## 2. Commutation and Power Loss Equalizing Degree of Coil Array
### Raising Power Loss Equalizing Degree by Convex Quadratic Optimisation Commutation  
Considering the commutated coil current vector \(\mathbf{i}_{\text{c}}\)  
\[\mathbf{i}_{\text{c}}=(i_{\text{c1}},\ i_{\text{c2}},\ \cdots,\ i_{\text{cn}})^{ \text{T}} \tag{19}\]  
its each element, \(i_{\text{cj}}\) (j = 1, 2, \(\ldots\), \(n\)), is a scalar and denotes the commutated coil current value, i.e., DC or AC RMS (alternative current root mean square). This means that the set of \(i_{\text{c}}\) is in the real domain: \(i_{\text{c}}\in\mathbb{R}^{n}\). Since \(\mathbb{R}^{n}\) is a convex set, \(i_{\text{c}}\) is a convex set, too [21].  
Further condidering the quadratic function \(f\) (\(i_{\rm c}\)):  
\[f(i_{\rm c})=\sum_{\rm j=1}^{n}i_{cj}^{2}=i_{c1}^{2}+i_{c2}^{2}+\cdots+i_{cn}^{2 }=i_{\rm c}^{\rm T}i_{\rm c} \tag{20}\]  
its defining domain, i.e., the value domain of \(i_{\rm c}\), is a convex set and it is twice differentiable on its defining domain. The Hessian matrix of \(f\) (\(i_{\rm c}\)), \(\
abla_{i_{\rm c}}^{2}f(i_{\rm c})\), can be twice differentiated as  
\[\
abla_{i_{\rm c}}^{2}f(i_{\rm c})=2\mathbf{I}_{n} \tag{21}\]  
where \(\mathbf{I}_{n}\) is a n-dimensional unit matrix. As a result of \(\mathbf{I}_{n}\) being a positive definite matrix, the Hessian matrix of \(f\) (\(i_{\rm c}\)), \(\
abla_{i_{\rm c}}^{2}f(i_{\rm c})\), is also a positive definite matrix. In light of the second order condition for function convexity [21],\(f\) (\(i_{\rm c}\)) is a strictly convex quadratic function.  
In order to generate the desired electromagnetic forces and torques, the commutated coil currents \(i_{\rm c}\) must first satisfy:  
\[K(\mathbf{q})i_{\rm c}=W_{\rm d} \tag{22}\]  
In addition, for the purpose of reducing the infinite norm of commutated coil current vector, \(1\,1\,i_{\rm c}\,1\,i_{\rm c}\,\), the absolute values of each element of \(i_{\rm c}\) an upper bound \(\rm I_{m}\) can be given:  
\[|i_{\rm c1}|\leq\rm I_{m},\ |\ i_{c2}|\leq\rm I_{m},\ \cdots,\ |i_{cn}|\leq\rm I _{m} \tag{23}\]  
where \(\rm I_{m}\) is a positive real number. When this upper bound \(\rm I_{m}\) is reduced, the infinite norm \(1\,1\,i_{\rm c}\,1\,i_{\rm c}\,\) is to be decreased and PLED can be raised according to (18).  
As a result of the discussions above, the problem of raising PLED by commutation can be summerized and abstracted as a convex quadratic optimisation problem:  
\[\underset{i_{\rm c}}{\text{minimize}}\ f(i_{\rm c})=i_{\rm c}^{\rm T}i_{\rm c },\ \text{subject to}\ \left\{\begin{array}{c}K(\mathbf{q})i_{\rm c}=W_{\rm d}\\ -\rm I_{m}\leq i_{c1}\leq\rm I_{m}\\ -\rm I_{m}\leq i_{c2}\leq\rm I_{m}\\ \cdots\cdots\\ -\rm I_{m}\leq i_{cn}\leq\rm I_{m}\end{array}\right. \tag{24}\]  
in which the optimisation variables are the commutated coil currents, \(i_{\rm c}\)=(\(i_{\rm c1}\), \(i_{\rm c2}\),..., \(i_{\rm cn}\))T; the optimisation objective is to minimize the quadratic function \(f(i_{\rm c})=i_{\rm c}^{\rm T}i_{\rm c}\); and the optimisation is subject to two constraint conditions: one is the linear equilty constraint as shown in Equation (22), and another is the boundary constraint, shown in Equation (23).