# Suppression of Harmonic Current in Magnetic Bearing-Rotor System with Redundant Structure
## 3 Characteristic and Control Strategy of Magnetic Bearings with Redundant Structures
### Force Mode of Eight-Pole Magnetic Bearings with Redundant Structures  
A radial magnetic bearing with eight-pole heteropolar redundant structures is depicted in Figure 3a, and the equivalent magnetic circuit model is illustrated in Figure 3b. The only sources of magnetic excitation in the bearing are the coils that are wound on each pole, and almost all circuit reluctance is due to the air gap associated with each pole.  
The relationship between the coil current and the magnetic flux can be written as  
\[R_{j}\phi_{j}-R_{j+1}\phi_{j+1}=\overline{N}_{j}I_{j}-\overline{N}_{j+1}I_{j+ 1},j=1,2,\cdots,8, \tag{4}\]  
\[R_{j}=\frac{g_{j}}{\mu_{0}A_{j}}, \tag{5}\]  
where \(R_{j}\) is the reluctance of the \(j\)-th pole gap, \(g_{j}\) is the length of the \(j\)-th air gap, \(\mu_{0}\) is the permeability of vacuum, \(j\) is the number of poles, and \(A_{j}\), \(\phi_{j}\), \(\overline{N}_{j+1}\), and \(I_{j}\) represent the area, magnetic flux, turns of the coil, and current in the \(j\)-th coil, respectively.  
The air gap \(g_{j}\) is variable while the rotor deviates from its equilibrium position, and it can be described as follows [6]:  
\[g_{j}=g_{0}-x\cos\theta_{j}-y\sin\theta_{j}. \tag{6}\]  
Meanwhile, the flux conservation is described as  
\[\sum_{j=1}^{n}\phi_{j}=0. \tag{7}\]  
Therefore, Equation (4) can be rearranged as  
\[R\Phi=\overline{N}I. \tag{8}\]  
Figure 3: Eight-pole bearing arrangement (**a**) and equivalent magnetic circuit (**b**).  
Assuming that the magnetic flux density is uniform in the gap, the flux conservation can be described by Equation (9).  
\[\mathbf{\Phi}=\mathbf{A}\mathbf{B}, \tag{9}\]  
where \(\mathbf{A}\) is the diagonal matrix of the pole area, and \(\mathbf{B}\) is the magnetic flux density matrix of the air gap, expressed as  
\[\mathbf{B}=\mathbf{A}^{-1}\mathbf{R}^{-1}\overline{\mathbf{N}}\mathbf{I}=\mathbf{V}\mathbf{I}. \tag{10}\]  
The current distribution matrix \(\mathbf{W}\) can be defined to describe the relationship between the current in each coil of the magnetic bearing and the logical control currents as shown in Equation (11).  
\[\mathbf{W}=\begin{bmatrix}\mathbf{w}_{b},\mathbf{w}_{x},\mathbf{w}_{y}\end{bmatrix},\]  
\[\mathbf{I}=\mathbf{W}\begin{bmatrix}C_{0}\\ i_{x}\\ i_{y}\end{bmatrix}=\mathbf{W}\mathbf{I}_{c}, \tag{11}\]  
where \(C_{0}\) is the bias coefficient, and \(i_{x}\) and \(i_{y}\) are the logical control currents in the \(x\)- and \(y\)-directions, respectively. The resultant forces in the \(x\)- and \(y\)-directions can be described as  
\[F_{x}(x,y)=\mathbf{I}_{c}{}^{T}\mathbf{W}^{T}M_{x}\mathbf{W}\mathbf{I}_{c}, \tag{12}\]  
\[F_{y}(x,y)=\mathbf{I}_{c}{}^{T}\mathbf{W}^{T}M_{y}\mathbf{W}\mathbf{I}_{c}, \tag{13}\]  
where  
\[\left\{\begin{array}{l}M_{x}(x,y)=-\mathbf{V}^{T}D_{x}\mathbf{V}\\ M_{y}(x,y)=-\mathbf{V}^{T}D_{y}\mathbf{V}\\ D_{x}=\frac{A_{1}}{2\mu_{0}}diag\left[\cos\theta_{j}\right]\\ D_{y}=\frac{A_{j}}{2\mu_{0}}diag\left[\sin\theta_{j}\right]\end{array}\right.. \tag{14}\]  
Furthermore, in order to ensure the linearization of the electromagnetic force, the current distribution matrix \(\mathbf{W}(x,y)\) should be obtained to satisfy Equation (15).  
\[\left\{\begin{array}{l}\mathbf{W}^{T}(x,y)M_{x}\mathbf{W}(x,y)-Q_{x}=0\\ \mathbf{W}^{T}(x,y)M_{y}\mathbf{W}(x,y)-Q_{y}=0\end{array}\right., \tag{15}\]  
where the matrices (\(Q_{x}\) and \(Q_{y}\)) are defined as  
\[Q_{x}=\left[\begin{array}{ccc}0&0.5&0\\ 0.5&0&0\\ 0&0&0\end{array}\right],Q_{y}=\left[\begin{array}{ccc}0&0&0.5\\ 0&0&0\\ 0.5&0&0\end{array}\right]. \tag{16}\]  
Through the calculation of the above theory, the electromagnetic forces can be linearized as  
\[\left\{\begin{array}{l}F_{x}(x,y)=C_{0}i_{x}\\ F_{y}(x,y)=C_{0}i_{y}\end{array}\right.. \tag{17}\]