## 3 Analysis of Torque
### The Effect of the Gap Shape on the Magnetic Circuit  
To investigate the effect of the gap shape on the magnetic circuit, a transmission drive with the same radial planar disc working gap size was used for comparison. \(r_{\text{a}}\) is the arc radius, \(\alpha\) is the arc circle angle, \(N\) is the number of turns of the excitation coil, and \(I\) is the coil current. The magnetic field distribution in both gaps when both the arc radius \(r_{\text{a}}\) and the working gap thickness \(h\) are 1 mm and the number of ampere-turns \(NI\) is 300 A is shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(R_{1}\) & \(R_{2}\) & \(R_{3}\) & \(R_{4}\) & \(R_{5}\) & \(L_{1}\) & \(L_{2}\) \\ \hline
87 & 77 & 65 & 35 & 25 & 15 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of the transmission device (mm). Figure 3: Material characteristic curve of the magnetorheological fluid (MRF-5). (**a**) Relationship between temperature, magnetic flux density, and shear yield stress; (**b**) magnetic curve. The upper and lower surfaces are the contact surfaces between the output housing, the input disc, and the MRF, respectively. We use the magnetic flux density in Region 2 as an example of the effect of the gap shape on the magnetic field. From Region 2 in Figure 5a,b, it can be found that the magnetic flux density distribution of the semi-circular gap is more uneven compared with the planar gap. The magnetic resistance at the transition of the two arcs is low. It can be observed from the magnetic boundary conditions that the direction of the magnetic flux density will be deflected after it passes through different magnetic materials [18] (pp. 252-253). The relationship between the deflections \(\theta_{1}\) and \(\theta_{2}\) of the magnetic flux density on both sides and the relative magnetic permeability of the material is  
\[\frac{\tan\theta_{1}}{\tan\theta_{2}}=\frac{\mu_{r1}}{\mu_{r2}} \tag{2}\]  
where \(\mu_{r1}\) and \(\mu_{r2}\) are the relative magnetic permeabilities of the materials on each side, respectively. The magnetic field boundary conditions are illustrated in Figure 6.  
Figure 4: Magnetic properties of the materials. Figure 5: Magnetic field distribution in the two working gaps. (**a**) Semi-circular gap; (**b**) planar gap; (**c**) schematic diagram of the gap shape. As can be seen in Figure 5, the magnetic saturation region appears only near the cusp of the model. However, there is no magnetic saturation in either the Steel-1010 or the Steel-1030 in the entire working gap region. When the magnetic field from the Steel-1010 passes through the MRF, \(\mu_{r1}\) and \(\mu_{r2}\) are 620 and 5 (\(NI=300\) A), respectively. Bringing the relative magnetic permeability into Equation (2) yields \(\theta_{2}\) as follows:  
\[\theta_{2}=\arctan(\frac{5\tan\theta_{1}}{620}) \tag{3}\]  
From Equation (2), the magnetic field will deflect at a lower inclination angle \(\theta_{2}\) when it crosses the boundary at any angle (except 90\({}^{\circ}\)), and the same applies to the deflection that occurs when the magnetic field penetrates the MRF gap. According to the Flux Continuity Theorem [18] (pp. 255-257), the magnetic induction line will pass through the path of least magnetic resistance. That is, the magnetic induction within the working gap is in the same direction as the radius of the lower surface arc. Define \(h_{z}\) as a coordinate from 0 to \(h=1\) mm along the normal axis, which indicates the distance between the arc plane where the MRF is located and the input disc. The relationship between the magnetic flux density at different thicknesses \(h\) within the MRF gap and the circular center angle \(\alpha\) of the arc is shown in Figure 7.  
As can be seen from Figure 7a, the magnetic flux density at the transition of the arc increases as \(h_{z}\) increases when the current is the same, which is the opposite trend to that at the center of the arc.