## 2 Design and Analysis Methods
### Multi-Objective Parametric Analysis  
Torque, speed, power, and efficiency curves are critical when selecting or designing an electric motor. These curves need to be analyzed under different operating conditions of the motor to understand its performance, determine if it is suitable for the application, and determine the optimum parameters for the desired performance. The optimum design in electrical machines is to obtain an acceptable design that fulfills all the requirements. Since electrical machines are composed of many parameters, the minimization or maximization of a set of objective functions is required for multi-objective parametric optimization [33; 34]. One or more variable sweep definitions, each defining a range of variable values, are defined in a parametric analysis. You can parameterize component values, for instance. At every variation, the design is solved using optimetrics. The performance of each design variant can then be ascertained by comparing the outcomes. Because parametric analyses aid in identifying an appropriate range of variable values for the optimization study, they are frequently employed as preludes to optimization solutions. Current optimization methods usually choose to optimize all design parameters simultaneously, and such optimization methods require a lot of computational time and are complex. Therefore, all design parameters are broken down into five phases and optimized individually based on the results of the sensitivity analysis to address the issue of the increase in calculation consumption brought on by the increase in design parameters. Figure 5 shows a suggested optimization framework for the investigated BLDC motor based on the multistep parametric optimization strategy. As shown, the optimization progress has been divided into 5 steps. Different objectives and constraints will be considered for each optimization level. The objectives are to minimize total loss (Pcore and Pcu), maximize torque (Timech), rate power (Pmech), and increase efficiency, as well as minimize cogging torque (Tcog). Equation (30) provides the generalized form of the multi-objective function utilized. In this case, \(y\) or \(f(x)\) are objective functions that reach their minimum or maximum value while preserving the ranges of the other parameters. \(x\) is a vector variable stating structural or sizing parameters in the iteration related to the machine size and restricted to take a value within a lower (\(x_{L}\)) and an upper (\(x_{H}\)) bound. The \(g_{m}(x)\) and \(h_{k}(x)\) are constraint functions, M is inequality, and K is equality constraint. The parameters are as follows:  
\(n_{cs}\) is the conductor per slot;  
\(L_{skt}\) is the stack length;  
\(h_{m}\) is magnet thickness;  
\(\alpha_{m}\) is the pole pitch/magnet pitch ratio;  
\(b_{s0}\) is slot opening. \[ \begin{array}{l}\max\textit{or}\min\left\{\begin{array}{l}y_{j}=f(\overset {\rightarrow}{x}),\quad j=1,2,\ldots,J\\ \overset{\rightarrow}{x}=[x_{1},x_{2},\ldots,x_{D}],\overset{\rightarrow}{x} \in R^{D}\end{array}\right\}\\ \textit{subject to}\quad\left\{\begin{array}{l}x_{L}\leq x_{i}\leq x_{H}, \ \ i=1,2,\ldots,N\\ g_{m}(x)\geq 0,\quad m=1,2,\ldots,M\\ h_{k}(x)=0,\quad k=1,2,\ldots,K\end{array}\right\}\end{array}  \tag{30}\]For example,  
\[\left\{ \begin{array}{l}\text{minimize}\to P_{core}=f(h_{m},\alpha_{m},L_{ skt},n_{cs},b_{s0})\\ \text{minimize}\to P_{cu}=f(h_{m},\alpha_{m},L_{skt},n_{cs},b_{s0})\\ \text{maximize}\to P_{mech}=f(h_{m},\alpha_{m},L_{skt},n_{cs},b_{s0})\\ \text{maximize}\to T_{mech}=f(h_{m},\alpha_{m},L_{skt},n_{cs},b_{s0})\\ \text{maximize}\to\eta=f(h_{m},\alpha_{m},L_{skt},n_{cs},b_{s0})\\ \text{minimize}\to T_{cg}=f(b_{s0})\\ \text{related to}\to\left\{\begin{array}{l}20 mm\leq L_{skt}\leq 60 mm,++1 mm\\ 1 mm\leq h_{m}\leq 5 mm,++0,2 mm\\ 0.5\leq\alpha_{m}\leq 1,++0.01\\ 6\leq n_{cs}\leq 18,++1\\ 2 mm\leq b_{s0}\leq 4 mm,++0.1 mm\end{array}\right\}\end{array} \right\} \tag{31}\]  
In addition, there are several constraints, such as  
\[\text{related to}\to\left\{