magnetic coupling between magnetic circuits. The maximum and minimum values are 1.22 T, 1. 17 × 10⁻⁴ T for the Ortho type, and 1.25 T, 1. 17 × 10⁻⁴ T for the Cross type. In condition of the same coil excitation currents, the maximum flux density of the Cross type is 2.4% larger than the Ortho type.

Fig. 7.

Flux density of two distinct winding configurations: (a) Ortho type; (b) Cross type.

4. Simulation and discussions

To investigate the differences of the two winding configurations, four different cases are analyzed for the AMB-rotor system. Rotor vibrations in case of horizontal static loads, vertical static loads and unbalance loads during the run-up are compared. The radial bearing capacity and associated power consumptions are simulated.

4.1. Comparison of horizontal static loads on system performance

The effects of horizontal static loads on the AMB-rotor system with two winding configurations are shown in Figs 8–10. The employed control parameters are K _p = 12,000, T _d = 872.2. The horizontal loads which imposed on the AMB at 0.09 s are 50 N, 100 N and 150 N, respectively.

Fig. 8.

Rotor displacements under horizontal static loads (F _x = 50 N/ 100 N/ 150 N and F _y = 0 N): (a) X displacement; (b) Y displacement.

Fig. 9.

Control currents of the AMB under horizontal static loads (F _x = 50 N/ 100 N/ 150 N and F _y = 0 N): (a) in X and ϵ directions; (b) in Y and δ directions.

Fig. 10.

Power consumptions of two winding configurations under horizontal static loads (F _x = 50 N/ 100 N/ 150 N and F _y = 0 N).

Figure 8 demonstrates the rotor vibrations in X (Fig. 8a) and Y directions (Fig. 8b). The rotor vibrations in X direction increase with the horizontal load for both winding configurations. However, the Y-axis displacement with the Ortho winding configuration is nearly zero, while that with the Cross type increases with the horizontal load (0.051 μm, 0.16 μm for 100 N, 150 N respectively).

Control currents are demonstrated in Fig. 9. The control currents in X∕ϵ directions increase with the horizontal loads for both winding configurations. However, currents of the Ortho type (in X direction) are larger than that the Cross type (in ϵ direction) all the time. In addition, the control currents in Y direction with the Ortho winding configuration remain the same when horizontal loads are exerted, while that of the Cross type in δ direction increase from a smaller value and surpass currents of the Ortho type under horizontal loads.

Figure 10 shows the current square of the AMB coils. The power consumption of the AMB is proportional to the square of continuous current rating (A²) and time (s) if the AMB winding resistance is assumed as a constant. Despite different control current distributions, the power consumptions are equal for both winding types.

System performances in cases of horizontal static loads show that although the Cross type has more evenly-distributed currents in both directions, the power consumptions with both winding configurations are identical under the same horizontal loads, bias currents and control parameters. However, the vibration responses reveal that the Cross type has a coupling effect in X and Y directions.

4.2. Comparison of vertical static loads on system performance

The effects of vertical static loads on the AMB-rotor system with two winding configurations are shown in Figs 11–13. The vertical loads which imposed on the AMB at 0.09 s are −50 N, −100 N and −150 N, respectively.

Fig. 11.

Rotor displacements under vertical static loads: (a,b) X and Y displacements with F _x = 0 N and F _y = −50 N/ −100 N/ −150 N; (c,d) X and Y displacements with F _x = 100 N and F _y = −50 N/ −100 N/ −150 N.

Fig. 12.

Control currents of the AMB under vertical static loads: (a,b) control currents in X∕ϵ and Y∕δ axes with F _x = 0 N and F _y = −50 N/ −100 N/ −150 N; (c,d) control currents in X∕ϵ and Y∕δ axes with F _x = 100 N and F _y = −50 N/ −100 N/ −150 N.

Fig. 13.

Power consumptions of AMBs under vertical static loads: (a) F _x = 0 N, F _y = −50 N/ −100 N/ −150 N; (b) F _x = 100 N, F _y = −50 N/ −100 N/ −150 N.

Figure 11 presents the rotor vibrations in X and Y directions. The first row (Figs 11a, 11b) excludes horizontal loads. The Ortho type fails to resume equilibrium position when the vertical load increases to −150 N. The second row (Figs 11c, 11d) imposes 100 N horizontal load at the same time. Both Ortho and Cross winding configurations fail in case of −150 N. The Ortho type has lower load capacity in gravity direction with the same control parameters.

Control currents are shown in Fig. 12. The currents of the Cross type are more even in ϵ∕δ directions. While loads in Y direction have no impact on currents in X direction for the Ortho type.

The power consumptions are illustrated in Fig. 13. Despite that the control current distributions in two directions are different, the power consumptions are identical for both winding types except for failure cases.

System performances in cases of vertical static loads show that despite the Ortho type prevails in vibration responses, it also suffers from lower load capacity in Y direction under the same parameters. The results also reveal that the radial load capacities of the AMB with the two winding configurations vary along the circumference.

