# Design of External Rotor Ferrite-Assisted Synchronous
## 2 Challenges and Design Considerations for High Power Density External Rotor Fe-PMaSyn RM
### Concerns with Reluctance Torque Portion in High Power Density External Rotor PMaSynRM  
The volume power density \(P_{v}\) of external rotor Fe-PMaSynRM, defined as the developed output power over the motor volume, can be expressed as follows:  
\[P_{v}=\frac{T_{em}}{\pi\,\frac{\omega_{m}}{r_{ro}^{2}\,L}} \tag{1}\]  
Tem is developed electromagnetic torque, \(\omega_{m}\) is the rotor's mechanical angular speed, \(L\) is the motor's axial length, and \(r_{ro}\) is the rotor's outer radius (which is also the outer radius of the motor). According to (1), for a fixed motor volume and fixed rotor speed, the volume power density of the motor (\(P_{v}\)) is proportional to the developed electromagnetic torque (\(T_{em}\)). For a five-phase PMaSynRM, the average torque \(T_{em}\) can be written as follows: [30]  
\[T_{em}=T_{PM}+T_{ reluctance}=\frac{5}{2}\ p\ \lambda_{PM}\ I_{d}+\frac{5}{2}\ p\ \lambda_{PM}\ I_{d}\left(\frac{L_{d}}{L_{d}}-1\right)L_{q}\ L_{d}\ I_{q} \tag{2}\]  
where \(T_{PM}\) is developed electromagnetic torque due to the PM, \(T_{ reluctance}\) is the developed electromagnetic torque due to rotor saliency, \(\lambda_{PM}\) is the permanent magnet flux linkage, p is the number of pole pairs, and \(L_{d}\), \(L_{q}\), \(I_{d}\), and \(I_{q}\) are the \(d\)- and q-axis inductances and currents, respectively. The reluctance torque is largely dependent on the saliency ratio (\(L_{d}\)/\(L_{q}\)), whereas the PM-torque is dependent on the permanent magnet flux linkage. The torque equation in (2) can be extended as follows:  
\[T_{em}=\frac{\pi}{4}\ B_{PM1}\ K_{d}\ D^{2}\ L+\frac{\pi}{4}\big{(}B_{d1}\ K_{q}-B_{q1}\ K_{d}\big{)}D^{2}L \tag{3}\]  
\(K_{d}\) and \(K_{q}\) are the d- and q-axis current density, \(B_{PM1}\), \(B_{d1}\), and \(B_{q1}\) are the fundamental of air-gap flux density due to PM, d-axis current, and q-axis current, respectively.  
According to (3), the developed torque in Fe-PMaSynRM increases in proportion to the bore diameter (\(D\)) square while the other terms are constant. To obtain higher torque in external rotor PMaSynRM, if the bore diameter (\(D\)) increases while the \(B_{PM1}\), and \(L\) are maintained constant by keeping almost the same dimension of the PM, then the PM-torque will increase with \(D^{2}\). However, the increase of \(D\) will hurt the reluctance torque component. The increased bore diameter increases the d-axis equivalent circuit reluctances due to decreased width and increased length of the d-axis flux path, which will decrease \(B_{d1}\). Simultaneously, the q-axis equivalent circuit's reluctances decrease as the q-axis flux paths become wider with increased \(D\), which will increase \(B_{q1}\). Therefore, the increase of bore diameter (\(D\)) in external rotor PMaSynRM might decrease the value of (\(B_{d1}\)\(K_{q}\)-\(B_{q1}\)\(K_{d}\)) and limit the reluctance torque. Overall, a higher bore diameter (\(D\)) in the external rotor motor tends to increase the PM-torque and decrease the reluctance torque. The design objective is to achieve higher average torque with a significant proportion of reluctance torque.