## 3 Topological Diversity and Structural Design in Magnetic Gear Systems
### Design Parameters of Magnetic Gears
Therefore, recent studies have highlighted the increasing relevance of analytical and semi-analytical approaches in evaluating torque and magnetic field distribution for coaxial magnetic gears (CMGs) featuring Halbach magnet arrays. These methods offer significant advantages in early-stage design due to their low computational cost and ability to provide closed-form insight into electromagnetic interactions. For instance, [PERSON] et al. [53] developed a comprehensive two-dimensional (2D) analytical model for Halbach-type CMGs, assessing torque transmission and magnetic field behavior. The model demonstrated excellent  
Figure 7: Halbach array (**a**) magnet orientations (**b**) flux lines. agreement with the FEA results, with torque deviation remaining below 5%. Similarly, [PERSON] et al. [52] introduced a semi-analytical method to approximate magnetic flux modulation in coaxial configurations, achieving notable improvements in both accuracy and computational efficiency. One of the pioneering studies in this field, conducted by [PERSON] and [PERSON] [54], presented one of the first analytical frameworks incorporating Halbach magnetization into magnetic gear modeling and achieved accurate torque estimations. Subsequently, [PERSON] et al. [55] developed a detailed 2D analytical model to investigate the effects of various magnetization patterns, including Halbach arrays, on CMG performance. More recently, [PERSON] et al. [56] extended these approaches to 3D models, incorporating end effects and core losses into their analyses. The effectiveness of such arrangements, which enhance the magnetic field concentration, is directly related to the appropriate selection of magnet materials. In magnetic gear systems, not only the type of arrangement but also the physical and magnetic properties of the magnets used are critical to improving efficiency. Among magnets produced from rare earth elements, neodymium-iron-boron (NdFeB) magnets offer the highest energy density and residual magnetic induction after samarium-cobalt (SmCo) alloys. NdFeB magnets are preferred in high-performance applications due to their high energy density and magnetic flux values per unit volume [57]. However, their primary disadvantages are their relatively low Curie temperatures and high cost [58]. A comparative summary of the advantages and disadvantages of NdFeB magnets versus other types, such as Alnico and SmCo, is provided in Table 1.  
As shown in Table 1, NdFeB magnets stand out due to their high energy density advantage. However, due to their low Curie temperature, they have limitations in high-temperature applications. The performance of permanent magnets is generally defined by their energy density, which is expressed as the maximum energy product (BHmax). These advantages and disadvantages serve as important references for guiding magnet selection based on the intended application. In particular, the low Curie temperature of NdFeB magnets should be carefully considered in high-temperature applications. Table 2 presents a comparative overview of the magnetic properties of different types of permanent magnets. As seen in Table 2, NdFeB magnets stand out with their high BHmax values and strong remanent magnetic induction, whereas SmCo magnets are notable for their high-temperature resistance and superior coercive force. These comparisons play a critical role in magnetic material selection, as they directly impact overall system performance. However, \begin{table}
\begin{tabular}{c c c} \hline
**Type** & **Advantages** & **Disadvantages** \\ \hline NdFeB & High energy and flux density per unit volume & Low Curie temperature and high cost \\ AlNiCo & High Curie temperature and high flux density & Low coercivity, brittleness, and costliness \\ SmCo & High coercive force, high flux density, and resistance to heat & High cost and fabrication complexity \\ \hline \end{tabular}
\end{table} 
Table 1: Advantages and disadvantages of NdFeB, AlNiCo, and SmCo magnets.