# Suppression of Harmonic Current in Magnetic Bearing-Rotor System with Redundant Structure
## 2 Mode of Mass Unbalance and Sensor Runout  
Periodic vibration is a typical problem in the system of a magnetically levitated rotor, and the vibration is mainly caused by the characteristics of the whole system. Therefore, it is necessary to analyze and express the motion characteristics and system noise of the structural parts of the system. In essence, the support between the magnetic bearing and rotor can be regarded as an elastic support with air gap. The mechanical structure uses its own material characteristics and mutual cooperation to form the electromagnetic field, and the electromagnetic force is formed between the rotor and the stator. The existence of machining error of the mechanical structure, uneven material, rotor imbalance, sensor noise, and nonlinear characteristics of each module of the control system can produce vibration with rich frequency components. Under the joint action of multisource vibration, the harmonic component disturbances of the fundamental and multiple frequencies are generated in the whole magnetic bearing-rotor system, and the current with the corresponding multi-frequency components is generated through the control system.  
Uneven rotor mass is the main factor causing rotor vibration. Rotor unbalance mainly includes static imbalance and dynamic imbalance. Static unbalance refers to the deviation between the inertia axis center and the geometric axis center of the rotor. Dynamic unbalance means that the inertia axis of the rotor is different from the geometric axis.  
The section of the single radial magnetic bearing-rotor system is shown in Figure 1. It is supposed that the geometric center plane of the rotor coincides with the inertial center plane, and the geometric center of the rotor \(O_{g}\) is concentric with the geometric center of the magnetic bearing, The center of inertia of the rotor is \(O_{i}\), and \(\varepsilon\) is the mass eccentricity. When the rotor rotates around the geometric center during operation, the unbalance forces \(F_{ex}\) and \(F_{ey}\) in the \(x\)- and \(y\)-directions can be expressed as  
\[\left\{\begin{array}{l}F_{ex}=m\varepsilon\omega^{2}\cos(\omega t+\zeta)\\ F_{ey}=m\varepsilon\omega^{2}\sin(\omega t+\zeta),\end{array}\right. \tag{1}\]  
where \(m\) is the mass of the rotor, \(\omega\) is the rotor speed, and \(\zeta\) is the initial phase.  
Due to the inevitable manufacturing errors in the rotor and stator of the magnetic bearing system shown in Figure 2, there is a lack of concentricity of the sensing surface and no uniform electrical or magnetic properties around the sensing surface. Therefore, while the rotor is in the process of rotation, the output signal of the displacement sensor contains the harmonic interference of synchronous and multiple frequencies of rotation speed. Thus, in a magnetically levitated rotor system with redundant structures, the signal of displacement sensors can be described as  
\[\left\{\begin{array}{l}x_{s}=x_{0}+xd(t)\\ y_{s}=y_{0}+yd(t),\end{array}\right. \tag{2}\]  
where \(x_{d}(t)\) and \(y_{d}(t)\) denote the sensor runout in the \(x\)- and \(y\)-directions of sensors, and the mathematical model can be described as  
\[\left\{\begin{array}{l}x_{d}(t)=\sum\limits_{i=1}^{n}a_{xi}\cos(i\omega t+ \phi_{xi})\\ y_{d}(t)=\sum\limits_{i=1}^{n}a_{yi}\cos(i\omega t+\phi_{yi})\end{array}\right.,i=1,2,\cdots,n, \tag{3}\]  
where \(i\) is the order of harmonics, \(a_{xi}\) and \(a_{yi}\) represent the sensor runout corresponding to the \(x\)- and \(y\)-direction sensors, and \(\phi_{xi}\) and \(\phi_{yi}\) are the initial phases.  
Figure 1: Section of single radial magnetic bearing–rotor system.  
Figure 2: Mechanism diagram of sensor runout.