# Suppression of Harmonic Current in Magnetic Bearing-Rotor System with Redundant Structure
## 3 Characteristic and Control Strategy of Magnetic Bearings with Redundant Structures
### Control Strategy and Analysis of Magnetic Bearings with Redundant Structures  
The essence of the current distribution matrix \(\mathbf{W}\) is the mapping relationship between the logic currents and the current of each coil. The current distribution control strategy is designed with the mapping relationship described in Equation (10), and the distribution control strategy can be treated as a part of the system control strategy in the magnetic bearing with redundant structures. The entire system controller is described in Figure 4.  
In Figure 4, the system controller includes the PID controller and current distribution controller. According to the error between the desired rotor position signal and the feedback of the actual rotor position, the PID algorithm controls the rotor position. With the control strategy described in Figure 4, the electromagnetic forces of the rotor while at the equilibrium position can be considered as a linear relationship with displacement and logic current. For a deep analysis of the relationship between logic control current and current of each coil in the magnetic bearing, we can use the inverse of the current distribution matrix and current of each coil to form the logic control current; this relationship can be described as  
\[\mathbf{I}_{c}=\mathbf{W}^{-1}\mathbf{I}. \tag{28}\]  
In this paper, the structural parameters of the eight-pole symmetrical radial magnetic bearing are shown in Table 1. The corresponding current distribution matrix \(\mathbf{W}\) is  
\[\mathbf{W}=\frac{g_{0}}{4N\sqrt{\mu_{0}A}}\left[ \begin{array}{ccc}2&2&0\\ -2&-\sqrt{2}&-\sqrt{2}\\ 2&0&2\\ -2&\sqrt{2}&-\sqrt{2}\\ 2&-2&0\\ -2&\sqrt{2}&\sqrt{2}\\ \end{array} \right]. \tag{29}\]  
On the basis of the structural parameters of the magnetic bearing and current distribution matrix, the position stiffness and current stiffness can be calculated using Equations (20)-(27), where \(K_{xx}=-40\),000 N/m, \(K_{xy}=K_{yx}=0\), \(K_{yy}=-40\),000 N/m, \(K_{ix}=4\) A/m, \(K_{xy}=K_{iyx}=0\), and \(K_{iyy}=4\) A/m. The results show that the essence of the bias current coefficient is current stiffness. For the eight-pole symmetrical radial magnetic bearing studied in this paper, the electromagnetic forces can also be described as  
\[ \begin{array}{l}F_{x}=-K_{xx}x+K_{ix}x_{x}\\ F_{y}=-K_{yy}y+K_{iyy}y_{y}\\ \end{array}  \tag{30}\]  
Furthermore, considering the unbalanced force and neglecting the torque of the rotor, the rotor dynamic equation in the \(x\)- and \(y\)-directions can be written as  
\[ \begin{array}{l}m\ddot{x}=F_{x}+F_{cx}\\ m\ddot{y}=F_{y}+F_{ey}\\ \end{array}  \tag{31}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Structure Parameter** & **Value** & **Unit** \\ \hline Pole area, \(A_{0}\) & \(5.4\times 10^{-5}\) & m\({}^{2}\) \\ Turns per coil, \(\overline{N}\) & 56 & / \\ Pole initial gap, \(g_{0}\) & \(4\times 10^{-4}\) & m \\ Pole angle, \(\theta_{j}\) & \((j-1)\pi/4\) & rad \\ Saturation magnetic-flux density, \(B_{sat}\) & 1.2 & T \\ Rotor weight, \(m\) & 0.8 & kg \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of magnetic bearing. Figure 4: Closed-loop control strategy in magnetic bearings with redundant structures. Through the above analysis, the equivalent control schematic diagram of the entire system can be depicted as shown in Figure 5.  
As shown in Figure 5, \(x^{\star}\), \(y^{\star}\), \(\hat{r}^{\star}x\), and \(\hat{r}^{\star}y\) denote the desired position and desired logic control current, respectively. \(G_{cx}\) and \(G_{cy}\) are the position control law in the \(x\)- and \(y\)-directions; \(G_{w}\) is the transfer function of the amplifier, where \(G_{w}=1\) A/V; \(K_{xx}\) and \(K_{yy}\) are the gain of sensors. \(I\)(s) denotes the current of each coil in magnetic bearing. From Figure 5, although the currents in the coils determine the displacement of the rotor in both \(x\)- and \(y\)-directions, in view of the logic control current, the motion of the rotor in the \(x\)- and \(y\)-directions can be considered decoupled.