# Modeling and Multi-Objective Optimization Design of High-Speed on/off Valve System
## 2 New High-Speed on/off Valve Structure and System Modeling
### System Dynamic Modeling  
The moving components of an HSV can be simplified into a second-order system. The differential equation of the movable components in the system during the opening process of the valve core can be expressed as follows:  
\[M\frac{d^{2}x}{dt^{2}}+D\frac{dx}{dt}=F_{e}(x)+P_{T}A_{s}-P_{a}A_{\rm s}-F_{s} (x) \tag{1}\]  
where \(M\) represents the mass of the moving components, \(x\) is the displacement of the valve core, \(D\) is the viscous damping, \(F_{e}\) is the output force of the electromagnetic coil as a function of the valve core displacement, \(P_{T}\) is the load pressure, \(P_{a}\) is the output pressure, and \(F_{s}\) is the steady-state hydraulic force function acting on the valve core when it is opening. The differential equation of the system's moving components when the valve core is closed can be expressed as follows:  
\[M\frac{d^{2}x}{dt^{2}}+D\frac{dx}{dt}=P_{s}A_{s}-P_{a}A_{\rm s}-F_{e2}(x)-F_{ s2}(x) \tag{2}\]  
\(F_{e2}\) represents the resistance encountered when cutting magnetic field lines through the armature, and \(F_{s2}\) is the steady-state hydraulic force function acting on the valve core when it is closing. By observing Equations (1) and (2), it can be understood that the valve is simultaneously affected by electromagnetic force, pressure, and hydraulic force during both opening and closing. Furthermore, these factors continuously change with the displacement of the valve core. Therefore, accurately calculating the opening and closing time requires comprehensive consideration of the mechanical, electromagnetic, and fluid effects within the system. Magnetic force exerted by a solenoid can be mathematically represented as follows:  
\[F_{e}=\frac{\varphi(i,x)^{2}}{2\mu_{0}S_{0}} \tag{3}\]  
where \(S_{0}\) is the cross-sectional area of the main air gap, and \(\mu_{0}\) is the vacuum permeability. The magnetic flux \(\varphi\) is a function of the coil current \(i\) and the position \(x\). \[\varphi=\frac{i\times n}{\sum_{j=1}^{k}R_{j}}\times C(i,x) \tag{4}\]  
where \(n\) is the number of turns in the coil. \(R_{j}\) represents the reluctance of each air gap in the magnetic path. Each reluctance can be expressed as follows:  
\[R_{j}=L/\mu_{0}S \tag{5}\]  
where \(S\) is the cross-sectional area of the magnetic path, and \(L\) is the length of the path. Based on the calculated flux at the equivalent air gap, considering errors due to the armature's geometric shape, core saturation, and leakage flux, the correction factor \(C(i,z)\) is calibrated using finite element analysis (FEA) results. Figure 2 compares the FEA results with the electromagnetic force model, showing that the model is accurate after calibration. The finite element simulations of HSVs were conducted using MAXWELL 2021R1, and the results are shown in Figure 3. It can be seen that the addition of permanent magnets effectively weakens the induction intensity in most magnetic circuits, effectively avoiding the occurrence of magnetic saturation. The voltage across the helical coil is induced by the applied power source, with its behavior being governed by the coil's inductance, as expressed in Equation (6).The equation  
Figure 3: FEA simulation schematic diagram. Figure 2: Validation of the electromagnetic force model. takes into account the resistance of the helical coil, denoted by \(r\). Equation (6) is composed of three terms: the second term represents the back-EMF due to the coil's self-inductance, and the third term signifies the back-EMF resulting from the helix's motion. \[u_{i}=i\cdot r+\frac{\partial\varphi}{\partial i}\dot{i}+\frac{\partial\varphi} {\partial s}\dot{x} \tag{6}\]  
A model for the static hydrodynamic force acting on the valve core is established. Static hydrodynamic force refers to the force generated by the change in fluid velocity when the flow area and direction through the valve opening change, causing a change in fluid momentum.