## 2 Static Performance Optimization of Solenoid Electromagnetic Actuator
### Structure and Simulation Model
The numerical relationship between \(H\) and the magnetic flux density \(B\) and the magnetic flux \(\Phi\) in the cross-sectional area of the yoke are obtained as follows:  
\[\overset{\rightarrow}{B}=\mu\overset{\rightarrow}{H} \tag{2}\]  
\[\phi=\overset{\rightarrow}{B}S_{e} \tag{3}\]  
\[S_{e}=\pi(\mathrm{f}1)^{2}-\pi(\mathrm{f}1-\mathrm{t})^{2} \tag{4}\]  
where \(\mu\) is the magnetic permeability of the material, \(S_{e}\) is the equal cross-sectional area of the yoke in the 2D model and t is the equal thickness of the yoke in the 2D model. Meanwhile, the cross-sectional area \(S_{e}\) is equal to the area of the yoke in the 3D model (45-mm width and 50-mm height rectangle with a hole of 21-mm radius). All components of the LPS in the 2D model are cylindrical axisymmetric structures, but the yoke of the LPS in the 3D model is a cube structure. A simplified 2D model is established, which has equal yoke cross-sectional area with the 3D model. From the basic theoretical Equations (1)-(3), it can be obtained that \(H\) in the yoke of these two models are equal in the same simulation condition. Moreover, \(B\) in the yokes are also equal by the same material parameter \(\mu\). Thus, when the cross-sectional area \(S\) of the two models are equal, the magnetic flux \(\Phi\) in the yoke of the two models will equal and the simplified 2D model will be considered suitable. The 2D-magnetostatic field simulator solver computes are derived from [PERSON]'s law and [PERSON]'s differential equation:  
\[\
abla\times\overset{\rightarrow}{H}=\overset{\rightarrow}{J} \tag{5}\]  
Figure 4:0  
Figure 4:1  
Figure 4:2  
where \(J\) is the current density of the direct current excitation of the coil. \(A\) is the magnetic vector potential. To calculate the magnetostatic field result of the 2D model and the 3D model, ANSYS MAXWELL (ANSYS, Inc., Canonsburg, PA, USA, 2019, version: ANSYS Electronics suite 2019R2) software is employed in this study. These simulation model are calculated by a workstation with two CPU (Intel E5-2650v2) and 32 GB DDR3 memory. The 2D FEA model with 3.6-mm air gap (A1) has 45,732 elements and the calculation time cost is 1 min and 30 s and the workstation can calculate 30 tasks at same time. While, the 3D FEA model with 3.6-mm air gap (A1) has 746,669 elements, and the calculation time cost is 31 min and 49 s, and the workstation only calculate one task. Therefore, replacing 3D FEA model with 2D FEA model can reduce the time cost by more than 99.8%. The boundaries of the 2D FEA model is balloon boundary and the boundaries of the 3D FEA model is zero tangential H field boundary. The current density of the coil excitation is \(5.56\times 10^{6}\) A/m\({}^{2}\). This study calculated a series of FEA model with different air gap (A1), which are distinguished by the interval of the 0.2-mm air gap. The FEA model use the conventional values of each components and pure iron as the soft magnetic material to calculate the static electromagnetic force of the armature. The property of pure iron are shown in Figure 4:3 The absolute values of the static electromagnetic force of the armature are shown in Figure 4a. In Figure 4a, the static force versus displacement curves of the experimental result, the 3D model simulation result and the 2D model simulation result of conventional LPS are illustrated. The push force of the 2D FEA model in Figure 4a is the armature static electromagnetic force along the negative direction of the Z coordinate axis in Figures 2 and 4c. Moreover, push force of the 3D FEA model in Figure 4a is the armature static electromagnetic force along the negative direction of Y coordinate axis in Figure 4b. By the comparison of these curves, the 2D model has same accuracy as the 3D model. It also can be confirmed that both 2D and the 3D model can guarantee equal accuracy on magnetostatic force calculation. In Figure 4b,c, the flux density field of the 3D model and the flux density field of the 2D model are illustrated, respectively.