## 2 Static Performance Optimization of Solenoid Electromagnetic Actuator
### Simulation and the Measurement Results  
All simulation resulted in this study were calculated by ANSYS MAXWELL (ANSYS, Inc., Canonsburg, PA, USA, 2019, version: ANSYS Electronics suite 2019R2). To ensure the accuracy of the measurement results, ZLDS100 (ZSY GROUP LTD) laser displacement sensor and U9C (HBM) force transducer were employed in the performance test device. The structural details of the test device are shown in Figure 8a; the experimental setup is shown in Figure 8b. For static performance test, a series of the static push-force versus displacement curves are depicted in Figure 9. By the comparison of the manufactured optimal LPS and conventional LPS in different coil excitation current, it can be seen that the static performance of the optimal LPS showed a high force generation and a well constant performance in low coil current. When the coil current was rated current (1980 ampere-turn), the numerical calculation error will increase with the influence of the magnetic saturation phenomenon in the soft magnetic material of the LPS. The comparison of the design objectives is shown in Figure 10. The average force shown in Figure 10a was calculated by the measured values in Figure 9. The results indicate that the optimal-type LPS can generate more push force than the conventional type. Moreover, the standard deviation of the force versus displacement curves is shown in Figure 10b. In high magnetomotive force condition (1980 and 1587 ampere-turn), the push force of the conventional-type LPS showed a better constant performance. However, in low magnetomotive force condition (1188, 792 and 396 ampere-turn), the optimal-type LPS showed a better constant performance. Moreover, considering the same push force output, the optimal-type LPS was more stable. As shown in Figure 10c, it can be seen that the optimal-type LPS showed a better force output efficiency. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Key Parameter and Key** & **Range of the Design** & **Conventional** & \multirow{2}{*}{**Optimal Type LPS**} & **Electrical Steel** \\
**Characteristic** & **Parameter (mm)** & **Type LPS** & & **Optimal Type LPS** \\ \hline length of the armature—a1 & 25 mm–34.4 mm & 31.4 mm & 32.3 mm & 33.2 mm \\ radius of the armature—a2 & correlated to \(\mathcal{\text{c}}\)2 and y6 & 9.02 mm & 9.65 mm & 10.15 mm \\ depth of the spring & \multirow{2}{*}{4 mm–20 mm} & \multirow{2}{*}{16.9 mm} & \multirow{2}{*}{4.7 mm} & \multirow{2}{*}{8 mm} \\ hole—a3 & & & & \\ thickness of the spring & \multirow{2}{*}{3 mm–8 mm} & \multirow{2}{*}{4.5 mm} & \multirow{2}{*}{5.2 mm} & \multirow{2}{*}{4.5 mm} \\ hole sleeve—a4 & & & & \\ thickness of the stator—y1 & \multirow{2}{*}{4 mm–8 mm} & \multirow{2}{*}{5.5 mm} & \multirow{2}{*}{5.5 mm} & \multirow{2}{*}{4.9 mm} \\ non-magnetic ring—y2 & \multirow{2}{*}{3.5 mm–3.9 mm} & \multirow{2}{*}{3.7 mm} & \multirow{2}{*}{3.6 mm} & \multirow{2}{*}{3.7 mm} \\ non-magnetic ring—y3 & \multirow{2}{*}{1.5 mm–5 mm} & \multirow{2}{*}{3 mm} & \multirow{2}{*}{2.5 mm} & \multirow{2}{*}{3.3 mm} \\ non-magnetic ring—y4 & \multirow{2}{*}{3 mm–6 mm} & \multirow{2}{*}{3.9 mm} & \multirow{2}{*}{3.4 mm} & \multirow{2}{*}{3.3 mm} \\ non-magnetic ring—y5 & \multirow{2}{*}{1 mm–3 mm} & \multirow{2}{*}{1 mm} & \multirow{2}{*}{2.6 mm} & \multirow{2}{*}{2.2 mm} \\ thickness of the armature & \multirow{2}{*}{1.5 mm–3 mm} & \multirow{2}{*}{2 mm} & \multirow{2}{*}{2.3 mm} & \multirow{2}{*}{1.9 mm} \\ sleeve—y6 & \multirow{2}{*}{1.5 mm–3 mm} & \multirow{2}{*}{2 mm} & \multirow{2}{*}{2.3 mm} & \multirow{2}{*}{1.9 mm} \\ length of the coil—c1 & & & & \\ thickness of the coil—c2 & \multirow{2}{*}{6 mm–8.5 mm} & \multirow{2}{*}{41.5 mm} & \multirow{2}{*}{43.8 mm} & \multirow{2}{*}{44.6 mm} \\ moving mass & & & & \\ rated average force & & & & \\ force response frequency & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The values and the design ranges of key design parameters and the key characteristics of the design results.