# Less Rare-Earth Electromagnetic Design for a High-Performance Permanent Magnet Motor
## 3 Proposed Magnetization Pattern for Banana-Shaped Magnets with Parallel Magnetization
\[\hat{B}_{g}=\frac{B_{r}}{1+\mu_{r}\frac{S_{m}}{S_{g}}\frac{g}{l_{m}}}\frac{S_{m }}{S_{g}}[T] \tag{1}\]  
\[C_{\varphi_{S}}=\frac{S_{m}}{S_{g}} \tag{2}\]  
Table 4 shows the magnet properties and the surface areas of the air gap (\(S_{m}\)) and magnet (\(S_{g}\)) for the single-banana case of Figure 7. So, applying Equation (2) in this specific geometric case tells us that the ideal maximum attainable \(C_{\varphi_{S}}\) is 1.2613 (see Figure 7). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Unit** & **Parallel Magnetization** & **Radial Magnetization** \\ \hline \(S_{m}\) & mm\({}^{2}\) & & 2669.568 & \\ \(S_{g}\) & mm\({}^{2}\) & & 2116.587 & \\ \(C_{\varphi_{S}}\) & - & 1.2613 & 1.2613 & \\ \(B_{m}\) & T & 1.0336 & 0.9448 & \\ \(H_{m}\) & kA/m & 85.480 & 154.439 & \\ \(\hat{B}_{g}\) & T & 0.8740 & 1.0733 & \\ \(\hat{B}_{g1}\) & T & 0.9631 & 1.1354 & \\ \(C_{\varphi_{B}}\) & - & 0.8456 & 1.1360 & \\ \(B_{r}\) & T & 1.1424 & 1.1424 & \\ \(\mu_{r}\) & - & 1.0180 & 1.0180 & \\ \(PC\) & - & 9.6712 & 4.8690 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Surface areas and magnetic properties of a single banana layer simulation. Figure 6: Fundamental harmonic air gap flux density vs. magnet length. However, it is possible to account for the magnetization pattern and leakage by computing the flux concentration factor using Equation (3) if these flux density values are obtained directly from FEM. \(B_{m}\) is the magnet operating point calculated as the mean value of the magnet's flux density distribution (same is done for the magnetic field: \(H_{m}\)). So, to check the difference, radial and parallel magnetization is applied as shown in Figure 7 and \(\hat{B}_{g}\) and \(B_{m}\) are obtained from FEM. The no-load simulation yields the results presented in Table 4. On the one hand, as expected, the results show that radial magnetization makes better use of the magnet weight for the same rotor structure, having a 9.93% decrease with the maximum attainable flux concentration factor. However, on the other hand, parallel magnetization decreased 32.95%, which translates into 17.89% less air gap flux density than the radial case. However, the permeance coefficient (\(PC\)) [25] (shown in Table 4) shows that radial magnetization leads to a lower magnet open circuit operating point than the parallel case, which could affect demagnetization. \[C_{\varphi_{B}}=\frac{\hat{B}_{g}}{B_{m}} \tag{3}\]  
The values in Table 4 show that radial magnetization provides better use of the magnet weight. Nevertheless, the main drawback to applying radial magnetization is its manufacturing process since a specific magnetic circuit with molds needs to be built. The baseline motor shown in Table 4 uses parallel magnetization [23], so it is opted to find a solution with parallel magnetization because the magnet unit cost is maintained for the same rotor structure since no change during the magnetization process is required. In practical terms, as seen in Figure 7, the geometry is limited only to parallel or radial magnetization. Therefore, the proposed method requires splitting the banana, as seen in the baseline motor in Figure 1 and its model in Figure 4. Although this will increase the leakage due to the new ribs, it can also allow the implementation of parallel magnetization differently. The hypothesis is that since radial magnetization provides the highest \(\hat{B}_{g1}\), there should be an optimum tilt angle that increases \(\hat{B}_{g1}\) compared to the conventional parallel magnetization (straight as seen in Figure 7a) for the same magnet weight. Hence, getting the most out of the magnet weight with parallel magnetization in a given rotor structure.