# Design of a New Bilayer Multipole Electromagnetic Brake System for a Haptic Interface
## 2 Modelling of a Magnetic Brake  
### Magnetic Force  
The brake contains a stator in which twelve electromagnetic poles are axially arranged in two layers. A solenoid coil is wound around each pole having a polarity opposite to its immediate neighboring poles. Six friction pads made from high friction coefficient metals are placed in spaces between these poles. A disc-shaped armature of magnetic material is placed coaxial to the stator with a tiny air gap between the two bodies, as shown in Figure 1. Because of the symmetry about the axis normal to the surface of the stator, the design is divided into six sections. Every symmetry section contains four poles with half of the actual cross-sectional area: two from inner layer and two from outer layer, as shown in Figure 2a. A schematic of a magnetic circuit is shown in Figure 3, which consists of closed loops connecting four solenoid coils through reluctances \(\mathcal{R}\) and flux \(\Phi\) of the magnetic path. Figure 1: Design of two-layer, 12-pole axially-actuated electromagnetic brake: (**a**) magnetic core poles surrounded by solenoid coils and the friction pad supported by poles; (**b**) rotary armature with a spring-connected disc. Figure 3: Schematic diagram of flux path interaction for one symmetry section. Figure 2: Layout of a symmetric section: (**a**) isometric view showing two inner and two outer poles; (**b**) side view showing a closed-path of magnetic flux. With \(N\) turns of the solenoid coil and \(I\) ampere current through the windings, the magneto-motive force \(\mathcal{F}\) for each pole is given by  
\[\mathcal{F}=NI=\mathcal{R}\Phi. \tag{1}\]  
Analyzing the [PERSON]'s circuit law in each closed loop of Figure 3, the superpositions of magnetic flux generated by the magneto-motive force in loops ABDCA, AEGCA, BDHFB and FEGHF, respectively, are given by the relations. \[ \begin{split}\mathcal{F}_{1}+\mathcal{F}_{2}&= \mathcal{R}_{\text{e1}}\[PERSON]_{1}+\mathcal{R}_{\text{e2}}\[PERSON]_{2}+\mathcal{R}_{ \text{1,2}}\[PERSON]_{1,2}+\mathcal{R}_{\text{2,1}}\[PERSON]_{2,1}\\ \mathcal{F}_{1}+\mathcal{F}_{3}&=\mathcal{R}_{\text{e1 }}\[PERSON]_{1}+\mathcal{R}_{\text{e3}}\[PERSON]_{3}+\mathcal{R}_{\text{1,3}}\[PERSON]_{1,3}+ \mathcal{R}_{\text{3,1}}\[PERSON]_{3,1}\\ \mathcal{F}_{2}+\mathcal{F}_{4}&=\mathcal{R}_{\text{e2 }}\[PERSON]_{2}+\mathcal{R}_{\text{e4}}\[PERSON]_{4}+\mathcal{R}_{\text{2,4}}\[PERSON]_{2,4} +\mathcal{R}_{\text{4,2}}\[PERSON]_{4,2}\\ \mathcal{F}_{3}+\mathcal{F}_{4}&=\mathcal{R}_{\text{ e3}}\[PERSON]_{3}+\mathcal{R}_{\text{e4}}\[PERSON]_{4}+\mathcal{R}_{\text{3,4}}\[PERSON]_{3,4}+ \mathcal{R}_{\text{4,3}}\[PERSON]_{4,3},\end{split}  \tag{2}\]  
where \(\mathcal{R}_{\text{e}i}\) is the equivalent reluctance of magnetic core and air gap in the _ith_ pole. \(\mathcal{R}_{i,j}\) and \(\mathcal{R}_{j,i}\) are the reluctances of armature and the base, respectively, and can be calculated using Figure 2b as  
\[\mathcal{R}_{ei}=\frac{L_{C_{i}}}{\mu_{0}\mu_{r}A_{C_{i}}}+\frac{L_{g_{i}}}{ \mu_{0}A_{g_{i}}}\text{, }\mathcal{R}_{i,j}=\frac{L_{i,j}}{\mu_{0}\mu_{r}A_{i,j}}\text{, }\mathcal{R}_{j,i}=\frac{L_{j,i}}{\mu_{0}\mu_{r}A_{j,i}}\text{,} \tag{3}\]  
where \(\mu_{0}\) and \(\mu_{r}\) are the permeability of air and relative permeability of magnetic material, respectively. The parameters \(A_{C_{i}}\), \(A_{g_{i}}\), \(A_{i,j}\) and \(A_{j,i}\) represent the cross-sectional areas of the pole, air gap, armature and base part of the magnetic circuit, respectively and their values can be calculated from the dimensional parameters illustrated in Figures 2b and 4 as. \[