# Less Rare-Earth Electromagnetic Design for a High-Performance Permanent Magnet Motor
## 3 Proposed Magnetization Pattern for Banana-Shaped Magnets with Parallel Magnetization
So, \(\hat{B}_{g1}\) is obtained for a sweep of tilt angles, \(\alpha_{1}\) and \(\alpha_{2}\), from \(0^{\circ}\) to \(90^{\circ}\) applied to each magnet, as seen in Figure 4. Table 5 shows the main results quantities of the sweep for \(\hat{B}_{g1}\) in per unit, normalized over \(\hat{B}_{g1}\) for radial magnetization (1.0934 T shown in the table, equivalent to one per unit) shows the resulting curve of the sweep for \(\hat{B}_{g1}\) in per unit. Figure 7: (**a**) Parallel magnetization; (**b**) Radial magnetization. The remarks for the baseline motor from the results shown in Figure 8 and Table 5 are:  
1. The maximum fundamental air gap flux density consists of a different tilt angle for each magnet (\(\alpha_{1}\): 36\({}^{\circ}\) and \(\alpha_{2}\): 19.4\({}^{\circ}\)), obtaining a 0.99291 per unit of the radial magnetization for this case study. 2. A second option is choosing the maximum air gap flux density that applies the same tilt angle to both magnets (\(\alpha_{1}\): 29.4\({}^{\circ}\) and \(\alpha_{2}\): 29.4\({}^{\circ}\)), leading to a slightly lower value than the maximum. In this case, 0.98067 per unit of the radial magnetization and the difference between the \(\hat{B}_{g1}\) maximum is 0.01224 per unit. 3. Straight magnetization (initial parallel magnetization at zero degrees) provides the smallest value of the three options. In this case, 0.84588 per unit of the radial magnetization. This pattern is the typical way of applying parallel magnetization. Figure 9a illustrates how to obtain the tilted desired pattern with parallel magnetization for the real magnet part. Figure 9b would illustrate the resulting magnetization pattern in the motor model if the magnet parts were magnetized in this way. The solution is to tilt the magnet part to the desired angle (29.4\({}^{\circ}\) in this case), which provides the manufacturing advantage of continuing to use parallel magnetization (illustrated by the red arrow in Figure 9a). Thus, the magnet part is tilted instead of adjusting or changing the magnetization device. Therefore, keeping the same parallel magnetization device provides the advantage of avoiding extra investment in a different magnetization device, as explained in Section 1. Figure 8: Peak fundamental harmonic air gap flux density for 0\({}^{\circ}\) to 90\({}^{\circ}\) tilt angles. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Magnetization Pattern** & \(\overset{\wedge}{B_{g1}}\) [\(\mathbf{p}\)u] & \(\mathbf{\kappa}_{1}\) [\({}^{\circ}\)] & \(\mathbf{\kappa}_{2}\) [\({}^{\circ}\)] \\ \hline \(\hat{B}_{g1}\) maximum with a different angle & 0.99291 & 36 & 19.4 \\ \(\hat{B}_{g1}\) maximum with same angle & 0.98067 & 29.4 & 29.4 \\ \(\hat{B}_{g1}\) straight angle & 0.84588 & 0 & 0 \\ \(\hat{B}_{g1}\) radial & & 1.0934 [T] & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Peak fundamental flux density result and corresponding tilt angles normalized to radial magnetization value. From here onward, the same tilt angle pattern for both magnets is selected for two reasons. The first reason is the slight difference of 0.01224 per unit between \(\hat{B}_{g1}\) maximum and the \(\hat{B}_{g1}\) maximum with the same tilt angle option for both magnets. The second reason is that it could open the possibility of magnetizing both magnets simultaneously, as seen in Figure 9a. Figure 10 presents the flux density distribution and flux lines in FEM at no load and ambient temperature with the initial magnetization (straight at zero degrees) and the tilted magnetization with the same angle pattern. The performance comparison between the baseline motor model with straight magnetization and tilted magnetization with the same angle for both magnets is given in Figure 10. The performance comparison targets a similar thermal performance (winding and magnet temperatures), as shown in the same table. Moreover, for this case study, a simple demagnetization analysis is carried out to check if there could be a risk of demagnetization at the operating temperature.