## 2. Commutation and Power Loss Equalizing Degree of Coil Array
### Power Loss Equalizing Degree and the Approach to Raise It  
For the sake of derivation convenience, the commutated coil currents and the equivalent resistances of each coil are respectively denoted by \(i_{\mathrm{Q}}\) (j = 1, 2,..., \(n\)) and \(R_{\mathrm{j}}\) (j = 1, 2,..., \(n\)). Then the power loss summation of all commutated coils, which is named as the global power loss, can be calculated and denoted by \(P_{\mathrm{g}}\):  
\[\mathbf{P}_{\mathrm{g}}=\sum_{\mathrm{j=1}}^{n}i_{\mathrm{c}}^{2}R_{\mathrm{j}} \tag{12}\]  
and the power loss of every coil, which is named as the local power loss, is denoted by \(P_{\mathrm{j}}\):  
\[\mathbf{P}_{\mathrm{j}}=i_{\mathrm{Q}}^{2}R_{\mathrm{j}}\left(\mathrm{j=1,\ 2,\ \cdots,\ n}\right) \tag{13}\]Then, defining an index, namely power loss equalizing degree (PLED) and denoted by \(E_{\text{P}}\):  
\[E_{\text{P}}=\frac{\mathbf{P}_{\text{g}}/n}{\max(\mathbf{P}_{\text{j}})}\ =\ \frac{\sum\limits_{\text{j}=1}^{n}i_{cj}^{2}R_{\text{j}}}{n\cdot\max(i_{\text{ cj}}^{2}R_{\text{j}})} \tag{14}\]  
This indicates the ratio of the average to the maximum of all the local power losses.  
Furthermore, letting the maximal square of commutated coil currents be denoted by \(S_{\text{jm}}\)  
\[S_{\text{jm}}=\max(i_{\text{q}}^{2}),\ (\text{j}=1,\ 2,\ \cdots,\ n\ ) \tag{15}\]  
and the square sum of commutated coil currents be denoted by \(S_{\text{g}}\)  
\[S_{\text{g}}=\sum\limits_{\text{j}=1}^{n}i_{cj}^{2} \tag{16}\]  
and then supposing the equivalent resistances of each coil are equal, PLED can be derived as:  
\[E_{\text{P}}=\frac{\sum\limits_{\text{j}=1}^{n}i_{cj}^{2}}{n\cdot\max(i_{\text {cj}}^{2})}=\frac{S_{\text{g}}}{n\cdot S_{\text{jm}}} \tag{17}\]  
It can be deduced from Equation (14) and Equation (17) that PLED varies between 0 to 1 and it is a dimensionless quantity; therefore, it can be identified as a normalization index to evaluate the power loss distribution of commutated coil array. The higher this normalization index is, the nearer the average and the maximum of all the local power losses are, and the more well-distributed the local power losses are.  
It can also be observed from Equation (17) that there are three possible selections to raise PLED. The first one is to increase the square sum of the commutated coil currents, \(S_{\text{g}}\). This is obviously inadvisable, because doing so will increase global power loss, \(P_{\text{g}}\). The second selection is to decrease the coil number \(n\). Confined to the high energy requirement and the precision motion requirement of MLPM, this selection is also inadvisable. Consequently, the only feasible selection is the third one, i.e., raising PLED by decreasing the maximal square of commutated coil currents, \(S_{\text{jm}}\). In order to decrease \(S_{\text{jm}}\), the maximal amplitude of commutated coil currents, \(\max|i_{\text{q}}|\) (j = 1, 2, \(\ldots\), \(n\)), should be reduced. Actually, this maximal amplitude is the infinite norm of commutated coil current vector:  
\[\left|\left|i_{\text{c}}\right|\right|_{\infty}=\max\left(\left|i_{\text{c1}} \right|,\ \left|i_{\text{c2}}\right|,\ \cdots,\ \left|i_{\text{cn}}\right|\right)=\max|i_{\text{q}}|,\ (\text{j}=1,\ 2,\ \ldots,n) \tag{18}\]  
Thus, the PLED can be raised by decreasing the infinite norm of commutated coil current vector \(\left|\left|i_{\text{c}}\right|\right|_{\infty}\).