## 2 Static Performance Optimization of Solenoid Electromagnetic Actuator  
### Structure and Simulation Model  
The structure parameters are the main factors of the static performance optimization. In order to simplify the complexity of optimization, this study only optimizes the parameters of the core components such as armature, stator, coil, yoke and non-magnetic ring. Moreover, a series of magnetostatic models are needed to describe the force versus displacement curve of the LPS static performance. These models are distinguished by different air gap, symbol A\({}_{1}\) in Figure 2, between armature and the stator. Moreover, the air gap of each model is expressed in 0.2-mm intervals. Therefore, at least 14 calculation models are required for entire working stroke (from 1-mm air gap to 3.6-mm air gap). The holes in the armature include shaft hole and the two oil through holes. Oil-through-hole is usually used to introduce oil and reduce moving mass of the LPS. In the 3D models, the influence of these two holes is considered in Figure 3 and the result is shown in Figure 4b. Considering the calculation time of the 3D model and the number of calculation models, a simplified two-dimensional (2D) model was used for structure parameter optimization. The description of this cylindrical axisymmetric 2D model (about axis Z) is shown in Figure 2a. Moreover, the mesh of the 2D FEA model is shown in Figure 2b. Figure 1: Overall description of this performance optimization strategy. Figure 4: \(Cont\). Figure 3: Geometry structure definition of the LPS. Figure 2: Simplified 2D finite element analysis (FEA) model of linear proportional solenoid (LPS). (**a**) geometry structure and the design parameters; (**b**) the mesh of the 2D FEA model. The definition of the LPS shape parameters are shown in Figure 2a. All parameters can be divided into four parameter arrays. Parameter array \(P_{\text{armature}}\){a1, a2, a3, a4} represents the design parameters of the armature, which includes the length and radius of the armature and the length of return spring hole and the thickness of the spring sleeve. Parameter array \(P_{\text{yoke}}\){y1, y2, y3, y4, y5, y6} includes the design parameters of the stator design parameter y1, the non-magnetic ring design parameters from y2 to y5 and the yoke parameter y6--thickness of the armature sleeve. Parameter array \(P_{\text{coil}}\){c1, c2} represents the length and thickness of the coil winding. Other fixed shape parameters (radius and the length of the 2D model, height of the stator and the three thickness parameters of the bobbin) are contained in array \(P_{\text{fixed}}\){f1, f2, f3, f4, f5, f6}. The conventional values of the shape design parameters array are \(P_{\text{armature}}\){31.4 mm, 9.02 mm, 16.9 mm, 4.5 mm}, \(P_{\text{yoke}}\){5.5 mm, 3.7 mm, 3 mm, 3.9 mm, 1 mm, 2 mm}, \(P_{\text{coil}}\){8.2 mm, 41.5 mm} and \(P_{\text{fixed}}\){26.8 mm, 67.5 mm, 23 mm, 1.4 mm, 1.1 mm, 1.2 mm}. The theoretical Equations of this simplified 2D model are shown as below. The magnetic field strength \(H\) is obtained by the [PERSON]'s law:  
\[\overset{\rightarrow}{H}=\frac{\overset{\rightarrow}{i}N}{\text{c}_{1}} \tag{1}\]  
Figure 4: Verification of simplified 2D FEA model: (**a**) push force comparison; (**b**) flux density field result of the 3D FEA model; (**c**) flux density field result of the 2D FEA model. where \(i\) represents coil excitation direct current, \(N\) is the number of turns of the coil, c\({}_{1}\) represents the length of flux path in the cross-sectional area of the yoke.