# Multi-Step Design Optimization for the Improvement of an Outer-Rotor Brushless Direct Current Motor
## 3 Taguchi Method and Response Surface Methodology (RSM)
### Respond Surface Methodology (RSM)  
Respond Surface Methodology (RSM) starts with a carefully planned series of experiments. The RSM offers advantages when optimizing problems with fewer than three design variables. It helps to determine optimal values by creating a model based on statistical data, allowing for an easy exploration of the interaction effects between variables [21]. These experiments are designed to systematically vary critical factors to understand how changes in these factors affect the outcomes or responses. The design often includes central composite design (CCD), Box-Behnken design (BBD), or factorial design as the selected design, and the workflow of the RSM is shown in Figure 8[11; 12; 22]. Among the options, we chose the CCD method, because it is often better for BLDC motor optimization due to its flexibility, ability to capture complex relationships (including quadratic effects), and efficient exploration of the response surface. It strikes a good balance between thoroughness and experimental count, making it suitable for scenarios where multiple factors and their interactions are critical [22]. After careful analysis of the FEM analysis outcomes and evaluation of the sample test for the optimized model derived via the Taguchi method, an enhanced design iteration was conducted employing RSM. This approach is particularly efficacious for optimizing fewer than three design variables and enables the determination of optimal responses for the objective function while accounting for interactive effects [13]. To effectively meet the performance objectives, a two-stage optimization design utilizing RSM was implemented, which capitalizes on the principal factors identified during the initial optimization phase, thus obviating the need for a comprehensive re-optimization of the entire array of motor variables. As depicted in Table 7, the design variables and their corresponding ranges for the RSM are outlined. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Performance**} & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **Results** \\ \hline \multirow{4}{*}{**Efficiency**} & Reference & 2 & 2 & 1 & 2 & 3 & 2 & 2 & 72.76\% \\  & Proposed & 2 & 3 & 3 & 2 & 3 & 2 & 3 & 81.93\% \\  & \multicolumn{4}{c}{Improvement (+12.6\%)} & & & & 9.17\% \\  & Reference & 2 & 2 & 1 & 2 & 3 & 2 & 2 & 26.06 W \\
**Output** & Proposed & 2 & 3 & 3 & 3 & 2 & 1 & 1 & 34 W \\  & \multicolumn{4}{c}{Improvement (+30.5\%)} & & & & 10 W \\  & Reference & 2 & 2 & 1 & 2 & 3 & 2 & 2 & 0.098 Nm \\
**Torque (Nm)** & Proposed & 3 & 2 & 1 & 1 & 3 & 2 & 3 & 0.130 Nm \\  & \multicolumn{4}{c}{Improvement (+32.6\%)} & & & & 0.032 Nm \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of results for control factors for reference and proposed model using Taguchi method. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \begin{tabular}{c} **Exp** \\ **No** \\ \end{tabular} & \begin{tabular}{c} **Thickness of Magnet** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Width of Stator Slot** \\ **Opening (mm)** \\ \end{tabular} & \begin{tabular}{c} **Number of Turns** \\ **Number of Turns** \\ \end{tabular} & \begin{tabular}{c} **\(\eta\) (\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Pout (W)** \\ \end{tabular} &
\begin{tabular}{c} **\(\tau_{\text{max}}\)** \\ **(Nm)** \\ \end{tabular} \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 7: FEM analysis of CCD of RSM. Figure 8: Organization of RSM procedure. A critical parameter, the magnet thickness, was singled out from the array of variables recognized in the Taguchi optimization stage due to its significant impact on motor performance. In addition to this, two other variables, namely, the number of turns and the width of the stator slot's opening, were incorporated to further scrutinize the operational scope and efficiency of the motor. Variables that would lead to an escalation in material costs and weight were deliberately excluded from this phase.