Electromechanical modeling of the dynamical behavior of magnetic bearings subjected to induced currents

Details

Written by virginie kluyskens

 

Abstract

Contextualisation

Magnetic bearings can prove to be useful in various situations, like high-speed or vacuum applications where classical roller bearings reach their limits.  

Magnetic bearings' stability is submitted to Earnshaw theorem, which implies that a static body in a magnetostatic field can not levitate in a stable way when the relative permeability of the whole system is greater than one.  Magnetic bearings can be classified by the way they achieve stability, despite this theorem.  On one hand, there are magnetic bearings which lay outside the scope of Earnshaw theorem, and are not submitted to this latter.  The first category of magnetic bearings is active magnetic bearings, in which control systems are used to command the electromagnetic forces.  Most common active magnetic bearings use an electromagnet in which the current intensity is commanded.  The second category of magnetic bearings is fully passive and includes diamagnetic bearings, superconducting bearings, permanent magnet bearings stabilized by a gyroscopic torque, and electrodynamic bearings.  On another hand, magnetic bearings using permanent magnet interaction or reluctance forces can not be stable, according to the Earnshaw theorem.  They are then combined with other kind of magnetic bearings for the remaining degrees of freedom, in order to be stable.  The third category of magnetic bearings is semi-active magnetic bearings which combine permanent magnet passive means of levitation with active means of levitation.  Similarly, hybrid magnetic bearings combine passive means of levitation with other kinds of bearings, like mechanical or hydraulic bearings.  In semi-passive magnetic bearings, the permanent magnet bearing is combined with a passive magnetic bearing.  The advantages of these bearings compared to active magnetic bearings result from the absence of electronics and powering: they are more reliable, have a lower power consumption, and are more cost effective.

   In all these magnetic bearings categories, eddy currents may play a significant role.  Eddy currents in magnetic bearings can be undergone negatively because they generate losses and unpredicted forces.  When not taken into account, this may even lead to a malfunction of the magnetic bearing;  But eddy currents in magnetic bearings may also be used positively, to generate desired forces, like in electrodynamic bearings.  However, these latter are very difficult to design.

This is why a dynamical model for the eddy current forces in electrodynamic magnetic bearings, but also in all kind of magnetic bearings and even electric machines, is important.  This model has to integrate the mechanical aspects linked to the rotor dynamics and the electromagnetic nature of the forces.

Objectives

The first objective of the electromechanical model developed in this project is to fully understand and explain the physical phenomena present in magnetic bearings, and in particular the consequences of forces due to induced currents which may lead to stable and unstable speed ranges.

The second objective is to have a practical tool which can be used to predict the dynamical behavior of passive magnetic bearings. This is necessary in order to design new stable magnetic bearings, or possibly to seak for solutions to stabilize existing magnetic bearings for a desired speed range.   These solutions may lay at the magnetic bearing level, by modifying its topology or by adding external damping, or at its support level, by mounting the complete bearing on a damper.

 

Modeling

The general parameterized electromechanical model allows to study the radial dynamical behavior of any radial magnetic bearings in which the induced currents play a significant role. A macroscopic point of view is chosen as to the electromagnetic phenomena involved in the system. This allows us to generalize the model to all kind of passive or semi-passive magnetic bearing: electrodynamic bearings, permanent magnet bearings...

 

These bearings are modeled by mechanical components, like springs, for reluctant forces, and dampers, for eddy current forces.  Indeed, on the one hand, the reluctance forces and the forces between permanent magnets can be modeled by stiffnesses, as they are proportional to the relative displacement, when these displacements remain small enough to linearize the forces. On the other hand, the Lorentz forces due to the interaction between induced currents, due to relative speed, and magnetic fields, can be modeled by introducing damping into the equations. Depending on which kind of motion the eddy currents result from, they are modeled by rotating damping (in the rotor) or by non-rotating damping (in the stator).

