# Effects of Imperfect Assembly and Magnetic Properties on the Three-Pole AMB System
## 2 System Description and Dynamic Modeling  
The system under study is the 3-pole AMB system shown in Figure 1. Figure 2 shows the relative position of the centers of the back-up bearing and stator. The dotted line indicates the allowable domain of the rotor motion. The AMB can provide magnetic forces in X and Y directions, supporting a 2-DOF disk-like rotor. It is assumed that the axial motion is constrained. The upper two magnetic poles share the same coil with opposite winding directions. The magnetic reluctances of the 3 air gaps between the rotor and the 3 magnetic poles are:  
\[ \begin{array}{c}R_{1}=\frac{1}{\mu A}(l_{0}+y_{r}+\Delta y);R_{2}=\frac{1}{ \mu A}(l_{0}+\frac{\sqrt{3}}{2}(x_{r}+\Delta x)-\frac{1}{2}(y_{r}+\Delta y))\\ R_{3}=\frac{1}{\mu A}(l_{0}-\frac{\sqrt{3}}{2}(x_{r}+\Delta x)-\frac{1}{2}(y_{ r}+\Delta y))\end{array}  \tag{1}\]  
where \(\mu\) is the magnetic permeability of the air, \(A\) is the pole face area, \(l_{0}\) is the nominal air gap, \((x_{r},y_{r})\) is the position of the rotor center, and \((\Delta x,\Delta y)\) is the deviation of the back-up bearing center from the stator center. The origin of the inertial coordinate system is set to be the center of the back-up bearing since it can be easily calibrated. Next, we assume that \(\phi_{i}=k_{i}\phi_{i},\ i=1\sim 3\), where \(\phi_{i}\) is the actual magnetic flux at each pole, \(\phi_{i}\) is the theoretical magnetic flux based on the assumption of uniform magnetic properties, and \(k_{i}\) represents t the non-uniformity of the magnetic fluxes. The theoretical magnetic fluxes can be obtained from the magnetic circuit analysis as:  
\[\hat{\phi}_{1}=N\frac{(R_{2}+R_{3})i_{1}+(R_{2}-R_{3})i_{2}}{R_{1}R_{2}+R_{2}R_{3 }+R_{3}R_{1}} \tag{2}\]  
\[\hat{\phi}_{2}=N\frac{-R_{3}i_{1}+(2R_{1}+R_{3})i_{2}}{R_{1}R_{2}+R_{2}R_{3}+R_{ 3}R_{1}} \tag{3}\]  
\[\hat{\phi}_{3}=N\frac{-R_{2}i_{1}-(2R_{1}+R_{2})i_{2}}{R_{1}R_{2}+R_{2}R_{3}+R_{ 3}R_{1}} \tag{4}\]  
where \(N\) is the number of coil turns on each magnetic pole and \(i_{1},~{}~{}i_{2}\) are coil currents. Then, the dynamic model of the system is given by  
\[\ddot{x}_{r}=c_{0}\[PERSON]_{1}\[PERSON]_{2}=c_{0}\hat{\[PERSON]}_{1}\hat{\[PERSON]}_{2}+\Delta f_ {x}\ddot{y}_{r}=\frac{c_{0}}{2}(\[PERSON]_{2}^{2}-\[PERSON]_{1}^{2})-g=\frac{c_{0}}{2}( \hat{\[PERSON]}_{2}^{2}-\hat{\[PERSON]}_{1}^{2})-g+\Delta f_{y} \tag{5}\]  
where \(c_{0}=\frac{4\mu AN^{2}}{3\mu}\),\(m\) is the rotor mass, and \(\[PERSON]_{1}\), \(\Phi_{2}\) are given by  
\[\Phi_{1}=\frac{3}{4\mu AN}(\phi_{3}+\phi_{2});\Phi_{2}=\frac{\sqrt{3}}{4\mu AN }(\phi_{3}-\phi_{2})\]  
From (1) to (4), \(\Phi_{1}\) and \(\Phi_{2}\) can be expressed as  
\[ \begin{array}{c}\left[\begin{array}{c}\Phi_{1}\\ \Phi_{2}\end{array}\right]=\frac{-1}{L}K\!\!\left[\begin{array}{cc}2l_{0}-(y _{r}+\Delta y)&\sqrt{3}(x_{r}+\Delta x)\\ (x_{r}+\Delta x)&\sqrt{3}(2l_{0}+(y_{r}+\Delta y))\end{array}\right]\left[ \begin{array}{c}i_{1}\\ i_{2}\end{array}\right];\\ K=\left[\begin{array}{cc}\frac{1}{2}(k_{2}+k_{3})&-\frac{\sqrt{2}}{2}(k_{2 }-k_{3})\\ -\frac{\sqrt{3}}{6}(k_{2}-k_{3})&\frac{1}{2}(k_{2}+k_{3})\end{array}\right] \end{array}  \tag{6}\]  
where \(L=4l_{0}^{2}-\left[(x_{r}+\Delta x)^{2}+(y_{r}+\Delta y)^{2}\right]\). Let \(\hat{L}=4l_{0}^{2}-(x_{r}^{2}+y_{r}^{2})\). Since, in general, \(|\Delta x|\), \(|\Delta y|\leq 0.1l_{0}\), it is assumed that \(\hat{L}/L\approx 1\). Then, in Equation (5), the nominal part of \(\Phi_{1}\), \(\Phi_{2}\)is given by  
\[ \begin{bmatrix}\hat{\Phi}_{1}\\ \hat{\Phi}_{2}\end{bmatrix} =\frac{-1}{\hat{L}}K\!\!\left[ \begin{array}{cc}2l _{0}-y_{r}&\sqrt{3}x_{r}\\ x_{r}&\sqrt{3}(2l_{0}+y_{r})\end{array} \right]\!\! \begin{bmatrix}i_{1}\\ i_{2}\end{bmatrix}