# Shape Optimization of Discontinuous Armature Arrangement PMLSM for Reduction of Thrust Ripple
## 3 Multipurpose Optimal Design of PMLSM Using the Design of Experiments
### Multi Objective Optimization Based RSM
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & \(p\)**-Value** \\ \hline Model & 6 & 4.53492 & 0.75582 & 216.38 & 0 \\ Linear & 3 & 3.25095 & 1.08365 & 310.23 & 0 \\ \(P_{h}\) & 1 & 1.9321 & 1.9321 & 553.12 & 0 \\ \(T_{h}\) & 1 & 0.03062 & 0.03062 & 8.77 & 0.005 \\ \(T_{w}\) & 1 & 1.28822 & 1.28822 & 368.79 & 0 \\ Square & 2 & 1.25157 & 0.62578 & 179.15 & 0 \\ \(P_{h}\ast P_{h}\) & 1 & 0.51278 & 0.51278 & 146.8 & 0 \\ \(T_{w}\ast T_{w}\) & 1 & 0.55114 & 0.55114 & 157.78 & 0 \\
2-Way & 1 & 0.0324 & 0.0324 & 9.28 & 0.004 \\ Interaction & 1 & 0.0324 & 0.0324 & 9.28 & 0.004 \\ \(P_{h}\ast T_{w}\) & 1 & 0.0324 & 0.0324 & 9.28 & 0.004 \\ Error & 39 & 0.13623 & 0.00349 & & \\ Lack-of-Fit & 34 & 0.13623 & 0.00401 & & \\ Pure Error & 5 & 0 & 0 & & \\ Total & 45 & 4.67115 & & & \\ \hline \hline \multicolumn{5}{c}{Model summary} \\ \hline S & R-sq & R-sq (adj) & R-sq (pred) \\
0.0591023 & 97.08\% & 96.63\% & 94.35\% \\ \hline \end{tabular}
\end{table} 
Table 7: Analysis of variance of the magnetic flux density. Figure 10: Main effect plot. For the optimization of each objective function, a multi-objective optimization-based RSM was adopted. As shown in Table 8, the objective functions were optimized by varying the target values for each case. In Case 1, if the value of the magnetic flux density is minimized, the detent force and THD value of the back EMF exceed the values in the initial model. Therefore, Case 2, in which the magnetic flux density is limited between 1.0 and 1.5 T, is selected as the optimal value. For this case, the derived results were as follows: the height of the permanent magnet was 3.3 mm; the height of the tooth was 25 mm; the tooth tip angle and tooth tip height had minimum values of 30\({}^{\circ}\) and 1 mm, respectively; and the tooth width was 6.1 mm. Figure 12 illustrates the detent force relative to the slot effect of the optimized shape. \begin{table}
\begin{tabular}{l l c c c} \hline \hline  & **Item** & **Case1** & **Case2** & **Target** \\ \hline \multirow{3}{*}{Setup} & Detent Force & Minimize & Minimize & Minimize \\  & BackEMF THD & Maximize & Maximize & Minimize \\  & Magnetic Flux & Minimize & 1.0 (T)–1.5 (T) & 1.3 (T) \\  & Density & Minimize & 1.0 (T)–1.5 (T) & 1.3 (T) \\ \hline \multirow{3}{*}{Response} & Detent Force & 2.56 & 0.74 & Minimize \\  & BackEMF THD & 6.94 & 1.16 & Minimize \\  & Magnetic Flux & 1.08 & 1.3 & 1.3 (T) \\ \cline{1-1}  & Density & & & \\ \cline{1-1}  & \(P_{h}\) (mm) & Min & 3.3 & - \\ \cline{1-1}  & \(T_{h}\) (mm) & Max & 18.6 & - \\ \cline{1-1}  & \(T_{a}\) (\({}^{\circ}\)) & Min & Min & - \\ \cline{1-1}  & \(T_{th}\) (mm) & Min & Min & - \\ \cline{1-1}  & \(T_{w}\) (mm) & Max & 6.1 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Control factors and targets for each cases. Figure 11: Response surface of an interaction term. (**a**) Detent force (**b**) Magnetic flux density (**c**) Back EMF THD. For the optimized model, a detent force of 0.62 N was generated, which was reduced by approximately 35% compared with the initial model. As shown in Figure 13a, the maximum value of the back EMF of the optimized shape increased from 27.3 V (initial model) to 30.89 V. Although the height of the permanent magnet decreased, it was considered that the back EMF increased according to the change in the air gap. In the optimized model, as shown in Figure 13b, the 3 rd-order back EMF harmonic component decreased. The THD decreased by 0.57%. As shown in Figure 14, the maximum magnetic flux density generated in the optimized circuit was 1.18 T, indicating that the circuit was adequately designed. Figure 14: Field distribution and flux density contours of Optimized Circuit. Figure 12: Detent force by slot effect of initial model and optimized model. Figure 13: No-load voltage back EMF: (**a**) Back EMF of initial model; (**b**) Harmonic components of back EMF.