## 2 Materials and Methods
### Finite Element Model and Experimental Setup  
According to this optimization, it has been determined that the parameters \(t_{L}\) = 3.0 mm, \(t_{D}\) = 0.2 mm, and \(t_{M}\) = 2.5 mm represent the dimension combination with the lowest value for the objective function for the design. Finite element analysis was performed for this dimension, and the magnetic field strength and magnetic flux density analyses are shown in Figure 7a,b. As seen in this Figure, the largest flux density is in the conical part of the solenoid and is of the order of 0.9 Tesla. Magnetic flux paths of the solenoid and are shown in Figure 8, below. Figure 8a shows the magnetic flux path on a large scale between \(2.62\times 10^{-9}\) and \(2.62\times 10^{-5}\) Weber. According to the results of the FEM analysis, approximately 54% of the equivalent magnetic flux lines pass through the added conical region, 23% pass through the flat face of the fixed pole, and the remaining 23% represent the fringing flux. If the lower limit of the scale is selected as \(8\times 10^{-6}\) Wb in the Maxwell program, then Figure 8b is obtained. Since magnetic flux lines lower than \(8\times 10^{-6}\) Wb are not shown in  
Figure 8: The magnetic flux path of the solenoid: (**a**) the overall magnetic flux on a large scale; (**b**) the magnetic flux path for a certain range. Figure 7: The solenoid designed for g = 0.1 mm: (**a**) magnetic field strength (H); (**b**) magnetic flux density (B) analysis. Figure 8b, the low flux passing through the flat part of the solenoid is not visible. Most of the useful magnetic flux passes through the conical part, as expected. The designed proportional solenoid was successfully manufactured. The CAD model and pictures of the manufactured solenoid are provided in Appendix Figures 10 and 11. A test bench was designed and manufactured for static and dynamic tests. The experimental setup was designed so that the plunger moved in a horizontal direction but not in a vertical direction, and the gravitational acceleration was neglected. The experimental setup designed for the experimental analysis of static and dynamic characteristics is shown in Figure 9. Figure 9a shows a general view of the experimental setup designed and implemented for static characteristic measurement. It consists of a load cell for measuring the force, a position sensor for measuring the air gap/position of the plunger, a power supply, and a current circuit driver. Figure 9b shows a detailed view of the current circuit driver (see Appendix A for the current circuit driver electronic scheme). The load cell used for force measurement in this setup was a Puls Elektronik STA Series Load Cell, manufactured by Puls Elektronik, based in Istanbul, Turkey, with a maximum capacity of 50 kg.. In the experimental setup, a Panasonic Laser Sensor HL-G105-A-C5, manufactured by Panasonic Corporation, headquartered in Osaka, Japan, was used to measure the position of the moving core of the solenoid. This sensor features a resolution of 1.5 \(\upmu\)m, a measuring range of 10 mm, and selectable sampling times of 200 \(\upmu\)s, 500 \(\upmu\)s, 1 ms, or 2 ms.. The current driver circuit was driven by 15 V DC electrical voltage; two LM741 opamps were used. In the experimental setup, linear motion was facilitated by a Proxxon KT70 model screw shaft mechanism module, manufactured by Proxxon GmbH, based in Fohren, Germany. This module provides a stroke of 134 mm in the X direction and 46 mm in the Y direction.. A feedback brand power supply providing \(\pm\)15 V DC electrical voltage was used to feed the circuit. The measurement conditions for static measurements included a test bench with a current applied in the range of 0.1 A to 1.5 A, and the air gap (position sensor) varying from 0.1 mm to 3.5 mm. The dynamic system response measurements were performed using the NI (National Instruments) PCI-6221 Multifunction Data Acquisition Board (DSUB). A signal generator block in MATLAB-Simulink was employed for dynamic system response measurements.