## 3 Results  
In this study, we designed and analyzed a proportional solenoid constituting the pilot stage of a electro-pneumatic brake valve for use in heavy vehicles. The design criteria were taken from a system used in the application, and a standard on-off-type solenoid design was followed during the preliminary design phase. By changing the geometry of the pre-designed on-off-type solenoid magnetic circuit, we obtained a solenoid design that provided proportional characteristics. Due to this, the diameter of the fixed pole forming the magnetic circuit was increased compared to the moving pole, and a conical projection was formed, in which the moving pole operated. In this way, the magnetic flux path was deflected in small air gaps, and a permanent magnetic force was obtained within a certain operating range. This allowed the magnetic force to be independent of the air gap in a certain working region and made it proportional only to the current change. It is not possible to derive a direct mathematical relationship between the solenoid magnetic circuit and the force-current variation. Therefore, the optimization of the geometry providing the proportional characteristics could be performed with the help of the ANSYS-Maxwell program, based on the finite element method. The analysis program was run more than 2000 times, and we determined the proportional solenoid geometry that would provide constant magnetic force and achieve the design criteria. In the analysis performed for the ANSYS-Maxwell solenoid model, the design of which was completed, it is shown that the force-air-gap characteristic provides the desired 42 N force in the range of 1-3 mm with a current of 1.3 A (Figure 12a). The force graph for the optimized solenoid for different currents at different positions is given in Figure 12b, where the current-force curves are superimposed on each other when the air gap is between 1.00 mm and 3.00 mm. However, for values outside 1.0-3.0 mm of the air gap (0.5 and 3.5 mm for this graph), the current-force curves are scattered up and down. In other words, it can be interpreted that this region corresponds to the uncontrollable region for the proportional solenoid. However, that was out of our scope; we considered only the \"working region\" that has proportionality. As can be seen in Figure 12a, in the working region (between 1.0 mm and 3.0 mm), the force values are almost constant. This means that when the current is stationary, the  
Figure 12: Magnetic force (FEM analysis): (**a**) various currents; (**b**) various air gaps. force does not change, even if the plunger position is changed inside the working region. Therefore, it can be concluded that the force does not rely on the position (in traditional electromagnetic actuators, the force is a function of the current and the air-gap/plunger position, as indicated in Equation (1)). For each different current value, different forces were obtained, regardless of the displacement. In this way, the force function for the form \(F_{m}=f(I,x)\) is dependent only on the current, taking the form \(F_{m}=f(I)\) and ignoring the displacement term \(x\). The force can only be controlled by changing the applied current. The current-force proportion constant value can be easily obtained from Figure 12b (inside the working region; this means that air-gap values of 0.5 mm and 3.5 mm are neglected in Figure 12b). When the on-off-type solenoid valve with the same geometry is compared with the designed proportional valve, the on-off-type solenoid provides a relatively high magnetic force, at 70 newtons in low air gaps, and drops to the order of 10 newtons in wide air-gaps, whereas the designed proportional solenoid provides a magnetic force of around 45 newtons at maximum current, but it can maintain this value throughout the movement. The force characteristic of the designed proportional solenoid is only a function of the current, independent of the displacement. It has been proven that the force increases approximately linearly with the current.