## 4 Design and Analysis of Magnetic Fields
### Simplified Model of the Magnetic Circuit  
According to the magnetic Ohm's Law [19], we performed a magnetic field analysis of the working gap of the transmission device. The simplified equivalent model of the transmission device and the equivalent model of the magnetic circuit of radii \(r_{1}\)-\(r_{7}\), the widths \(L_{1}\)-\(L_{4}\), magnetoresistances \(R_{1}\)-\(R_{9}\), etc., are shown in Figure 4 and Table 2. \begin{table}
\begin{tabular}{c c c c} \hline
**Volume Fraction/\%** & **Additives Ratio/\%** & **Base Fluid** & **Zero Field Viscosity/Pa-s** \\ \hline
5 & 1.4 & silicone oil & 0.12 \\ \hline \end{tabular}
\end{table} 
Table 1: Magnetorheological fluid (MRF-1) performance parameters. Figure 3: Rheological characteristic curve of MRF. (**a**) Magnetic curve of MRF-1, (**b**) characteristics of viscosity. The input disk, output disk and support shell of the MR drive are made of Steel-1020, the material of the drive shaft and driven shaft are Q235, the surrounding is set as an air boundary. and the thickness of the working gap is \(h\). The thickness of the air gap _h_\({}^{\prime}\) at \(R_{3}\) and \(R_{8}\) was 0.35 mm. According to the magnetic Ohm's Law, the magnetomotive force \(F_{\rm m}\) generated by the excitation coil is [(20) (pp. 256-257)]  
\[F_{\rm m}=NI \tag{27}\]  
where \(N\) is the number of turns of the coil (t) and \(I\) is the current passed into the excitation coil (A). The ratio of the magnetomotive force \(F_{\rm m}\) to the magnetic flux \(\Phi\) is defined as the magnetic reluctance \(R\). From Figure 4, the magnetic circuit can be divided into a total of eight parts, labelled 1-9 [(19)], and the magnetoresistance of each part is  
\[ \begin{array}{llll}R_{1}=f_{4}^{r_{1}}\frac{dr}{2\pi r\mu_{1}L_{1}}=\frac{1 }{2\pi r\mu_{1}L_{1}}\ln\frac{r_{1}}{r_{4}}&R_{2}=\frac{L_{1}}{\pi\mu_{1}(r_ {1}^{2}-r_{2}^{2})}&R_{3}=\frac{h^{\prime}}{\pi\mu_{0}\left(r_{1}^{2}-r_{2}^{ 2}\right)}\\ R_{4}=\int_{r_{6}+L_{5}}^{r_{7}}\frac{dr}{2\pi r\mu_{1}L_{3}}=\frac{2\pi r\mu_{1 }L_{3}}{2\pi r\mu_{1}L_{3}}\ln\frac{r_{7}}{r_{6}+L_{5}}&R_{5}=\frac{h^{\prime} }{\pi\mu_{2}\left(r_{2}^{2}-r_{3}^{2}\right)}&R_{6}=\int_{r_{4}}^{r_{1}}\frac{ dr}{2\pi r\mu_{1}L_{3}}=\frac{1}{2\pi r\mu_{1}L_{3}}\ln\frac{r_{1}}{r_{4}}\\ R_{7}=\int_{r_{4}}^{r_{6}}\frac{dr}{2\pi r\mu_{1}L_{3}}=\frac{1}{2\pi r\mu_{1}L_ {3}}\ln\frac{r_{6}}{r_{4}}&R_{8}=\frac{h^{\prime}}{\pi\mu_{0}\left(r_{5}^{2}-r _{4}^{2}\right)}&R_{9}=\frac{L_{1}}{\pi\mu_{1}\left(r_{5}^{2}-r_{4}^{2}\right) }\end{array}  \tag{28}\]  
where \(\mu_{1}\) and \(\mu_{2}\) are the absolute permeability of Steel-1020 and MRF, respectively, and \(\mu=\mu_{0}\mu_{r}\). The total magnetoresistance of the magnetic circuit \(R_{\rm M}\) is obtained by summing up the magnetoresistance of each segment as follows  
\[R_{\rm M}=R_{1}+R_{2}+R_{3}+R_{4}+R_{5}+R_{6}+R_{7}+R_{8}+R_{9} \tag{29}\]  
Therefore, the total magnetic flux of the whole magnetic circuit is  
\[\phi=\frac{NI}{R_{\rm M}} \tag{30}\]  
Figure 4: Equivalent model of transmission device. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(r_{1}\) & 135 & \(r_{7}\) & 125 \\ \(r_{2}\) & 129 & \(L_{1}\) & 10 \\ \(r_{3}\) & 79 & \(L_{2}\) & 26 \\ \(r_{4}\) & 73 & \(L_{3}\) & 12 \\ \(r_{5}\) & 83 & \(L_{4}\) & 25 \\ \(r_{6}\) & 92 & \(L_{5}\) & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main dimensions of transmission device (mm). According to Kirchhoff's Law, the magnetic flux density in the working gap is  
\[B=\frac{\phi}{A}=\frac{\phi}{\pi\big{(}r_{2}^{2}-r_{3}^{2}\big{)}} \tag{31}\]  
Substituting Equation (30) into Equation (31), the relationship between magnetic induction and current can be obtained as  
\[B(I)=\frac{NI}{\pi R_{\text{M}}\big{(}r_{2}^{2}-r_{3}^{2}\big{)}} \tag{32}\]  
From the conversion relationship between magnetic flux density \(B\) and the magnetic field strength \(H\)[20] (pp.