# Shape Optimization of Discontinuous Armature Arrangement PMLSM for Reduction of Thrust Ripple
## 3 Multipurpose Optimal Design of PMLSM Using the Design of Experiments  
### Screening  
The primary parameters that affected each objective function were selected to minimize the detent force and the THD of the back EMF due to the slot effect and to prevent the saturation of the circuit. Through screening, the parameters with insignificant main effects were pooled. The fractional factorial design method was adopted to identify the main effect, and a small number of interaction effects with a small number of simulations and 7 shape parameters were screened to identify 7 main effects and 14 two-factor interaction  
Figure 4: No-load voltage back EMF: (**a**) Back EMF of initial model; (**b**) Harmonic components of back EMF. Figure 5: Field distribution and flux density contours of the circuit. Figure 3: FEM result of detent force at no load: (**a**) Detent force caused by end effect; (**b**) Harmonic components of detent force. effects. Figure 6 presents an overview of the optimization process. The constraints of the shape parameters are presented in Table 2.  
Constraints on the shape of permanent magnets and armature were set in consideration of the distance between the mover and the stator, the distance between the stators, the ratio between the slot and the pole, and the winding area. In this study, 33 simulation runs were performed using fractional factorial design. The simulation conditions and results for each shape parameter are presented in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Lower Bound** & **Reference** & **Upper Bound** \\ \hline Height of PM, \(P_{h}\) (mm) & 2 & 10 & 15 \\ Width of PM, \(P_{w}\) (mm) & 28 & 35 & 45 \\ Slot Opening, \(S_{o}\) (mm) & 1 & 2 & 5 \\ Height of tooth, \(T_{h}\) (mm) & 10 & 20 & 25 \\ Tooth tip angle, \(T_{a}\) (\({}^{\circ}\)) & 30 & 90 & 70 \\ Tooth tip height, \(T_{th}\) (mm) & 1 & 3 & 5 \\ Tooth width, \(T_{w}\) (mm) & 1 & 7 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design variables and boundary of the stator and the mover. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Run** & \multirow{2}{*}{\(P_{h}\)} & \multirow{2}{*}{\(P_{h}\)} & \multirow{2}{*}{\(T_{h}\)} & \multirow{2}{*}{\(S_{o}\)} & \multirow{2}{*}{\(T_{a}\)} & \multirow{2}{*}{\(T_{th}\)} & \multirow{2}{*}{\(T_{w}\)} & **Detent** & **Back EMF** & **Magnetic Flux** \\
**Order** & & & & & & & & **Force (N)** & **THD (\%)** & **Density (T)** \\ \hline
1 & 28 & 15 & 25 & 5 & 30 & 1 & 8 & 1.03 & 0.50 & 2.31 \\
2 & 28 & 15 & 10 & 5 & 70 & 5 & 1 & 0.43 & 28.78 & 2.44 \\
3 & 28 & 2 & 25 & 1 & 30 & 1 & 8 & 0.69 & 1.63 & 0.83 \\
4 & 28 & 2 & 10 & 5 & 30 & 1 & 1 & 0.20 & 23.88 & 2.04 \\
5 & 45 & 2 & 25 & 1 & 30 & 5 & 1 & 0.82 & 63.65 & 1.87 \\  & & & & & & & & & \\
29 & 28 & 15 & 25 & 1 & 70 & 5 & 8 & 0.51 & 0.54 & 1.27 \\
30 & 45 & 15 & 10 & 1 & 30 & 5 & 8 & 0.31 & 1.83 & 1.75 \\
31 & 45 & 15 & 25 & 5 & 30 & 5 & 1 & 16.02 & 50.31 & 2.41 \\
32 & 28 & 15 & 10 & 1 & 70 & 1 & 8 & 0.51 & 0.65 & 1.26 \\
33 & 45 & 15 & 25 & 1 & 30 & 1 & 8 & 0.12 & 4.98 & 1.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fractional factorial design and results. Figure 6: Optimization process flow chart. The parameters not significant to the detent force or the THD of the back EMF due to the slot effect, as well as the magnetic flux density of the tooth, i.e., the objective functions, to the error terms were pooled. Subsequently, a Pareto chart was plotted, as shown in Figure 7. The main effect based on an analysis of variance (ANOVA) is illustrated in Figure 8.  
The tooth width, tooth height, tooth tip height, and tooth tip angle exhibited significance with respect to the detent force, and interactions existed between the permanent magnet height, and the tooth width, tooth tip height, and tooth tip angle. For the THD of the back EMF, the tooth width, tooth height, and tooth tip angle exhibited significance, and an interaction existed between the tooth height and tooth width.