## 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
The regression prediction model of the response value of the design parameters can derive the prediction equation from the results of the face-centered design, which can be obtained with the quadratic regression model for the independent variable K as a quadratic function of the face-centered design, using Equation (3). \[\mathrm{Y}=\beta_{0}+\sum_{i=1}^{k}\beta_{i}X_{i}+\sum_{i=1,j\leq i}^{k}\beta_{ ij}X_{i}X_{j} \tag{3}\]  
where \(X_{i},~{}X_{j}\) are the independent variables (design parameters), \(Y\) is the dependent variable (resulting value), and \(\beta_{i},~{}\beta_{ij}\) are coefficients, which can be obtained by least squares [34; 35]. From Equation (3) and the face-centered design analysis results, the regression prediction equation for the electromagnetic force can be expressed by Equation (4). \[\mathrm{(Q1)F}= -0.3809\mathrm{MT}^{2}~{}-0.1838\mathrm{PT}^{2}~{}-0.3712\mathrm{ YP}^{2}~{}-0.5113\mathrm{YT}^{2}~{}-0.4012\mathrm{MT}~{}\times~{}\mathrm{PT}+0.673 9\mathrm{MT}~{}\times~{}\mathrm{YP}~{}+0.1760\mathrm{MT}~{}\times~{}\mathrm{ YT}~{}+0.3744\mathrm{PT}~{}\times~{}\mathrm{YP}~{}+0.0819\mathrm{PT}~{}\times~{} \mathrm{YT}~{}+3.323\mathrm{MT} \tag{4}\] \[+2.142\mathrm{PT}~{}-2.221\mathrm{YP}~{}+3.954\mathrm{YT}~{}-4.22\]  
The distance between the PT and cap must have the minimum value necessary to avoid vibrational interference. Therefore, their geometric relationship can be derived from Figure 5 as expressed by Equation (5), where the height of the coil was set to a constant value of 10.1 mm. \[\mathrm{(Q2)~{}D}=10.1+\mathrm{YP}-\mathrm{MT}-\mathrm{PT} \tag{5}\]  
where D is the distance between magnetic flux collecting plate and cap base, YP is the vertical distance between the coil and yoke, MT is the magnet thickness, and PT is the thickness of the magnetic flux collecting plate. Moreover, the volume of the actuator is correlated with its thickness, with its diameter fixed, and the thickness can be determined using Equation (6) derived from the relationship shown in Figure 5, where the height of the coil is set at a constant value of 10.1 mm. \[\mathrm{(Q3)~{}T}=10.1+\mathrm{YP}+\mathrm{YT} \tag{6}\]  
where T is the magnetic circuit thickness, YP is the vertical distance between the coil and yoke, and YT is the yoke thickness. The coefficient of determination (R\({}^{2}\)) in Equation (4) indicates that the closer the value is to 1, the higher the degree of coincidence of the prediction model of the experiment with the actual value [36]. In this regard, [PERSON] and [PERSON] [37] proposed a coefficient of determination of 0.8, or higher, to establish a good fit of the prediction model. The coefficient of determination of the design parameters for the electromagnetic force was 0.9985 in the regression model, and the adjusted coefficient of determination (R\({}^{2}\)Adj) was 0.9974. Therefore, the model fit of the derived prediction regression model can be established according to the [PERSON] and [PERSON] criteria. The electromagnetic force and magnetic circuit thickness are in a trade-off relationship, and among the objective functions of this trade-off relationship, we used a multi-objective function as an optimization method. As an algorithm for the multi-objective function, we used a desirability function, which is a method proposed by [PERSON] and [PERSON] (1980) [38]. After calculating the individual desirability for each response, the composite desirability was derived. The composite desirability function D is the geometric mean of the values of the desirability function (di), as expressed by Equation (7):  
\[\mathrm{D}=\left(\mathrm{d}_{1}\times\mathrm{d}_{2}\times\mathrm{d}_{3}\cdot \cdot\cdot\cdot\cdot\times\mathrm{d}_{n}\right)^{1/n} \tag{7}\]  
(D = composite desirability, d\({}_{i}\) = individual desirability, and n = total number of measured responses). As shown in the above equation, the composite desirability is calculated as the geometric mean because if one of the response variables is not satisfied, the composite desirability is zero [39].