# 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} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline Model & 7 & 12,361.8 & 1765.97 & 203.37 & 0.000 \\ Linear & 4 & 9598.1 & 2399.53 & 276.33 & 0.000 \\ \(P_{h}\) & 1 & 333.8 & 333.78 & 38.44 & 0.000 \\ \(T_{h}\) & 1 & 230.8 & 230.81 & 26.58 & 0.000 \\ \(T_{a}\) & 1 & 66.7 & 66.72 & 7.68 & 0.009 \\ \(T_{w}\) & 1 & 8966.8 & 8966.81 & 1032.63 & 0.000 \\ Square & 1 & 2484.4 & 2484.36 & 286.1 & 0.000 \\ \(T_{w}\)*\(T_{w}\) & 1 & 2484.4 & 2484.36 & 286.1 & 0.000 \\
2-Way & 2 & 279.3 & 139.64 & 16.08 & 0.000 \\ \(T_{h}\)*\(T_{w}\) & 1 & 178.5 & 178.47 & 20.55 & 0.000 \\ \(T_{a}\)*\(T_{w}\) & 1 & 100.8 & 100.82 & 11.61 & 0.002 \\ Error & 38 & 330 & 8.68 & & \\ Lack-of-Fit & 33 & 330 & 10 & & \\ Pure Error & 5 & 0 & 0 & & \\ Total & 45 & 12,691.7 & & & \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c} \hline \hline S & R-sq & R-sq (adj) & R-sq (pred) \\
2.94678 & 97.40\% & 96.92\% & 95.28\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Analysis of variance of the back-EMF THD. \begin{table} \begin{tabular}{c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline Model & 4 & 348.641 & 87.16 & 13.41 & 0 \\ Linear & 2 & 186.199 & 93.1 & 14.32 & 0 \\ \(T_{th}\) & 1 & 50.729 & 50.729 & 7.8 & 0.008 \\ \(T_{w}\) & 1 & 135.47 & 135.47 & 20.84 & 0 \\ Square & 1 & 53.534 & 53.534 & 8.23 & 0.006 \\ \(T_{w}\)*\(T_{w}\) & 1 & 53.534 & 53.534 & 8.23 & 0.006 \\
2-Way & 1 & 108.907 & 108.907 & 16.75 & 0 \\ \(T_{th}\)*\(T_{w}\) & 1 & 108.907 & 108.907 & 16.75 & 0 \\ Error & 41 & 266.56 & 6.501 & & \\ Lack-of-Fit & 36 & 266.56 & 7.404 & & \\ Pure Error & 5 & 0 & 0 & & \\ Total & 45 & 615.2 & & & \\ \hline \hline \end{tabular} \begin{tabular}{c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline Model & 4 & 348.641 & 87.16 & 13.41 & 0 \\ Linear & 2 & 186.199 & 93.1 & 14.32 & 0 \\ \(T_{th}\) & 1 & 50.729 & 50.729 & 7.8 & 0.008 \\ \(T_{w}\) & 1 & 135.47 & 135.47 & 20.84 & 0 \\ Square & 1 & 53.534 & 53.534 & 8.23 & 0.006 \\ \(T_{w}\)*\(T_{w}\) & 1 & 53.534 & 53.534 & 8.23 & 0.006 \\
2-Way & 1 & 108.907 & 108.907 & 16.75 & 0 \\ \(T_{th}\)*\(T_{w}\) & 1 & 108.907 & 108.907 & 16.75 & 0 \\ Error & 41 & 266.56 & 6.501 & & \\ Lack-of-Fit & 36 & 266.56 & 7.404 & & \\ Pure Error & 5 & 0 & 0 & & \\ Total & 45 & 615.2 & & & \\ \hline \hline \end{tabular} 
\begin{tabular}{c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline Model & 7 & 12,361.8 & 1765.97 & 203.37 & 0.000 \\ Linear & 4 & 9598.1 & 2399.53 & 276.33 & 0.000 \\ \(P_{h}\) & 1 & 333.8 & 333.78 & 38.44 & 0.000 \\ \(T_{h}\) & 1 & 230.8 & 230.81 & 26.58 & 0.000 Figure 10 shows the effect of each design variable. For detent force, tooth width and tooth tip height and tooth width had an effect, and tooth tip height had an effect in the form of a quadratic function. For back-EMF THD, all design variables except tooth tip height were affected among the five variables, and tooth width was affected in the form of a quadratic function. For magnetic flux density, permanent magnet height, tooth height, and tooth width were affected, and permanent magnet height and tooth width were affected in the form of quadratic functions. Figure 11 shows the response surface for terms with interactions. As shown in Figure 11a, the detent force decreased as the tooth tip height was lower and the tooth width was wider. In addition, the detent force was decreased even when the tooth tip height was high and the tooth width was narrow. As shown in Figure 11b, both tooth width and permanent magnet height showed a curvature effect on magnetic flux density. As shown in Figure 11c, the back-EMF THD decreased as the tooth height decreased and the tooth tip angle increased, and showed a curvature effect on the tooth width.