## 2 Materials and Methods  
In this study, firstly, a flat-faced piston-type electromagnet that provides 50 N of force under certain limitations for an electromechanical pneumatic brake valve was pre-designed [17; 18]. A conical-type solenoid was achieved by making some modifications to the flat-faced piston-type solenoid. Then, by changing the geometric parameters using FEA (finite element analysis) on the conical region, as in the literature [3; 5], we optimized a proportional solenoid design that provided sufficient constant force in the desired working region. A mathematical model for the whole system (electrical, magnetic, and mechanical subsystems) was introduced. The solenoid was modeled in the MATLAB-Simulink environment. The magnetic data (such as the magnetic flux or current) were fed from the magnetic FEA results by means of Look-Up Tables. For experimental validation, a test bench was designed and manufactured. Static tests for the actuator were performed for various current and plunger positions. Dynamic measurements were performed by measuring the current response. The theoretical results for both static and dynamic cases were validated through the experiments. The solenoid pre-design was carried out using rough empirical formulations. By following the classical electromagnet design steps [17], we obtained the solenoid dimensions (Figure 1) for providing the desired force according to the size limitations, as shown in Table 1.  
The magnetic force of the pre-designed electromagnet is expressed as follows:  
\[F_{m}=\frac{\mu_{0}A_{c}(NI)^{2}}{2\left(x_{t}-x\right)^{2}} \tag{1}\]  
As seen in Equation (1), the magnetic force varies directly with the square of the applied current and is inversely proportional to the amount of the result in mechanical movement. Accordingly, the magnetic force is a nonlinear function of the current and displacement. The displacement-magnet-force characteristics for a constant current for this electromagnet were obtained using the magnetic finite element method in the Maxwell module of the ANSYS program. It is known that for on-off-type solenoids, the magnetic force is high in the range where the distance between the fixed pole and the moving pole is very small, whereas the magnetic attraction force decreases suddenly as the distance increases. Therefore, a nonlinear characteristic is seen in the force curve corresponding to the change in distance for a given current [19; 20]. This type of solenoid is suitable for on-off-type valves but not for proportional valves. As seen in Figure 2, in proportional solenoids, it is possible to provide a magnetic force proportional to the current change by creating a region that provides a constant force in a certain operating range between the position and the force. It is possible to obtain a proportional solenoid by changing the geometric structure of the classical on-off-type solenoid, and there are various studies and applications on this subject in the literature [3; 21]. In this study, a proportional solenoid design was developed using the ANSYS Maxwell solenoid model, and new geometric dimensions were defined. As can be seen from Equation (1), the magnetic force, \(F_{m}=f(i,x)\), in standard solenoids is a nonlinear function of the current and displacement. Conversely, in proportional solenoids, the magnetic force can be expressed as \(F_{m}=f(i)\), independent of the amount of movement in the working region and only depending on the current change. Generally, in solenoid-controlled valves, the moving element works against a spring force, and the balance of these forces can be expressed as follows:  
\[F_{m}+F_{spring}=f(i)-kx=0\Rightarrow x=(1/k)f(i) \tag{2}\]  
where \(F_{m}\) is the magnetic force, \(F_{spring}\) is the spring counter force, \(k\) is the spring constant, \(x\) is the displacement, and \(i\) is the exciting current. The \(f(i)\) function is linear and can be controlled by varying the current.