## 2. Commutation and Power Loss Equalizing Degree of Coil Array
### Commutation of Coil Array  
The commutation of MLPM starts with the modeling of the electromagnetic forces and torques. As Figure 3 shows, there are two coordinate systems. One is the fixed coordinate system \(O\)-\(XYZ\) and is defined on the surface of the permanent magnet array. In this coordinate system, the three-dimension magnetic flux density distribution of the permanent magnet field can be represented analytically. The other one is the moving coordinate system \(o\)-\(xyz\) and is set up at the bottom plane of the coil array. In this coordinate system, each coil is defined. Two coordinate systems are parallel. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parts** & **Parameters** & **Symbol** & **Unit** & **Value** \\ \hline \multirow{8}{*}{Permanent magnet surface residual magnetism} & \(M_{0}\) & Tesla & 1.23 \\  & Permanent magnet array manufacture pole pitch & \(\tau\) & mm & 15.0 \\  & Permanent magnet array motion pole pitch & \(\tau_{n}\) & mm & 10.6 \\  & Permanent magnet array length & \(L_{m}\) & mm & 200.0 \\  & Permanent magnet length & \(b_{m1}\) & mm & 10.0 \\  & Permanent magnet width & \(b_{m2}\) & mm & 5.0 \\  & Permanent magnet thickness & \(h_{m}\) & mm & 10.0 \\ \hline \multirow{8}{*}{Coil array stage mass} & \(M\) & kg & 1.02 \\  & Number of coils & \(N\) & \(4\times 5\) \\  & Number of coil turns & \(N\) & \(52\pm 1\) \\  & Coil array length & \(L_{c}\) & mm & \(144.0\pm 2.0\) \\  & Coil length & \(l_{c}\) & mm & \(63.6\pm 1.0\) \\  & Coil width & \(w_{c}\) & mm & \(8.4\pm 0.2\) \\  & Coil turn width & \(b_{c}\) & mm & \(6.0\pm 0.2\) \\  & Coil thickness & \(l_{c}\) & mm & \(5.3\pm 0.2\) \\  & Coil resistance (Room temperature) & \(R\) & \(\Omega\) & \(0.6\pm 0.05\) \\  & Coil inductance (AC 1 kHz) & \(L\) & mH & \(0.1\pm 0.01\) \\  & Coil specified current (an hour, \(<\)90 \({}^{\circ}\)C) & \(l_{s}\) & A & 3.15 \\  & Coil specified voltage (DC) & \(l_{s}\) & V & \(\pm 24\) \\  & Coil specified power & \(P_{s}\) & Watt & 75.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Related parameters and dimensions of MLPM (magnetically levitated planar motor) under investigation. Figure 2: Configurations and parameters of MLPM (magnetically levitated planar motor) under investigation: (**a**) Halbach permanent magnet array; (**b**) coil array. Thus, the position and attitude that the mover locates in the stator can be denoted by \(q\):  
\[q=[x,\ y,\ z,\ 0,\ 0,\ 0]^{\rm T} \tag{1}\]  
and the position and attitude that coil C\({}_{\rm j}\) locates in the moving coordinate system _o-xyz_ can be denoted by \(q_{\rm cj}\):  
\[q_{\rm cj}=[x_{\rm cj},\ y_{\rm cj},\ z_{\rm cj}]^{\rm T} \tag{2}\]  
Given that the coil C\({}_{\rm j}\) is commutated by unit current, 1A, the _x-_, _y-_direction propulsion force \(F_{\rm xj}\) (_q_), \(F_{\rm yj}\) (_q_), _z-_direction levitation force \(F_{\rm zj}\) (_q_), and _x-_, _y-_, _z-_axis torques \(T_{\rm xj}\) (_q_), \(T_{\rm yj}\) (_q_), \(T_{\rm zj}\) (_q_), can be integrated by using the Lorenz force volume integral method [7]. These forces and torques are denoted by:  
\[W_{\rm j}(q)=[F_{\rm xj}(q),\ F_{\rm yj}(q),\ F_{\rm zj}(q),\ T_{\rm xj}(q),\ T_{\rm yj}(q),\ T_{\rm zj}(q)]^{\rm T} \tag{3}\]  
then the electromagnetic forces and torques generated from all coils of \(n\) number, which is commutated by unit current, respectively, can be listed as a matrix of 6 \(\times\)\(n\) dimensions. This matrix is denoted by \(K\) (_q_):  
\[K(q)=\left( \begin{array}{cccc}F_{x1}(q)&F_{x2}(q)&\ldots&F_{xn}(q)\\ F_{y1}(q)&F_{y2}((q)&\ldots&F_{yn}(q)\\ F_{z1}(q)&F_{z2}(q)&\ldots&F_{zn}(q)\\ T_{x1}(q)&T_{x2}(q)&\ldots&T_{xn}(q)\\ T_{y1}(q)&T_{y2}(q)&\ldots&T_{yn}(q)\\ T_{z1}(q)&T_{z2}(q)&\ldots&T_{xn}(q)\end{array} \right) \tag{4}\]  
It can be seen from (4) that \(K(q)\) is a matrix of 6\(\times\)\(n\) dimension and its six elements of each column represent three forces and three torques, respectively, when the coil C\({}_{\rm j}\) is commutated by unit current  
Figure 3: Coordinate systems for modeling of the electromagnetic forces and torques. (1A).