## 3 Magnetic Circuit Optimization of a Virtual Engine Sound Actuator
### Design Parameter Optimization for the Magnetic Circuit  
#### 3.2.1 Selection of Design Parameters for the Magnetic Circuit and Factorial Design  
We set the selection criteria for actuator design parameters as geometric factors that can minimize the volume necessary for the package while maximizing the electromagnetic force. As geometric variables affecting the performance [32], we selected the magnet thickness (MT), plate thickness (PT), yoke position (YP; distance between yoke and coil), and yoke thickness (YT), as shown in Figure 5, with the size based on the outer diameter (\(\Phi=65\) mm) fixed.  
Figure 4: Flow chart of design of experiment (DOE).  
Figure 5: Design cross-section and design parameters of vertical magnetized structure.  
The range of design of the selected parameters was set as the range of design boundaries and the direction minimizing the magnetic circuit thickness, as defined in Table 2. The minimum yoke position was limited to 2.5 mm, which is the minimum separation distance necessary to prevent interference with other components.  
Among the selected design parameters, the one with the greatest effect was identified using factor analysis of the factorial design methodology [33]. As outlined in Table 3, the factorial design of the selected parameters was implemented as a total of 17 combinations consisting of 16 combinations of a four-factor two-level full factorial design, based on a two-level factorial design for the maximum and minimum values, and one center point of curvature analysis, expressing the electromagnetic force obtained by electromagnetic field analysis as response values.  
The analysis of variance (ANOVA) of the factorial design of the selected parameters is the adjusted sum of squares (Adj SS), as expressed by Equation (2).  
\[\text{Adj SS }=\frac{(\text{T}_{1}-\text{T}_{0})^{2}}{\text{N}}, \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Parameter** & **Initial Value** & **Low Limit Value** & **Upper Limit Value** \\ \hline Magnet thickness (MT) & 7.8 & 7.0 & 9.0 \\ \hline Plate thickness (PT) & 4.5 & 3.5 & 5.0 \\ \hline Yoke position (YP) & 4.9 & 3.9 & 5.9 \\ \hline Yoke thickness (YT) & 4.5 & 3.5 & 5.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Range of design variable dimensions for vertical magnetized structure. (unit: mm).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No** & **MT (mm)** & **PT (mm)** & **YP (mm)** & **YT (mm)** & **Force (N)** \\ \hline  
\hline \end{tabular}
\end{table}
Table 3: Full factorial design matrix and results of vertical magnetized structure.  
where T1 and T0 are the sums of the response values for the high-level and low-level factors, respectively, and N is the number of experiments. Adj MS is the mean square obtained by dividing Adj SS by the degree of freedom (DF) presented in Table 4.  
The effect of the design parameters can be verified via the analysis presented in Table 4, based on the importance of the response values for the electromagnetic force. The F-value can be obtained by dividing the Adj MS of each factor by the Adj MS of the error, whereby the higher the value, the higher the effect of the related factor on the response value. The \(p\)-value is inversely correlated with the F-value, and the significance is considered established at 0.05 or lower. Accordingly, Table 4 lists the significant probability values for the lack of fit of the response values for the electromagnetic force with regard to the magnet thickness (MT), plate thickness (PT), yoke position (YP), and yoke thickness (YT). The analysis revealed that all four design parameters were statistically significant at the 5% significance level. Figure 6 shows the results of the corresponding Pareto analysis. Its nonlinearity suggests that not only the main factors but also their interactions have significant effects.