# Design of a New Bilayer Multipole Electromagnetic Brake System for a Haptic Interface
## 2 Modelling of a Magnetic Brake
### Magnetic Force
\begin{split} A_{C_{1}}=A_{C_{2}}=2(\text{Sector oab}-\Delta \text{ocd})=2\Big{(}\frac{1}{2}r_{\text{i}_{\text{max}}}^{2}\partial_{C}- \frac{1}{2}r_{\text{i}_{\text{min}}}^{2}\tan\theta_{C}\Big{)}\\ A_{C_{3}}=A_{C_{4}}=2(\text{Sector oef}-\Delta\text{ogh})=2\Big{(} \frac{1}{2}r_{\text{o}_{\text{max}}}^{2}\theta_{C}-\frac{1}{2}r_{\text{o}_{ \text{min}}}^{2}\tan\theta_{C}\Big{)}\\ A_{1,2}=A_{2,1}=\Big{(}r_{i\text{max}}-r_{i\text{min}}\Big{)}t_{C} \\ A_{3,4}=A_{4,3}=\Big{(}r_{\text{o}_{\text{max}}}-r_{\text{o}_{ \text{min}}}\Big{)}t_{C}\\ A_{1,3}=A_{3,1}=A_{2,4}=A_{4,2}=\frac{\theta_{5}}{2}r_{\text{e} }r_{\text{e}}r_{\text{e}}\\ A_{g_{i}}=A_{C_{i}}f_{i}\text{,}\end{split}  \tag{4}\]  
where \(r_{i\text{min}}\), \(r_{i\text{max}}\), \(r_{\text{o}_{\text{min}}}\) and \(r_{\text{o}_{\text{max}}}\) are the minimum and maximum radial dimensions of inner and outer poles, respectively. \(\theta_{C}\) and \(\theta_{S}\) are the angular dimensions of the pole cross-section and the symmetry sector, respectively. The parameter \(t_{c}\) defines the thickness of the armature and the bottom of the core body as shown in Figure 2b. The average radial distance between inner and outer poles is given by \(r_{av}=\big{(}r_{i\text{min}}+r_{i\text{max}}+r_{\text{o}_{\text{min}}}+r_{ \text{o}_{\text{max}}}\big{)}/4\). The factor \(f_{i}=1+\big{(}L_{g_{i}}\ln\big{(}2L_{C_{i}}/L_{g_{i}}\big{)}/\sqrt{A_{C_{i}}} \big{)}\) considers the losses due to fringing effect of magnetic flux in the air gap and depends on the geometry of the magnetic pole [18]. The parameters \(L_{C_{i}}\), \(L_{g_{i}}\), \(L_{i,j}\) and \(L_{j,i}\) are the effective lengths of the pole, air gap, armature and base part of the magnetic circuit, respectively, as calculated from Figures 2b and 4 as  
\[ \begin{split} L_{1,2}=L_{2,1}=\theta_{S}\bigg{(}\frac{r_{i\text{ min}}+r_{i\text{max}}}{2}\bigg{)}\\ L_{3,4}=L_{4,3}=\theta_{S}\bigg{(}\frac{r_{i\text{min}}+r_{i \text{max}}}{2}\bigg{)}\\ L_{1,3}=L_{3,1}=L_{2,4}=L_{4,2}=\bigg{(}\frac{r_{\text{o}_{\text{ min}}}+r_{\text{o}_{\text{max}}}}{2}\bigg{)}-\bigg{(}\frac{r_{i\text{min}}+r_{i \text{max}}}{2}\bigg{)}\text{.}\end{split}  \tag{5}\]  
The nodal analysis at node A of magnetic circuit in Figure 3 yields that  
\[\Phi_{1}=\Phi_{1,2}+\Phi_{1,3}+\Phi_{1,10}. \tag{6}\]Because of the structural symmetry, 1,10 is same as 1,2, and therefore, Equation (6) can be re-written into  
\[\Phi_{1}=2\Phi_{1,2}+\Phi_{1,3}. \tag{7}\]  
Similarly, the nodal analysis of magnetic flux at nodes B, C, D, E, F, G and H, respectively, yields  
\[ \begin{gathered}\Phi_{2}=2\Phi_{1,2}+\Phi_{4,2},\ \Phi_{1}=2\Phi_{2,1}+\Phi_{3,1},\ \Phi_{2}=2\Phi_{2,1}+\Phi_{2,4}\\ \Phi_{3}=2\Phi_{4,3}+\Phi_{1,3},\ \Phi_{4}=2\Phi_{4,3}+\Phi_{4,2}\\ \Phi_{3}=2\Phi_{3,4}+\Phi_{3,1},\ \Phi_{4}=2\Phi_{3,4}+\Phi_{2,4}. \end{gathered}  \tag{8}\]  
The linear Equations (2), (7) and (8) can be solved for the unknown magnetic flux in each part of the magnetic circuit. The magnetic energy stored in the volume \(V_{g_{i}}=L_{g_{i}}A_{g_{i}}\) of the air-gap between the poles and the armature is given by the relationship  
\[W_{mag}=\frac{1}{2}\sum_{i=1}^{4}B_{g_{i}}H_{g_{i}}V_{g_{i}}, \tag{9}\]  
where \(B_{g_{i}}\) and \(H_{g_{i}}\) are the magnetic flux density and magnetic field strength in the air gap of \(i^{th}\) pole, respectively, and are related by the relation \(H_{g_{i}}=B_{g_{i}}/\mu_{0}\). The magnetic flux continuity relationship \(\Phi_{i}=B_{C_{i}}A_{C_{i}}=B_{g_{i}}A_{g_{i}}\) tells that the flux passing through the core and air gap is the same, hence, (9) can be re-written in the form as  
\[W_{mag}=\frac{L_{g_{i}}}{2\mu_{0}}\sum_{i=1}^{4}\left(\frac{\Phi_{i}^{2}}{A_{g _{i}}}\right). \tag{10}\]  
The symmetry section of magnetic brake allows only a half of the magnetic flux \(\Phi_{i}/2\) to pass through a half of the cross-sectional area \(A_{g_{i}}/2\) of each air gap. This magnetic flux observes a variation in  
Figure 4: Nomenclature of radial dimensions for one symmetry section.