## 2 Design and Analysis Methods
### Main Dimensions  
Design specifications, including machine type, construction type (axial flux, radial flux, external rotor, and internal rotor), number of phases, rated power, rated rotational speed, power factor, and rated voltage, are the first step in the design process [25; 26]. In [25; 26], the design procedure of electrical machines is presented in detail. Table 1 presents reference parameters for the present machine. Although the design of electrical machines is computer aided today, obtaining the basic size parameters for all electrical machines, from micromotors to the largest motors [24], is essential. The fundamental question in electric motor design is what volume a motor must be to produce the required torque. This can be expressed using Equations (1)-(9) for radial flux motors [23; 24; 27; 28]. Current density and airgap magnetic flux density, also expressed as specific electrical and magnetic loading, are the most critical factors affecting the size and shape of the machine. Some key parameters that are independent variables in the design of electrical machines are given in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Rated power & 2.5 kW \\ \hline Maximum output power & 3.8 kW \\ \hline Rated speed & 750–900 rpm \\ \hline Rated torque & \textgreater{}26 Nm \\ \hline Targeted efficiency & \textgreater{}85\% \\ \hline DC voltage Vdc (automobile battery) & 96 V \\ \hline Stator material & M330-50A \\ \hline Rotor material & AISI/SAE 4340 Alloy Steel \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reference data of the BLDC motor. Figure 3: Different rotor designs are used in BLDCs. In radial flux motors, the flux crosses the airgap radially, and the magnetic loading is determined by the average radial flux density in the airgap [24]. The first stage is to choose the specific magnetic and electric loadings on electric machines' general or optimum design. These loadings are described, respectively, as follows [28]:  
\[B_{mg}=\frac{\Phi_{g}}{\alpha_{i}\tau_{p}L_{e}}=\frac{2p\Phi_{g}}{\pi\alpha_{i}L _{e}D_{g}},\quad A_{m}=\frac{2m\sqrt{2}N_{1}I_{a}}{\pi D_{g}} \tag{1}\]  
where \(\Phi_{g}\) is the airgap flux with the armature reaction taken into account. \(D_{g}\), \(L_{e}\), \(N_{1}\), and \(2p\) are the airgap diameter of the machine or the stator outer diameter (for an external rotor motor), the axial length of the machine, the number of turns per phase, and the number of poles, respectively. In Equation (2), \(b_{p}\) is the pole shoe width. The effective pole arc coefficient \(\alpha_{i}\) (pole arc/pole pitch ratio) and pole pitch \(\tau_{p}\) are expressed as follows. \[\alpha_{i}=\frac{b_{p}}{\tau_{p}};\quad 0.55<\alpha_{i}<0.75,\quad\ \ \tau_{p}=\frac{\pi D_{g}}{2p} \tag{2}\]  
In PMSMs, magnets are usually placed on the rotor surface, and the mean flux can be defined by the relative magnet width \(\alpha_{PM}=w_{PM}/\tau_{p}\). Thus, \(\alpha\)PM is used instead of \(\alpha_{i}\) in Equation (2). The machine's angular speed, frequency, and synchronous speed can be related as in Equation (3). \[\omega_{e}=2\pi f_{e}\to n_{s}=\frac{120f_{e}}{2p} \tag{3}\]  
where \(n_{s}\), \(\omega_{e}\), and \(f_{e}\) are synchronous speed (rpm), angular speed, frequency, and pole number, respectively. Thus, the airgap apparent electromagnetic power can also be written as [28; 29; 30]  
\[ \begin{array}{l}S_{elm}=m_{1}E_{f}I_{a}=\pi\sqrt{2}fN_{1}k_{\omega 1}\phi_{f} \frac{m_{1}\pi D_{g}A_{m}}{2m_{1}\sqrt{2}N_{1}}\\ =\left(\frac{\pi^{2}}{120}\right)(n_{s}2p)k_{\omega 1}\frac{D_{g}^{2}L_{e}}{2p}B_{ mg}A_{m}\\ =\left(\frac{\pi^{2}}{120}\right)k_{\omega 1}D_{g}^{2}L_{e}n_{s}B_{mg}A_{m} \end{array}  \tag{4}\]  
where \(k_{\omega 1}\) is the winding factor. If Equation (4) is rearranged by power coefficients \(\cos\varphi\) and \(\cos\theta\) by including efficiency, i.e., rotor iron and copper loss and rotational losses, electromagnetic torque and output (shaft) power are obtained as given in Equations (5) and (6), respectively.