## 3 Sensitivity Analysis
### Definition of the Sensitivity Analysis  
The aim of a sensitivity analysis is to deduce how much each of the input parameters of a model contributes to the uncertainty or variation of its output response. There are different types of sensitivity analyses, but a general distinction can be made between local and global analyses. In local sensitivity analyses, which are usually based on gradients or partial derivatives, the sensitivity of an input parameter is studied in a small range while keeping the rest of the inputs constant. In global sensitivity analyses, all the input factors can be varied simultaneously, which allows obtaining the sensitivity of the inputs and their interactions in a wider range. The negative aspect of global analyses is that their computational cost is higher as they are based on the dispersion between the response of all the samples defined in the input parameter range. In this work, even though a certain machine geometry is employed as a reference, there are no particular design restrictions, which means that the original dimensions can be modified when optimizing the design. For that reason, a global sensitivity analysis procedure was selected. Among the different types of global techniques that are used in the literature, the variance-based sensitivity analysis is one of the most popular ones as it allows measuring the sensitivity across the entire input space, to consider non-linear responses and to measure interactions between parameters [33]. A model can be considered as a function \(Y=f(X)\), being \(X\) the vector of \(d\) input parameters \(\{X_{1},X_{2},...,X_{d}\}\). The function \(f(X)\) can be decomposed, as shown in Equation (27) [34]:  
\[f(X)=f_{0}+\sum_{s=1}^{d}\sum_{i_{1}<...<i_{s}}^{d}f_{i_{1},...,i_{s}}(X_{i_{1} },...,X_{i_{s}}) \tag{27}\]  
Assuming that the function is square integrable:  
\[\int f^{2}(X)\mathrm{d}X-f_{0}^{2}=\sum_{s=1}^{d}\sum_{i_{1}<...<i_{s}}^{d} \int f_{i_{1},...,i_{s}}^{2}\mathrm{d}X_{i_{1}},...,\mathrm{d}X_{i_{s}} \tag{28}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Description** & **Value [mm]** \\ \hline \(D_{\text{ri}}\) & Internal diameter of the rotor & 202.4 \\ \(h_{\text{m}}\) & Height of the magnets & 5.3 \\ \(g\) & Length of the air gap & 1 \\ \(h_{\text{t}}\) & Height of the teeth & 25 \\ \(w_{\text{t}}\) & Width of the teeth & 7.71 \\ \(w_{\text{sl}}\) & Width of the slots & 8 \\ \(h_{\text{sy}}\) & Width of the stator yoke & 17.5 \\ \(D_{\text{se}}\) & External diameter of stator & 300 \\ \(L_{\text{s}}\) & Length of the stator & 175 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical dimensions of the electric machine. The left part of the equation is the variance of \(Y\) (Equation (29)), and the terms on the right part are the variance terms of the inputs (Equation (30)). \[V=\int f^{2}(X)\mathrm{d}X-f_{0}^{2} \tag{29}\]  
\[V_{i_{1},...,i_{k}}=\int f_{i_{1},...,i_{k}}^{2}\mathrm{d}X_{i_{1}},..., \mathrm{d}X_{i_{k}} \tag{30}\]  
According to those definitions, the global sensitivity indices can be defined as:  
\[S_{i_{1},...,i_{k}}=\frac{V_{i_{1},...,i_{k}}}{V} \tag{31}\]  
It is important to notice that all the \(S_{i_{1},...,i_{k}}\) indices are higher than or equal to 0, and that their sum equals 1. Moreover, two different indices can be distinguished: the first-order indices and the total-effect indices. The first-order sensitivity index (Equation (32)) represents the effect of modifying a certain input parameter alone. On the other hand, the total-effect sensitivity index (Equation (33)) represents the effect of modifying a certain input parameter, including the variances caused by its interactions with the other input parameters.