## 3 Magnetic Circuit Optimization of a Virtual Engine Sound Actuator
### Design Parameter Optimization for the Magnetic Circuit
#### 3.2.2 Design Parameter Analysis Using a Response Surface Design  
Given the nonlinearity of the design parameters of factorial design analysis and the significant characteristics of their interactions, we optimized the performance for volume (thickness, distance) and electromagnetic force using a response surface design conducive to two-level or higher nonlinear prediction, thereby applying the face-centered design generally used to optimize nonlinear design parameters. In the face-centered design, which is a type of response surface methodology, a total of 25 DOE combinations are arrayed, and the analysis results for the electromagnetic force, magnetic circuit thickness, and cap distance are outlined in Table 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & \(p\)**-Value** \\ \hline
**Model** & 11 & 56.1689 & 5.1063 & 437.67 & 0.000 \\ \hline
**Linear** & 4 & 44.1369 & 11.0342 & 945.77 & 0.000 \\ \hline MP & 1 & 2.3782 & 2.3782 & 203.84 & 0.000 \\ \hline PT & 1 & 1.7056 & 1.7056 & 146.19 & 0.000 \\ \hline YP & 1 & 20.491 & 20.491 & 1756.33 & 0.000 \\ \hline YT & 1 & 19.562 & 19.562 & 1676.71 & 0.000 \\ \hline
**Interaction** & 6 & 10.5352 & 1.7559 & 150.5 & 0.000 \\ \hline MP \(\times\) PT & 1 & 1.4488 & 1.4488 & 124.18 & 0.000 \\ \hline MP \(\times\) YP & 1 & 7.266 & 7.266 & 622.78 & 0.000 \\ \hline MP \(\times\) YT & 1 & 0.4957 & 0.4957 & 42.49 & 0.001 \\ \hline PT \(\times\) YP & 1 & 1.2618 & 1.2618 & 108.15 & 0.000 \\ \hline PT \(\times\) YT & 1 & 0.0604 & 0.0604 & 5.18 & 0.072 \\ \hline YP \(\times\) YT & 1 & 0.0025 & 0.0025 & 0.21 & 0.664 \\ \hline
**Curvature** & 1 & 1.4968 & 1.4968 & 128.3 & 0.000 \\ \hline
**Error** & 5 & 0.0583 & 0.0117 & & \\ \hline
**Total** & 16 & 56.2272 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Full factorial design ANOVA of vertical magnetization model. The main effects were characterized by the negative correlations of MT and PT and the positive correlations of YT and YP with the electromagnetic force. The interaction effects were characterized by the positive correlations between the value of YP \(\times\) YT and electromagnetic force, with the correlations of other parameters showing contradictory tendencies. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**No.** & **MT** & **PT** & **YP** & **YT** & **Force** & **Thickness** & **Distance** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 5: Response surface design matrix and results of vertical magnetization model. Figure 8: Interaction effects plot for electromagnetic force. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline
**Model** & 13 & 71.3403 & 5.4877 & 1184.54 & 0.000 \\ \hline
**Linear** & 4 & 49.3348 & 12.3337 & 2662.27 & 0.000 \\ \hline MT & 1 & 2.6293 & 2.6293 & 567.54 & 0.000 \\ \hline PT & 1 & 1.836 & 1.836 & 396.3 & 0.000 \\ \hline YP & 1 & 22.7527 & 22.7527 & 4911.24 & 0.000 \\ \hline YT & 1 & 22.1168 & 22.1168 & 4773.98 & 0.000 \\ \hline
**Square** & 4 & 11.4728 & 2.8682 & 619.11 & 0.000 \\ \hline MT \(\times\) MT & 1 & 0.3764 & 0.3764 & 81.26 & 0.000 \\ \hline PT \(\times\) PT & 1 & 0.0277 & 0.0277 & 5.99 & 0.026 \\ \hline YP \(\times\) YP & 1 & 0.3575 & 0.3575 & 77.17 & 0.000 \\ \hline YT \(\times\) YT & 1 & 0.6784 & 0.6784 & 146.42 & 0.000 \\ \hline
**Interaction** & 5 & 10.5327 & 2.1065 & 454.7 & 0.000 \\ \hline MT \(\times\) PT & 1 & 1.4488 & 1.4488 & 312.72 & 0.000 \\ \hline MT \(\times\) YP & 1 & 7.266 & 7.266 & 1568.39 & 0.000 \\ \hline MT \(\times\) YT & 1 & 0.4957 & 0.4957 & 107 & 0.000 \\ \hline PT \(\times\) YP & 1 & 1.2618 & 1.2618 & 272.36 & 0.000 \\ \hline PT \(\times\) YT & 1 & 0.0604 & 0.0604 & 13.04 & 0.002 \\ \hline
**Error** & 17 & 0.0788 & 0.0046 & & \\ \hline
**Total** & 30 & 71.419 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Response surface design dispersion analysis.