# Failure Type Prediction Using Physical Indices and Data Features for Solenoid Valve +
## 2 Failure Mechanism of High-Speed Solenoid Valve and Modeling
### Modeling  
The working process of the solenoid valve can be divided into five stages: suction touching stage, suction motion stage, power-on holding stage, release touching stage, and release motion stage [(29)]. Reference [(30)] analyzed the touching and moving stages of the solenoid valve and simplified the suction process into three mathematical equations: circuit equation, magnetic circuit equation, and mechanical equation. If we do not consider the influence of temperature on resistance, the circuit equation [(31)] is  
\[U=I(R+R_{L})+(L+L_{1})\frac{dI}{dt}+I\frac{d(L+L_{1})}{dt} \tag{1}\]  
where \(U\) is the excitation voltage of the coil, \(I\) is the coil current, \(R\) is the coil resistance, \(R_{L}\) is the additional resistance of coil circuit, \(L\) is the coil inductance, and \(L_{1}\) is the additional inductance of coil circuit. The magnetic circuit equation [(31)] is  
\[L(x)=\frac{\mu_{0}\,\pi D^{\,2}\,N^{\,2}\,l_{\,v}}{4l_{\,v}\,(l_{\,0}-x)+rD} \tag{2}\]  
where \(L\) is the coil inductance, \(\mu_{0}\) is the permeability of vacuum, \(D\) is the diameter of spool, \(N\) is turns of coil, \(l_{\,v}\) is the length of armature part of valve core, \(l_{0}\) is the maximum width of working air gap, \(r\) is the length of working air gap, and \(x\) is the displacement of valve element. By deriving \(L(x)\), we can get the following:  
\[\frac{dL}{dx}=\frac{4\mu_{0}\,\pi D^{\,2}\,N^{\,2}\,l_{\,2}^{\,v}}{(4l_{\,v} \,(l_{\,0}-x)+rD)^{\,2}}=\frac{4L^{\,2}\,(x)}{\mu_{\,0}\,\pi D^{\,2}\,N^{\,2}} \tag{3}\]  
Based on the mechanical analysis of the movement of the valve core, the mechanical equation can be obtained as follows:  
\[F_{\,e}=\frac{1}{2}I^{\,2}\,\frac{dL(x)}{dt} \tag{4}\]  
\[F_{\,e}-kx-\left(C_{\,v}+C_{\,f}\right)\frac{dx}{dt}=m\frac{d^{\,2}\,x}{dt^{\, 2}} \tag{5}\]  
where \(F_{e}\) is electromagnetic force, \(k\) is the force coefficient of spring, \(C_{\,v}\) is the dynamic friction factor of valve element, \(C_{\,f}\) is the viscous damping coefficient, and \(m\) is the quality of valve element. By combining Equations (3)-(5), we could get the following equation [(28)]:  
\[\left(\frac{2L^{\,2}\,(x)I^{\,2}}{\mu_{\,0}\,\pi D^{\,2}\,N^{\,2}}-\left(C_{ \,v}+C_{\,f}\right)\right)\frac{dx}{dt}-kx=m\frac{d^{\,2}\,x}{dt^{\,2}} \tag{6}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Failure Type** & **Representation** \\ \hline Aging of coil & the resistance of the coil decreases the steady state current increases the movement resistance increases \\ Drying up of lubricant & spool speed reduces \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The impact of each type of failure. According to Equation (6), the displacement of the solenoid valve core is closely related to the change process of driving current. The dynamic driven current and core displacement of the solenoid valve are shown as Figure 2.  
There are three indices related to the driving current and two related to the spool movement: trigger current \(i_{1}\), stable current \(i_{2}\), current drop \(\Delta i\), reaction time \(t_{1}\), and action time \(t_{2}\). The movement of the valve core and the driving current curve of the solenoid valve are different under each fault condition. The relationship between physical indices and failure factors is shown in Table 2.  
The coil aging is mainly caused by aging of the insulation layer, resulting in a smaller coil resistance. Therefore, the stable current \(i_{2}\) will increase. And after the insulation layer is aged, part of the wire contact is equivalent to the reduction of the number of coil turns, which causes the coil self-induction to become smaller. Consequently, the current drop \(\Delta i\), reaction time \(t_{1}\), and action time \(t_{2}\) will be reduced. Drying up of lubricant will increase resistance when the spool is moving, so that the electromagnetic force to overcome the resistance is required to increase, which means the trigger current \(i_{1}\) and the reaction time \(t_{1}\) will increase.