# Effects of Imperfect Assembly and Magnetic Properties on the Three-Pole AMB System
###### Abstract  
This study is concerned with a three-pole active magnetic bearing (AMB) system with assembly error and non-uniform flux distribution. The assembly error, which is the result of the misalignment of the back-up bearing and the stator of AMB, induces strong nonlinear uncertainty in the AMB dynamics. The non-uniform flux distribution, which is mainly due to non-uniform material properties, manufacturing errors, etc., makes the magnetic force model more complicated. A stable-levitation controller is designed in consideration of the above factors. The controller is designed using the method of feedback linearization and integral sliding mode control (ISMC). Both simulation and experimental results indicate that the rotor can be levitated to the center of the back-up bearing, verifying the effectiveness of the proposed stable-levitation controller.  
active magnetic bearing; assembly error; integral sliding mode control +
Footnote †: journal: Applied  
and linear magnetic field. In practical applications, such assumptions may not be satisfied. As a result, the uncertainties are generally inevitable in the magnetic force model. In addition to the magnetic force model, the uncertainties in the AMB system can also come from several other sources, including sensor noise, external disturbances, manufacturing and assembly errors, and imperfect magnetic properties.  
There have been many studies in the literature considering the uncertainty in an AMB system, such as sensor noise, un-modeled dynamics, magnetic saturation, etc. [13; 14; 15; 16; 17]. However, there are two factors that can exist in practical AMB applications but which are seldom studied. One is the assembly error. It is usually assumed that the back-up bearing and AMB are concentric at the steady state. In other words, the rotor will be stabilized at the center of the back-up bearing so that the air gap at each magnetic pole is the same. However, due to assembly errors, the center of the back-up bearing may be biased from the stator center. Another factor is the non-uniform magnetic properties among magnetic poles due to non-uniform material properties, manufacturing errors, etc., which will cause non-uniform flux distribution.  
This study is concerned with the modeling issue and controller design for a 3-pole AMB system with these two factors. The 3-pole AMB system possesses many advantages over the commonly used 8-pole AMB system [18]. Firstly, for a 2-degrees-of-freedom (DOF) AMB system, only two power amplifiers are required, while the 8-pole AMB system needs 4 power amplifiers. Next, it does not need large bias current for the linearization of the magnetic force, leading to lower bias currents. In addition, it has more space for winding, sensor installation, and heat dissipation. Finally, the re-magnetization frequency of the 3-pole AMB system is half of that for the 8-pole one, implying lower eddy-current loss and hysteresis loss. In summary, the 3-pole AMB system can save more costs compared to the 8-pole configuration. The difficulties associated with the 3-pole system is that there exist strong magnetic coupling effects among the 3 magnetic poles. As a result, the magnetic force model is strongly nonlinear and cannot be linearized. There have been extensive studies on the modeling and controller design for the 3-pole AMB system [19; 20]. In this paper, the effects of imperfect assembly and magnetic properties on the 3-pole AMB system will be discussed, which have never been studied in the literature. Due to these effects, the conventional magnetic force model of the 3-pole AMB system needs to be modified. These issues are more significant for the 3-pole AMB system since the magnetic fluxes among the magnetic poles are coupled and the magnetic force model is strongly nonlinear [12; 18]. The dynamic model of a 3-pole AMB system with the two factors will be derived and a stable-levitation controller will be designed.