## 4 Results and Discussion
### Analysis of Forces
Figure 16: Magnetic force for different clearance. Figure 15: Control pressure for different clearance. Figure 17 shows that the growth rate of the magnetic force increases with the clearance height while that of the viscous force and that of the spring force remain unchanged before the spool starts to move. As the magnetic force is much larger than the other two forces, the positive acceleration of the valve with higher clearance is larger than that with smaller one (Figure 18), which would induce longer displacement of the spool (Figure 19). During spool motion, the control pressure grows with the valve opening area, leading to the increasing pressure in the feedback chamber connected with the control port combining Figures 15 and 19. Hence, the feedback force and its growth rate both rise with the growing clearance height (Figure 17). Since the feedback force hinders the spool movement, the negative acceleration goes up with the increasing clearance height. According to [PERSON]'s second law, the value of the spool acceleration could be quite high because the spool is only 7.39 g.  
Figure 17: (**a**) Forces on the valve and the spool displacement in 10 \(\upmu\)m (**b**) Forces on the valve and the spool displacement in 20 \(\upmu\)m (**c**) Forces on the valve and the spool displacement in 30 \(\upmu\)m.  
Figure 18: Spool acceleration for different clearance. The force of inertia mainly influences the stability and the responsiveness of the valve during spool movement. If the force of inertia is too small, the spool will be very sensitive to the change of the resultant force, resulting in frequent changes of the spool motion state. The force of inertia helps the spool maintain a relatively stable movement trend and prevent vibration when the dynamic forces acting on the spool are constantly changing. In the hydraulic control circuit of an automatic transmission, the vibration of the spool can easily give rise of the pressure fluctuation which greatly reduces the shifting quality. In addition, it is hard for the hydraulic control circuit to converge because of the superposition of spool vibration. On the contrast, if the force of inertia is too large, the motion response of the valve will be delayed because the spool is insensitive to the change of the resultant force. The delayed motion response will result in both delayed control response and the increase of the overshoot in the hydraulic circuit. Therefore, the force of inertia is of great importance because it directly affects the control quality on the spool motion state. Thus, the maximum control pressure is enhanced when the displacement increased. The maximum control pressure will increase from 1.2 MPa to 1.5 MPa if the displacement grows from 0.9 mm to 1.1 mm. It could be easily deduced from the analysis that the magnetic force plays a main role in all the forces which could influence the control pressure of the valve. The starting time of the control pressure delays 0.1 s in the 30 \(\upmu\)m comparing to that in the 10 \(\upmu\)m as shown in Figure 15. As the movement of the valve starts only if the force balance is reached, the valve with small clearance height starts earlier than that with bigger one. The time that all forces reach to a balance is when the control pressure appears because the control port has been opened due to the displacement of the spool. Therefore, both the leakage and the pressure response should be taken into consideration when designing the clearance.