 

However, these Lorentz forces can not be modeled by a simple rotating damping coefficient.  Actually, when an electromotive force is induced on a conducting piece, the generated currents are subjected to inductive end resistive effects. Since the amplitude and the direction of the Lorentz forces depend on these induced currents, these forces are also subjected to these resistive and inductive effects.  These effects will be felt on the orientation of the force. The induced currents undergo a phase shift relative to the electromotive force function of the inductive and resistive effects. Our work shows that this phase shift results in an angle shift between the displacement and the forces.  The resistive and inductive effects also have consequences on the norm of the force. Indeed, the norm of these induced currents depends on the norm of the electromotive force through the impedance, and then, through the resistive and inductive effects.

Assumptions The model is based on the following assumptions: . the spinning speed  is constant; . there is no unbalance; . relative displacements remain small; . the electromagnetic variables vary in a way that may be approximated by a sinusoidal behavior; . there are only eddy currents due to the rotor spinning motion; . the resistive effects taking place in the conductor can be represented by a global resistance; . the inductive effects taking place in the conductor can be represented by a global inductance.

 

Identification of the model parameters

The  parameters of the model have to be identified for each particular magnetic bearing configuration.  This can be done on the basis of a few experimental measurements, or on the basis of a quasi-static finite element model of the bearing. The parameters are independent of the center shift.

We showed that the model gains in accuracy when a dependancy of the global resistance and the global inductance with the spin speed is considered.  This results from the skin effect.  The skin effect is a phenomenon which tends to concentrate currents on the surfaces of conductors nearest to the field producing them. This redistribution of the current produces an evolution of the apparent resistance and inductance.  At low frequencies, the impedance evolution with the frequency is generally insignificant because the skin depth is much higher than the  characteristic dimensions of the conductor. At high frequencies, when the skin effect becomes predominant, the frequency evolution of the resistance and the inductance is generally well-known and is not constant anymore. Magnetic bearings are susceptible to work with a broad frequency spectrum and pass gradually from a situation where the skin effect is negligible to a situation where it becomes predominant. We developped thus different models of the impedance dependence with respect to skin depth

Validation

Experimental qualitative validations show the existence of a stability threshold which depends of the amount of non rotating damping present in the system

Finite element quantitative validation show a good agreement between the FEM forces estimation and the electromechanical model forces predictions.  

Examples of exploitation of the model

The developped model has been used to analyze the influence of different parameters, model parameters and physical parameters, on the induced stiffness produced by an electrodynamical bearing and on the unstabilty speed range. The model has also been used to be integrated in more complex systems, where the rotor presents an eccentricity and where the stator is connected to the ground via a parallel spring-damper system.

 

References

V. Kluyskens, B. Dehez, "Comparison between models predicting the evolution of the electrical impedance with frequency", International Journal of Circuit Theory and Applications, April 2010, DOI 10.1002/cta.680

V. Kluyskens, B. Dehez, "Parameterized electromechanical model for magnetic bearings with induced currents", Journal of System Design and Dynamics (Special Issue on the Eleventh international Symposium on Magnetic Bearings), April 2009, vol 3, n°4, pp453-461, DOI 10.1299/jssd.3.453

V. Kluyskens, B. Dehez, ''Analysis of a dynamical electromechanical model'',  proceedings of ISMB11 conference, Nara, Japon, 26-29 août 2008

V. Kluyskens, B. Dehez, ''Computation of the Forces acting on a Magnetic Bearing due to Eddy Currents'',  COMSOL conference 2007, Grenoble, France,23-24 Octobre 2007

V. Kluyskens, B. Dehez, et H. Ben Ahmed, ''Dynamical electromechanical model for passive magnetic bearings'', IEEE Transaction on Magnetics, no. 43, pp. 3287-3292.

V. Kluyskens, B. Dehez, ''Induced currents in magnetic bearings: electromechanical model and skin effect'',  proceedings of ISMB10 conference, Martigny, Suisse, 21-23 août 2006

To contact us

 Virginie Kluyskens