## 4 Design and Analysis of Magnetic Fields
### Analysis of Magnetic Field  
From Equations (26) and (33), it can be seen that the magnetic field intensity is related to the current, the working gap size, and the permeability of materials. The calculations were performed for the above variables to determine the magnetic field strength in the working gap. The minimum magnetic field intensity \(H_{\text{min}}\) = 67.82 kA/m within the MRF gap under a rated load was obtained by bringing \(\tau\) = 38 kPa into Equation (26). Assuming that no magnetic saturation occurs in all parts of the MR drive, and the materials of the input disc, output disc and support shell are Steel-1020, the relative permeability of the housing and MRF are 1200 and 5, respectively. From Equation (29), the magnetoresistance is 1.06 \(\times\) 10\({}^{5}\) H\({}^{-1}\). The design value of the coil parameter _NI_ is 1500 A\(\cdot\)t for a magnetic induction intensity \(H\) greater than 67.82 kA/m in the working gap of the MRF. The relative magnetic permeability \(\mu_{\text{r}}\) of the material changes with the magnetic field intensity \(H\) when the current of the excitation coil changes. In order to improve the accuracy of the magnetic field analysis, the Finite Element Method (FEM) was used to simulate the magnetic field in the whole working gap. Set the number of turns of the coil \(N\) = 500 t, the coil was placed through the currents from 1-4 A, respectively. The 2D axisymmetric model in Figure 4 was divided into 8909 cells using a 2D triangular mesh. We analyzed the influence of the copper gaskets on the transmission using the FEM, obtaining the results shown in Figures 6 and 7. As presented in Figures 6 and 7, the overall magnetic flux distribution within the working gap of the MR transmission is uniform, and the working gap can be divided into four segments according to the variations in the magnetic field intensity. When the working gap radial distance \(r\) is from 0 to 5 mm, the magnetic field intensity increases rapidly to extremum with increases to the radial length. As the working gap radial distance is located in the two segments of \(r\) = 5-20 mm and \(r\) = 30-45 mm, the magnetic field intensity curve changes relatively smoothly and the magnetic field is more stable. With the radial distance from 20 mm to 30 mm, the magnetic field intensity curve decreases rapidly to 10 kA/m in the packing seal barrier interval. While \(r\) ranges from 45 to 50 mm, the magnetic field declines exponentially from the stable value. In the absence of copper gaskets, the magnetic field in the working gap of the MRF on both sides is asymmetric. The maximum values of magnetic field intensity were 29.51 kA/m, 42.62 kA/m, 49.47 kA/m and 54.67 kA/m when the currents were 1.0 A, 2.0 A, 3.0 A and 4.0 A, respectively. The differences between the maximum values of the magnetic field strength on both sides were 3.28 kA/m, 3.87 kA/m, 4.0 kA/m and 6.1 kA/m, respectively. The magnetic field strength increased gradually with the increase of current, but the magnetic field was not uniform on both sides. The magnetic field in the working gap was more uniform and the mean value was larger when the copper gaskets are added, and the maximum values of the magnetic field intensity were 43.0 kA/m, 59.26 kA/m, 67.25 kA/m and 73.74 kA/m when the currents were 1.0 A, 2.0 A, 3.0A and 4.0 A, respectively. The differences between the maximum values of the magnetic field intensity on both sides were 0.25 kA/m, 0.29 kA/m, 0.05 kA/m and 0.03 kA/m. The magnetic field intensity increased gradually with the increase of coil current and it reached the design requirement at 3A. Furthermore, the magnetic field strength increased by 45.71%, 39.04%, 35.94% and 34.88%, compared to the values obtained without the isolation washer, respectively, when the coil current was the same. In summary, for optimal torque characteristics, the working gap thickness of MRF \(h\) is 1.0 mm, the number of turns of the coil \(N\) is 400 t, and the current \(I\) is 0-3 A.