# Design of a New Bilayer Multipole Electromagnetic Brake System for a Haptic Interface
## 4 FEM-Based Magnetostatics Simulations  
The performance of the optimal design of BLMP brake has been validated using finite element method (FEM) based simulation software, named COMSOL Multiphysics. The magnetostatics simulation was carried in a 3-dimensional brake model. A magnetic material with a known magnetization curve was selected for the core domain surrounded by the air with relative permeability equal to 1. The same value of DC current was supplied to all the multi-turn solenoid coil domains of individual layer that result in the generation of magnetic flux moving along the magnetic circuit of Figure 3. Upon controlling the actuation of each layer individually, there is a superposition of magnetic flux in the armature domain that connects the flux of each pole with surrounding poles. The direction of magnetic flux has been shown in Figure 7, when applying a current of 1 A to each coil of the layer. Due to the actuation of inner layer only, the magnetic flux flows through the inner poles of the brake due to the shortest distance in between them, as shown in Figure 7a. Because of having a greater number of turns and thicker cross-section poles at the outer layer, the magnetic flux connects both the inner and outer layer poles when actuating the outer layer only, and further strengthens the same path when the combination of two layers is actuated, as shown in Figure 7b,c, respectively. The simulation results validate the proposed arrangement of solenoid coils in the brake design and the sequence of actuating individual layer to maximize the distribution of magnetic flux in the material.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Design** & \(\theta_{C}\) & \(R_{i_{\text{min}}}\) & \(R_{i_{\text{max}}}\) & \(R_{o_{\text{min}}}\) & \(R_{o_{\text{max}}}\) & \(w_{i}\) & \(w_{o}\) & \(f(x)\) \\
**Variables** & (deg) & (mm) & (mm) & (mm) & (mm) & (mm) & (mm) & (mm) & \(f(x)\) \\
**Initial values** & 15 & 14 & 21.2 & 25.3 & 32.7 & 1.7 & 1.7 & 0.3 \\
**Optimal values** & 17 & 15 & 21 & 30 & 35 & 2.5 & 3.5 & 0.69 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Initial and optimal values of design variables for the new bilayer multipole (BLMP) brake.  
Figure 6: Design evolution during optimization procedure. (**a**) Initial design; (**b**) optimal design.  
The thick cross-sectional poles help in generating larger magnetic flux densities proportional to the input solenoid current unless the magnetic material gets saturated. The presence of large number of tiny air gaps in the magnetic circuit can delay the saturation stage for higher values of input currents; therefore, a wide range of magnetic forces is available. For an air gap of 0.1 mm, a comparison of magnetic flux density distribution is shown in Figure 8 when applying a current of 1 A to each coil of the layer. The simulation results describe the uniform distribution of magnetic flux densities along the poles and the armature during all the actuation sequences, and hence, validate the optimal design of BLMP brake.  
Figure 9 shows a comparison of the analytical model of a brake with simulation results, wherein the magnetic forces are plotted as a function of air gap during actuation of the inner and combined layer for an input current of 1 A to each coil. The result shows that a gradual pulling force can be achieved from the brake design by controlling the current in each layer, sequence of actuation of the layers and the air gap between the armature and the friction pad.  
Figure 8: Magnetic flux density distribution along the cross-section of design with the actuation of the (**a**) inner layer only; (**b**) the outer layer only; (**c**) a combination of the inner and outer layers.  
Figure 7: Flow direction of magnetic flux in the armature domain with the actuation of (**a**) Inner layer only; (**b**) Outer layer only; (**c**) Combination of inner and outer layer.