4.3. Comparison of bearing capacities of radial AMBs

Radial bearing capacity distributions of the AMB with the two winding types in the same conditions of control parameters and bias currents are shown in Fig. 14. The bearing capacity curves of Cross type and Ortho type around the circumference are similar but have a slight difference of a 45-degree angle. The bearing capacities of Cross type at 180.0° and 90.0° degrees are 48.1% and 29% higher than the Ortho type, whereas the bearing capacities at 225.0°, 135.0° and 270.0° are relatively decreased by 28.2%, 28.2% and 87.1% compared with the Ortho type.

Fig. 14.

Bearing capacity distribution of the radial AMBs with Ortho- and Cross- type winding configurations under the same control parameter K _p = 120,000 and bias current I ₀ = 1.2 A.

Figure 15 illustrates the power consumption corresponding to the radial bearing capacity of the AMB. When the bearing capacities of Cross type and Ortho type are the same (e.g. at 0°, 157.5°, 202.5° angles), the power consumptions of the AMB are equal. The power loss of the AMB correlates positively with the external force.

Fig. 15.

Power consumption distribution of the radial AMB with Ortho- and Cross- type winding configurations under the same control parameter K _p = 120,000 and bias current I ₀ = 1.2 A.

From the above simulations, the Cross type has a larger reserve in the gravity direction in the same conditions of control parameters and bias currents, whereas the Ortho type leads to better performance at ±45° angles inclined to the gravity direction.

4.4. Comparison in run-up tests with unbalance mass

The load capacity in gravity direction is −135 N of the Ortho type and −200 N of the Cross type under the same control parameters K _p = 12,000, K _d = 100. To bear larger gravity load (−200 N) of the Ortho type, the control parameters for the Ortho type become K _p = 2,800,000, K _d = 4162.4, while that of the Cross type remain the same.

The run-up simulations with larger gravity load (−200 N) and unbalance mass are conducted. As shown in Fig. 16, the rotor runs up to 20,000 rpm with constant acceleration (about 2500 rpm/s). The unbalance mass weighs 0.002 kg and offsets 0.01 m from the geometric journal center.

Fig. 16.

Rotor speed over time during running up simulation.

Figure 17 shows that as the speed goes up, the rotor vibrations of the AMB-rotor system applying the Cross-type winding configuration increase steadily, yet are less than 0. 8 μm at 20,000 rpm. However, the rotor vibration with the Ortho type is less than 0. 04 μm even at 20,000 rpm.

Fig. 17.

Vibration amplitudes of the rotor with Ortho and Cross winding configurations during running up process under vertical load F _y = −200 N, unbalance mass m = 0.002 kg, and eccentric distance u _m = 0.01 m.

Figure 18 presents that with increasing rotating speed, the energy consumption trend vibrates with the Ortho type, while steadily goes down with the Cross type. The average power losses of the Cross type are reduced by 12.7% compared with the Ortho type during the run-up process, and by 17.2% at steady state (20,000 rpm).

Fig. 18.

Power consumptions of the AMBs with Ortho and Cross winding configurations during running-up process under vertical load F _y = −200 N, unbalance mass m = 0.002 kg, and eccentric distance u _m = 0.01 m.

The run-up tests with an unbalance mass and a heavy gravity load (−200 N) reveal that the Cross-type winding configuration contributes to less energy consumption due to larger load capacity in the gravity direction and therefore smaller control parameters. However, the Ortho type prevails in rotating accuracy. The first reason is that the rotor vibration of the Cross type has a coupling effect in X and Y directions, which increases the control complexity. The second reason is the much smaller control parameters of the Cross configuration, leading to smaller bearing stiffness and therefore larger vibrations.

5. Conclusions

The comprehensive performance assessment of two winding configurations has been demonstrated in this paper. The main results are as follows:

The magnetic fluxes are distributed in two major perpendicular axes which have a 45-degree angle difference with similar structures. The magnetic density of the Cross type is 2.4% larger than that of the Ortho.

The rotor displacements of the Cross type have a coupling effect in X and Y directions. The currents in ϵ∕δ directions of the Cross type are more evenly distributed. In the same conditions of bearable loads, control parameters and bias currents, the power losses are identical for both winding types.

With the same bias currents and control parameters, the Cross type has more reserve for gravity direction load (−135 N for the Ortho and −200 N for the Cross). Thus, the Cross type saves power losses with a percentage of 17.2% at steady state and 12.7% considering unbalance mass at the run-up test under a −200 N gravity load.

In general, the AMB with the Ortho winding configuration possesses better rotating accuracy. However, the Cross type saves power losses and has larger load capacity in the gravity direction, and therefore has its unique practical value.

Footnotes

Acknowledgements

The authors are grateful for the supports of Program of the National Natural Science Foundation of China (No. 51836009) and China Scholarship Council (CSC).

Nomenclature

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