Field–circuit model of the radial active magnetic bearing system
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Abstract
Paper presents a mathematical model of the radial active magnetic bearing, which is indented for the simulation of the magnetic bearing dynamic response. The circuit model of the bearing is based on differential equations. The circuit model has incorporated results of magnetic field analysis, which led to the creation of the field–circuit model. Presented model of the magnetic bearing takes into account the necessary control system. The experimental results are presented to validate the proposed model.
Keywords
Radial active magnetic bearing An electromagnetic actuator Finite element method Magnetic field analysis The control system1 Introduction
Magnetic bearings (MBs) represent an alternative support of the rotor in comparison with traditional bearings, i.e., ball or journal ones. MBs have found applications in many industrial devices, for example, in high-speed turbines, energy storage flywheels, turbomolecular pumps, turbogenerators, machine tool spindles and compressors [1, 2, 3]. The benefits of using magnetic bearings are well known [1]. Owing to the contactless operation of the rotor, the bearing provides a lack of friction, absence of lubricating substance, good vibration damping, online monitoring of the operation and reduced maintenance and operation costs.
Magnetic suspension dedicated to electric machine usually consists of two radial and one axial electromagnetic actuators and a control system. The actuator of the radial active magnetic bearing (RAMB) comprises two elements—a stator and rotor. The interaction between the stator and rotor is based on the principle of the electromagnetic interaction. The current flowing in the windings causes the pull of the movable ferromagnetic material. Unfortunately, the stable levitation of the RAMB rotor is only achievable by using position controllers.
In this paper, a field–circuit model of the RAMB system dedicated to the simulation of the transient states is described. The model is based on a set of the differential equations implemented in MATLAB/Simulink software. The main parameters of the RAMB were obtained from the magnetic field analysis. The model also includes the necessary control system with PID controllers for the rotor position and PI controllers for currents excited in windings. The presented simulation model was compared with the real object.
The aim of this paper is to present an effective and fast model of the RAMB system, which can be used to test various controllers as well as determine its parameters.
2 Structure of the active magnetic bearing
The main parameters of the RAMB actuator
Parameter | Value |
---|---|
The outer diameter of the stator | 53 mm |
Length of the stator | 45 mm |
Width of the air gap | 1 mm |
The turn number of one winding | 114 |
Bias current | 5 A |
Maximal current | 10 A |
3 Mathematical model of the active magnetic bearing
Magnetic field distribution was obtained from the 2D finite element method (FEM). The field analysis is based on the calculation of the magnetic vector potential \( \vec{A} \) [4]. Although the 2D FEM model neglects the end effects of the magnetic field, they have a minor impact on integral parameters of the magnetic field [5]. In numerical model, the eddy currents effects were neglected and the magnetic field was considered to be stationary. However, the impact of the manufacturing process was taken into account in the model, which changes the magnetic properties of narrow layers of the stator and rotor sheets [6, 7]. Hence, the air gap in the simulation model was increased in comparison with the real object.
The finite element mesh was carefully selected, in order to obtain the balance between accurate results and short time of computation. The stator and rotor subregions were discretized with the fine mesh. The discretization of the air gap has a significant impact on the force calculation results. Therefore, the air gap was divided into two subregions.
The calculated magnetic flux distribution was used to estimate the magnetic force generated by all four electromagnets. The magnetic force was calculated from Maxwell stress tensor. The calculations were executed for various cases, over the entire operating range of the magnetic bearing currents I_{1}, I_{2}, I_{3}, I_{4} ∈ (0, 10 A) and the various positions of the rotor shaft x, y ∈ (− 400, 400 μm).
Value of the constant parameters for the mathematical model
Parameter | Value |
---|---|
Windings resistance, R_{1}, R_{2}, R_{3}, R_{4} | 1.4 Ω |
Mass, m | 2.6 kg |
Eccentricity, e_{S} | 40 μm |
The pole placement method was used to obtain values of parameters for the current and position controllers [11].
Value of the coefficients for the position controllers
K_{P} (A/m) | K_{I} (A s/m) | K_{D} (A/m s) |
---|---|---|
17,417.4 | 839,446.9 | 74.8 |
Value of the coefficients for the current controllers
K_{P} (A^{−1}) | K_{I} (s/A) |
---|---|
0.1006 | 66.84 |
4 Simulation results
The value of the control current, for both axes in a steady state, is equal to 1.3 A. For this state, the electromagnets generate only the forces to balance the weight of the rotor. The settling time t_{s} of the MB system equals 36.8 ms.
5 Experimental verification of the simulations
One can find a good agreement between magnetic flux density values obtained from the field simulation and from measurements. Differences between the calculated and measured results do not exceed a few percents.
The values of control currents i_{cy}, i_{cx} in a steady state are approximately equal to 1.5 A, and they are slightly higher for the real object in comparison with those obtained from the computer simulation. This difference could be the result of omitting the eddy currents in the finite element model. The eddy currents are responsible for decreasing the values of the current and position stiffness. The settling time of the MB system is equal to 28.8 ms and its value is lower than in the computer simulation.
Values of the transient performance indicators
Transient performance indicator | Measurement | Calculation |
---|---|---|
t_{Sx} (ms) | 29.1 | 36.8 |
t_{Sy} (ms) | 28.5 | 36.8 |
J_{1x} (mm^{2} s) | 4.1·10^{−4} | 5.1·10^{−4} |
J_{1y} (mm^{2} s) | 4.3·10^{−4} | 5.1·10^{−4} |
J_{2} (μm) | 73.9 | 69.7 |
Parameter J_{2} denotes maximal deviation of the rotor center.
One can notice that the settling time and the integral of the square error took higher values for the simulation model than for the real object. It could be due to the higher value of damping in the physical model than in the numerical simulation.
6 Conclusion
In the paper, the field–circuit model of the active magnetic bearing system including its control system was presented. The correctness of the presented dynamic model was confirmed by measurements of the selected parameters under operating condition. They were calculated, for the lifting of the shaft as well as for the rotor rotation with the speed of 6000 rpm.
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