## 4 Discussion and Conclusions  
In this study, we successfully designed, manufactured, and tested a proportional solenoid, achieving the objectives set within the scope of the research. A test setup was developed specifically to determine the static and dynamic characteristics of the solenoid, enabling comprehensive evaluation. The results demonstrated sufficient agreement between the simulation and experimental findings regarding the dynamic characteristics of the solenoid, as well as the force characteristic of the solenoid for the static case. It was determined that the solenoid has a response time of approximately 15 ms, which is suitable for the target application. Furthermore, this response time is particularly effective for heavy vehicle braking systems, where proportional solenoid valves are commonly used.  
The solenoid design was found to meet the established design criteria, and the theoretical and experimental results showed reasonable compatibility within the study's scope. The proposed method uses Look-Up Tables generated through electromagnetic finite element method (FEM) analysis, effectively reducing the computation time for solving the differential equations of the dynamic model.  
However, this study faces certain limitations. Firstly, gravitational acceleration was neglected in both the theoretical design and experimental setup, limiting the method's applicability in scenarios where gravitational effects are significant. Secondly, the solenoid force range was confined to values below 70 N. Additionally, the Look-Up Table approach requires detailed data, including air-gap, flux linkage, and current values over a wide range, derived from FEM analysis. The absence of such data could limit the method's applicability.  
In conclusion, this research provides a validated approach to the design and analysis of proportional solenoids, demonstrating the feasibility of utilizing FEM-based Look-Up Tables for dynamic modeling. The findings contribute to advancing proportional solenoid applications and offer a foundation for further improvements.  
Conceptualization, E.D.; methodology, E.D.; software, E.D. and G.S.; validation, E.D.; formal analysis, E.D. and G.S.; investigation, E.D.; resources, E.D. and G.S.; data curation, E.D. and G.S.; writing--original draft preparation, E.D.; writing--review and editing, E.D. and G.S.; visualization, E.D.; supervision, G.S.; project administration, G.S.; funding acquisition, G.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Science, Industry, and Technology of Turkiye, grant number 0700. STZ. 2014.  
Figure 14: Current characteristics of measurement and theoretical models.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.  
**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
**Acknowledgments:** The authors would like to thank mechanical engineer [PERSON] from M.A.Y. Brake Systems (MAY Fren Sistemleri San. ve Tic. A.S.) for his technical support in the manufacturing of the proposed solenoid.  
**Conflicts of Interest:** The authors declare no conflicts of interest.  
**Nomenclature**  
\begin{tabular}{l l l}
**Symbol** & **Description** & **SI Units** \\ _AC_ & Cross-section area & m\({}^{2}\) \\ \(B\) & Magnetic flux density & T \\ \(H\) & Magnetic field & A/m \\ _\(\mu\)_ & Magnetic permeability & H/m \\ _\(\phi\)_ & Magnetic flux & Wb \\ \(F\) & Force & N \\ \(k\) & Spring constant & N/m \\ \(x\) & Displacement & m \\ _\(\mathfrak{R}\)_ & Magnetic reluctance & H\({}^{-1}\) \\ \(R\) & Electrical resistance & \(\Omega\) \\ \(L\) & Inductance & H \\ \(m\) & Mass & kg \\ \(b\) & Damping coefficient & Ns/m \\ \end{tabular}  
**Appendix A**  
\begin{tabular}{l l l}
**Figure A1.** Driver circuit.  
Figure A2: Solenoid CAD model.