## 3 Computation of MEC of Proposed Motor
### Torque and Performance Evaluation Index  
The electromagnetic torque \(T\) can be calculated by the total magnetic energy stored in the motor without considering the magnetic saturation, and the calculation equation is,  
\[T(t,\ \theta_{r})=-\frac{\partial W_{m}(t,\ \theta_{r})}{\partial\theta_{r}}, \tag{35}\]  
where \(W_{m}\) is the total magnetic energy stored by the proposed rotor with cross-shaped flux-barriers structure of SynRM. The assumption is made that the magnetic permeability of the core is much greater than that of the air, that is, the core does not store magnetic energy, the total magnetic energy stored in the motor can be divided into two parts, one is the magnetic energy stored in the magnetic barrier \(W_{mb}\), the other is the magnetic energy stored in the air-gap \(W_{mg}\). Then the total magnetic energy \(W_{m}\) can be expressed by the following equation:  
\[W_{m}(t,\ \theta_{r})=W_{mb}(t,\ \theta_{r})+W_{mg}(t,\ \theta_{r}). \tag{36}\]  
The magnetic energy stored in the magnetic barrier \(W_{mb}\) is expressed as,  
\[W_{mb}(t,\ \theta_{r})=p\Big{[}R_{b51}\Phi_{b51}(t,\ \theta_{r})^{2}+R_{b41} \Phi_{b41}(t,\ \theta_{r})^{2}+R_{eqb3}\Phi_{eqb3}(t,\ \theta_{r})^{2}+R_{eqb2}\Phi_{eqb2}(t,\ \theta_{r})^{2}+R_{eqb1}\Phi_{2}(t,\ \theta_{r})^{2}\Big{]}. \tag{37}\]  
The magnetic energy stored in the air-gap \(W_{mg}\) is expressed as,  
\[W_{mg}(t,\ \theta_{r})=\frac{L(D_{r}-g)g}{4\mu_{0}}\int_{0}^{2\pi}B_{g}(t, \theta,\theta_{r})^{2}d\theta. \tag{38}\]Through the above steps, the torque of the proposed SynRM with cross-shaped fluid-barriers can be predicted. Next the torque ripple \(\Delta T\) is defined as the ratio of the difference between the maximum value of torque \(T_{\text{max}}\) and the minimum value of torque \(T_{\text{min}}\) to the average value of torque \(T_{avg}\):  
\[\Delta T=\frac{T_{\text{max}}-T_{\text{min}}}{T_{avg}}. \tag{39}\]  
The sensitivity [14] is also used in this paper as one of the indicators to evaluate the performance of the motor. The method can be exploited when a discrete representation of the research domain is available [29]. Firstly, a perturbation space of \(n_{v}\)-dimensional hypercube with design vector in the center is established to estimate the maximum rate of change of torque pulsation, and the dimension of the parameter space is equal to the number of parameters, which is two in this paper. The parameters of the geometry in a generic solution are defined by the vector \(\overrightarrow{f}=\left(f_{1},f_{2},\ldots,f_{n}\right)\), where \(f_{1}=k_{arc}\) and \(f_{2}=k_{seg}\), and both values range from 0 to 0.8 in this study. The distance between the vector \(\overrightarrow{f}^{\rightarrow}\) and the vector \(\overrightarrow{f}^{\rightarrow\prime}\) is calculated by the following equation:  
\[d\left(\overrightarrow{f}^{\rightarrow},\overrightarrow{f}^{\rightarrow \prime}\right)=\left[\sum_{i=1}^{n_{v}}\left(f_{i}^{\prime}-f_{i}^{\prime \prime}\right)^{2}\right]^{\frac{1}{2}}. \tag{40}\]  
Sensitivity analysis of solutions in space requires the determination of a radius of an hypersphere \(R\), which must be adequately small to properly assess the sensitivity of each solution, but also adequately large to find other solutions around each one [30]. The number of solutions in this study is 65, and \(R\) is 0.2 after comprehensive consideration. For the solution with the minimum torque ripple, the parameters of the geometry is defined by the vector \(\overrightarrow{\tilde{f}}=\left(\widetilde{f}_{1},\widetilde{f}_{2},\ldots, \widetilde{f}_{n}\right)\), and then the fluctuation of the torque ripple is computed according to the variation of the geometry with respect to the optimal solution \(\overrightarrow{\tilde{f}}\), so as to evaluate the rate of change of the ripple with the deviation from the optimal solution \(\overrightarrow{\tilde{f}}\).