## 3 Dynamic Performance Improvement of Solenoid Electromagnetic Actuator
### The Analysis of the Magnetic Materials  
Except the coil winding parameters, the soft magnetic material of the LPS is another important factor on the static and dynamic performance of the LPS. Due to the high saturation magnetization, pure iron is usually used in the LPS to generate large static push-force. However, the dynamic performance of the LPS requires low remanence and low coercivity characterizes of soft magnetic material to generate less power loss. Thus, selecting the appropriate soft magnetic material of each part of the LPS is an improvement method for dynamic performance. The theoretical principle is shown as below. According to the empirical Bertotti Equation for the core loss calculation, the total power is presented as:  
\[P_{total}=P_{mechanical}+P_{coreloss}+P_{coreloss}+P_{solidloss} \tag{18}\]  
\[P_{coreloss}=P_{h}+P_{c}+P_{e}=k_{h}fB^{2}{}_{m}+k_{c}(fB_{m})^{2}+k_{e}(fB_{ m})^{1.5} \tag{19}\]  
\[P_{solidloss}=\int\limits_{Vol}\frac{l_{e}^{2}dVol}{\sigma} \tag{20}\]  
\[P_{coreloss}=i^{2}R \tag{21}\]  
where \(P_{total}\) represents the total power consumption, \(P_{mechanical}\) represents the mechanical power, \(P_{coreloss}\) represents the power loss generated by time varying magnetic field, \(P_{coreloss}\) represents the power loss generated by coil of the LPS and \(P_{solidloss}\) represents the resistive loss in the conductor volume of the LPS [23]. \(k_{h}\), \(k_{c}\) and \(k_{e}\) represent the coefficients related to hysteresis loss, eddy current loss and excessive loss, respectively. \(k_{h}\), \(k_{c}\) and \(k_{e}\) are determined by the measurement of specific magnetic material. \(f\) represents the frequency of the magnetic field, \(B_{m}\) represents the maximum amplitude of  
Figure 14: Frequency response of the push force at fixed air gap. the magnetic flux density. \(J_{e}\) represents the eddy current density, \(\sigma\) represents the conductivity of the magnetic materials and _Vol_ represents the volume of the magnetic materials of the LPS. By the coil winding parameters improvement strategy, the \(P_{\mathit{comperloss}}\) of conventional-type LPS and the optimal-type LPS are equal. Therefore, the mechanical power \(P_{\mathit{mechanical}}\) is correlated with \(P_{\mathit{total}}\), \(P_{\mathit{corloss}}\) and \(P_{\mathit{salidloss}}\). According to the expression of \(P_{\mathit{corloss}}\) and \(P_{\mathit{salidloss}}\), when \(P_{\mathit{total}}\) is constant, \(P_{\mathit{mechanical}}\) is related to materials of the LPS. Taking the armature into account--the only moving part of the LPS--electrical steel is employed as the soft magnetic materials of the armature in this study. The electrical steel armature is made by cold-drawn non-oriented silicon steel rod, which meet IEC 60404-8-6 class c1 standard. The comparisons of the magnetic hysteresis loops of pure iron and electrical steel are shown in Figure 15 and the test magnetic fields include the DC magnetic field and 20-Hz-, 50-Hz- and 100-Hz-AC magnetic field. It can be seen that the pure iron has large maximum relative permeability, low remanence and low coercivity in DC magnetic field, but when the frequency of the AC magnetic field increased, the electrical steel shows a better dynamic performance than pure iron. As a specific electromagnetic actuator used in SPV, LPS is required both high dynamic performance and high static performance. Therefore, based on the high dynamic performance requirement, electrical steel is more suitable as the metrical of the armature. Moreover, for large push-force requirements, pure iron is employed as the material of the other non-moving components. In order to verify the improvement of electrical steel, an electrical steel optimal-type LPS was redesigned and the manufactured. Considering the differences of Magnetic Hysteresis Loops between electrical steel and pure iron, the static optimization strategy in present study was employed to redesign the structure of electrical steel optimal-type LPS.