## 3 Dynamic Performance Improvement of Solenoid Electromagnetic Actuator
### The Analysis of the Coil Winding Parameters  
Obviously, the resistance and the inductance of the coil winding will directly affect the coil current response and finally affect the push force response property. Thus, the analysis of different rated excitation current is employed to improve the dynamic performance of the LPS. The theoretical principle is shown as below. According to the [PERSON]'s formula, the magnetomotive force-F can be described as:  
\[\mathrm{F}=\phi R_{m} \tag{10}\]  
where \(R_{m}\) is the magnetic reluctance. The definition of \(R_{m}\) is expressed as:  
\[R_{m}=\frac{\delta}{\mu S} \tag{11}\]  
Moreover, according to the definition of the magnetic flux density \(B\) and [PERSON]'s law, F is expressed as:  
\[\mathrm{F}=Ni \tag{12}\]  
The definition of the inductance \(L\) is expressed as follow:  
\[L=\frac{N\phi}{i} \tag{13}\]  
Combining the Equations (12) and (13), the inductance of the coil-\(L\) can be expressed as:  
\[L=\frac{N^{2}}{R_{m}} \tag{14}\]  
The simplified equivalent circuit of the coil drive control system of the LPS is shown in Figure 11:  
Since the pulse-width modulation (PWM) signal of the controller is assumed to be 100% duty cycle with ignoring the voltage drop of the other components of the controller, the controller can be simplified as 24 V DC power supply. Thus, it can be clearly obtained from Figure 11 that the LPS drive circuit is a resistor-inductor circuit, and the current response is determined by the Equation:  
\[i(t)=\frac{V(1-e^{-Lt/R})}{R} \tag{15}\]where \(i(t)\) represents the time-dependent coil excitation current, \(V\) is the drive voltage of the coil (24 V DC), \(R\) represents the resistance of the coil. Since the LPS is a direct current (DC) electromagnetic actuator with 24 V DC power supply, the drive voltage-V is constant. Thus, the current response time related with \(R\) and \(L\). Theoretically, according to the value of V and \(R\), the coil current can rise to the maximum value. But it is obvious that the power consumption of the coil and the controller are too high under maximum current. Thus, a rated power loss of the coil is limited in this study. The rated power loss \(P_{nted}\) can be expressed as:  
\[\mathrm{P_{rated}}=l_{rated}^{2}R \tag{16}\]  
where \(I_{nted}\) is the rated current of the coil. According to the Equations (12),(14)-(16), the response time of the rated current-\(t_{nted}\) can be described as follow Equation:  
\[t_{nted}=\frac{\mathrm{F_{rated}^{2}}}{R_{m}\mathrm{P_{rated}}}\ln\left( \frac{\mathrm{V}I_{nted}}{\mathrm{V}I_{natal}-\mathrm{P_{rated}}}\right) \tag{17}\]  
\(t_{natal}\) represents the time cost of the coil current from 0 to \(I_{natal}\). In this study, the rated magnetomotive force-\(\mathrm{F_{rated}}\) is 1980 ampere-turn, the drive voltage of the LPS-V is 24 V DC and the power loss of the coil-\(\mathrm{P_{rated}}\) is 32 W, these rated parameters are constant in the coil parameter analysis. Moreover, \(R_{m}\) is correlated with structure parameters of the LPS and it is unchanged with coil winding parameters. Thus, the response time of the rated current-\(t_{nted}\) is only related to coil rated current-\(I_{natal}\). With the increasing of excessive excitation current, the power consumptions of drive components in the controller and the coil winding of the LPS are increased rapidly. For the requirement of actually use, the rated current increased from 3A to 5A in this study. Correlatively, the turns number of the coil winding is reduced from 660 to 396 and the resistance of the coil winding is reduced from 3.5 \(\Omega\) to 1.3 \(\Omega\). The resistance of the coil is measured by multimeter (FLUKE). Thus, the copper loss of the LPS and the rated magnetomotive force can stay constantly. By the magnetic circuit calculation Equations in reference [21], the \(R_{m}\) calculation result of the conventional-type LPS is \(2.08145\times 10^{6}\) H\({}^{-1}\) at end position (3.6-mm air gap) of the working stroke.