# Suppression of Harmonic Current in Magnetic Bearing-Rotor System with Redundant Structure
## 3 Characteristic and Control Strategy of Magnetic Bearings with Redundant Structures
### Stiffness of Magnetic Bearing with Redundant Structures  
It is considered that the air gap between the poles and rotor is 0.4 mm while the rotor is at the equilibrium position, and the displacement of the rotor is tiny when the rotor is suspended by the magnetic bearing; thus, the electromagnetic force forms a linear relationship with the displacement and the logical control currents \(i_{x}\) and \(i_{y}\). By using Taylor series expansion, the resultant forces in the \(x\)- and \(y\)-directions can be linearized as \[F_{x}(x,y,i_{x},i_{y})\approx\frac{\partial F_{x}}{\partial x}\left|\begin{array}{c} x=0\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\right.\begin{array}{c}x+\frac{\partial F_{x}}{\partial y} \\ i_{x}=0\\ i_{y}=0\end{array}\right.\begin{array}{c}y+\frac{\partial F_{x}}{\partial i_{x} }\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}x=0\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{x}+\frac{\partial F_{x}}{\partial i_{y} }\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\] (18) \[F_{y}(x,y,i_{x},i_{y})\approx\frac{\partial F_{y}}{\partial x} \left|\begin{array}{c}x=0\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}x+\frac{\partial F_{y}}{\partial y}\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}y+\frac{\partial F_{y}}{\partial i_{x} }\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{x}+\frac{\partial F_{y}}{\partial i_{y} }\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{y}.\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}x=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{y}.\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}y=0\\ i_{y}=0\end{array}\begin{array}{c}y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{y}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{y}.\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}x=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{y}=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{y}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{y}.\\ y=0\end{array}\begin{array}{c}\\ x=0\\ i_{x}=0\\ i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{x}=0\\ i_{y}=0\end{array}\begin{array}{c}i_{y}.\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}\] (19) \[K_{xx},K_{xy},K_{yx},K_{yx},\text{and}K_{yy}\] (20) \[K_{xx},K_{xy},K_{yx},\text{and}K_{yy}\] (21) \[K_{xx},K_{xy},K_{yx},\text{and}K_{yy}\] (22) \[K_{xx},K_{xy},K_{yx},K_{yx},K_{yy},K_{yy},K_{yy},K_{yy},K_{yy},K_{yy},K_{yy},K _{yy},K_{yy}  
The model of current stiffness can be calculated as  
\[K_{ixx}=\frac{\partial F_{x}}{\partial I_{c}}\frac{\partial I_{c}}{\partial i_{x}} \begin{array}{c}\\ x=0\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}=\mathbf{w}_{x}^{\mathrm{T}}M_{x}\mathbf{w}_{b}C_{0}\\ +\mathbf{w}_{b}^{\mathrm{T}}M_{x}\mathbf{w}_{x}C_{0}\\ x=0\\ y=0\end{array}, \tag{24}\]  
\[K_{iy}=\frac{\partial F_{x}}{\partial I_{c}}\frac{\partial I_{c}}{\partial i_{y}} \begin{array}{c}\\ x=0\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}=\mathbf{w}_{y}^{\mathrm{T}}M_{x}\mathbf{w}_{b}C_{0}\\ +\mathbf{w}_{b}^{\mathrm{T}}M_{x}\mathbf{w}_{y}C_{0}\\ x=0\\ y=0\end{array}, \tag{25}\]  
\[K_{iyx}=\frac{\partial F_{x}}{\partial I_{c}}\frac{\partial I_{c}}{\partial i _{y}}\begin{array}{c}\\ x=0\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}=\mathbf{w}_{x}^{\mathrm{T}}M_{y}\mathbf{w}_{b}C_{0}\\ x=0\\ y=0\end{array}+\mathbf{w}_{b}^{\mathrm{T}}M_{y}\mathbf{w}_{x}C_{0}\\ +\mathbf{w}_{b}^{\mathrm{T}}M_{y}\mathbf{w}_{x}C_{0}\\ x=0\\ y=0\end{array}, \tag{26}\]  
\[K_{iyy}=\frac{\partial F_{x}}{\partial I_{c}}\frac{\partial I_{c}}{\partial i _{y}}\begin{array}{c}\\ x=0\\ y=0\\ i_{x}=0\\ i_{y}=0\end{array}=\mathbf{w}_{y}^{\mathrm{T}}M_{y}\mathbf{w}_{b}C_{0}\\ x=0\\ y=0\end{array}+\mathbf{w}_{b}^{\mathrm{T}}M_{y}\mathbf{w}_{y}C_{0}\\ x=0\\ y=0\end{array}. \tag{27}\]  
According to the above analysis, the decoupled linearized magnetic forces can be expressed with position stiffness and current stiffness, and they can be used to design a control law to ensure the stability of the system.