# Investigation of Segmented Halbach Array Design for Passive Magnetorheological Brake System
## 3 Segmented Halbach Array Design
### Magnetic Flux Density  
The simulation of the Halbach configuration through permanent magnets explains the distribution of magnetic flux density (B) with the flux concentration near the poles of the magnet and its surroundings. To conduct a simulation, we used common software such as COMSOL Multiphysics (version 5.5), which can deal with complex magnetic fields and flux patterns. First, for the shape of the magnets, a rectangle was chosen, and properties such as remanence (Br) and coercivity (Hc) of the magnets were specified. Boundary conditions of the simulation space that contain a magnetic field were defined. The results exhibit the magnetic flux density across the defined geometry in the form of flux lines, which shows the concentration of magnetic flux and its direction of flux flow. To analyze the magnetic flux density across the number of magnet pairs, the mathematical expressions are discussed for a cylindrical Halbach configuration.  
In the PM cylindrical Halbach configuration, magnetic flux density influenced by the number of magnets in the target regions can be estimated by the following expression [19]:  
\[\mathrm{B}=\mathrm{B}_{\mathrm{r}}\left(1-\mathrm{e}^{-krN}\right) \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value Range (mm)** \\ \hline Length of smaller magnet (\(L\)) (\% of length & 5 (25\%), 10 (50\%), 15 (75\%), 20 (100\%) \\ compared to larger magnets) & 5 (25\%), 10 (50\%), 15 (75\%), 20 (100\%) \\ \hline Gap width between the magnets (\(W\)) & 5–10 \\ \hline Position of the smaller magnets (\(P_{1}\) and & 5–5 \\ \(P_{2}\)) from center of larger magnet axis & 5–5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Design variable for simulation.  
Figure 5: Variation of the adjustable track: (**a**) \(L\)—length of the magnets, (**b**) \(W\)—gap width between the magnets, (**c**) \(P_{1}\) and \(P_{2}\)—position of the magnets relative to stationary magnets in a Halbach structure.  
Br is the remanence (residual magnetic flux density) of each magnet. \(k=\frac{2\pi}{\lambda}\) is the angular wave number. The spatial wavelength \(\lambda\) is the distance over which the magnetization direction completes one full rotation of \(360^{\circ}\), after which the pattern repeats. In cylindrical configuration, \(r\) is the inner radius, which is the distance from the center to the inner edge of the magnets, and \(N\) is the number of magnets in the array. By increasing \(N\), the array creates a more gradual and continuous transition of magnetization across the Halbach configuration, which enhances the field uniformity. Furthermore, the practical limits of the material properties, such as magnet remanence and coercivity, begin to dominate, and the magnetic flux density on the strong side approaches a saturation value. After performing the parametric analysis of \(N\) and magnet dimensions to quantitatively evaluate their impact on the magnetic flux density and performance of the Halbach array, ultimately, optimizing the number of magnets while considering these physical constraints ensures the best performance from the Halbach array.