## 4 Results and Discussion
### Analysis of Forces  
The magnetic force is the main driving force acting on the solenoid valve which would significantly influence the pressure response time. Meanwhile the response time will change with the variation of the magnetic force. The critical structure of the magnetic part that impacts on the magnetic force was analyzed using the model proposed in this paper. The air gap in the magnetic part transfers forces generated by the magnetic part to the mechanical part. Different diameters of the air gap are set in the model and the results are shown in Figure 11.  
It can be obviously observed that smaller air gap leads to bigger peak value and faster change of the magnetic force. The magnetic forces are almost accordant at the beginning and the ending of the time. However, the force in 10 mm rises up earlier than that in 11 mm. Higher maximum value in 10 mm gives rise to larger slope of the magnetic force curve. Figure 12 presents the static characteristics of the magnetic force in different current. It indicates that the magnetic force is almost unchanged with the displacement at the same current. Slight skewing is confined within 0.5 N. Because the magnetic force is generated by the electric current and has no connection with the displacement, slight skewing increases with the growing current. However, the motion of the armature will have a small influence on the magnetic field which could result in the slight change of the magnetic force during the spool movement. Figure 11: Magnetic force in different air gap. Figure 12: The static characteristics of the magnetic force as a function displacement for different electric current. The static viscous force appears when the spool has the tendency of motion. And the dynamic force derives from the relative movement between the spool and the sleeve when the fluid flows through. Although small, the viscous force cannot be neglected because the force balance which determines the pressure response characteristics will change with it. Figure 13 illustrates the effect of the viscous force on the control pressure. The control pressure almost equals to 0 before 0.37 s when considering the viscous force, which means that the viscous force hinders the rise of the control pressure. Different clearance height is considered in the model and the results are shown in Figures 14-16. The static viscous force grows with the increase of the clearance height. The static viscous force in 10 \(\upmu\)m is 0.32 N while the viscous force in 30 \(\upmu\)m is 0.94 N which indicates the viscous force will increase when the size of the clearance is increased. Meanwhile the dynamic viscous force will rapidly increase or decline when there is sudden change in the spool velocity. However, the greater the spool velocity changes, the smaller the dynamic viscous force varies because fast spool velocity thins the oil film. Figure 14: Viscous force for different clearance. Figure 13: Control pressure in the valve. One noticeable observation is that the magnetic force response becomes slower with the increase of the clearance. On the contrary, the height of the clearance contributes to the maximum value of the magnetic force. The same is true for the pressure response and the maximum pressure. Comparing the magnetic force with the pressure response in the same clearance, it can be found that both of them have the same variation tendency. Hence, the magnetic force has a direct influence on the control pressure. Figures 15 and 17 show the relation between the feedback force and the control pressure. The feedback force and its growth rate rise with the growing clearance height which indicates that the initial velocity and opening area of the valve are augmented due to the variation of the equilibrium point of the resultant force. What is more, the spool displacement rises which can also lead to larger valve opening area. Therefore, the clearance height is a key factor that influences on the control pressure, because the control pressure is in direct proportion to the valve opening area.