## 2 Design and Analysis Methods
### Two-DimensionalFem Analysis  
More analytical sizing and optimization studies are needed to conclude the design. With electromagnetic analysis, the flux density and flux distributions in the whole model, as well as the operating point of the magnet according to the demagnetization characteristic should be investigated. The 2D motor model is time stepped as a function of the magnetic field solutions created by the windings fed with the external voltage source and the permanent magnets. Figure 7 shows the B-H curve of the stator and rotor cores respectively. Figure 8 shows the flux distribution and effective flux line of the entire motor surface, the field strength in the magnets, current density, phase currents, and output torque for the 15.48-degree rotor \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Initial Design** & **Optimum Design** \\ \hline Rated power & 2.5 kW & 2.5 kW \\ \hline Rated speed & 1284 rpm & 873 rpm \\ \hline Rated torque & 18.58 Nm & 27.36 Nm \\ \hline Efficiency & 88\% & 91.74\% \\ \hline Armature current density & 9.15 A/mm\({}^{2}\) & 4.83 A/mm\({}^{2}\) \\ \hline Stack length & 35 mm & 38 mm \\ \hline Stator inner diameter & 196.88 mm & 193 mm \\ \hline Magnet thickness & 2 mm & 3 mm \\ \hline Stator tooth width & 12.19 mm & 9 mm \\ \hline Rotor yoke height & 6.64 mm & 5.5 mm \\ \hline Stator yoke height & 4.66 mm & 6.6 mm \\ \hline Slot opening & 2.5 mm & 3 mm \\ \hline Conductors per slot & 11 & 13 \\ \hline Wire diameter & 1.93 mm & 2.58 mm \\ \hline Slot fill factor & 30.11\% & 44.9\% \\ \hline \end{tabular}
\end{table} 
Table 3: Comparison data of the BLDC motor. Figure 6: Effects of some variables on 2.5 kW constant power outer runner BLDC motor speed, efficiency, rated torque, and cogging torque; (**a**) conductors per slot, (**b**) magnet thickness, (**c**) pole arc to pole pitch ratio, (**d**) stack length, and (**e**) slot opening. position, respectively. The rotor yoke flux density is low at the center of the pole arc. In contrast, at the junction of the two poles, it is around 1.7 T. While the flux density of the stator teeth is about 1.4 T, with a maximum of about 2 T on the shoe tips, the average flux density on the entire stator surface is 0.72 T. As shown in Figure 7, and these flux values, the saturation does not occur in the stator steel. The magnetic field lines are closed circuits through expected paths, and the periodicity conditions are appropriate. As is apparent from Figure 8c, the magnet demagnetization does not occur at operated conditions with the ambient temperature. In actual working conditions, excessive load will increase the temperature. Irreversible demagnetization may occur in permanent magnets with the temperature. Therefore, temperature analysis should be performed [1; 35]. Figure 8: Two-Dimensional FEM results; (**a**) flux density, (**b**) flux line, (**c**) magnetic field strength, and (**d**) current density. Figure 7: Material properties: B-H curve of M330-50A and 4340 Alloy steel. The current density presented in Figure 8d for overall conductor regions is a maximum of 4.68 A/mm\({}^{2}\). It is an acceptable current density value for this scale machine. Figure 9a-e show the simulated phase currents, back EMF, currents FFT, cogging torque, and corresponding output torque. The rms value of the currents was obtained at 34 A. When the FFT of the current is examined, it is observed that the 5 th and 7 th harmonic values are high. This is an expected situation in BLDC motors. The average value of the output torque was obtained at 36 Nm, which provides the objective function for the optimization design. To improve the torque quality, the cogging torque value must be low. The least common multiple (LCM) and slot opening width affect the cogging torque value [36]. A very low gear torque can be obtained if the slot and pole numbers are selected, so the LCM between these numbers is enormous. The cogging torque for a design with 32 poles and 36 slots is represented in Figure 9d.