###### Abstract  
This study presents the design and analysis of a proportional solenoid used in electro-pneumatic brake systems for heavy vehicles. The solenoid was designed using a traditional method, and its static and dynamic characteristics were investigated both theoretically and experimentally. ANSYS 2024 R1 Maxwell was employed for theoretical static analysis, focusing on the effects of the geometric dimension parameters in the fixed and moving pole contact regions on the force-displacement characteristics. The optimal dimensions for proportionality were determined under constraint parameters. The static analysis results provided the magnetization curve data, which were used to create Look-Up Tables for a dynamic model in MATLAB R2024b-Simulink, and this method reduced the simulation time and increased the dynamic simulation accuracy. Following static analysis, a prototype electromagnet was manufactured and tested. The solenoid achieved a constant magnetic force of 45 \(\pm\) 3 N with a current of 1.3 A over a working range of 1-3 mm. The dynamic model, incorporating data from ANSYS, yielded results that closely matched the experimental findings. electromagnetic; actuator; FEM; proportional solenoid actuator; constant force characteristics; proportional solenoid valve; solenoid; dynamic model; look-up table; optimization +
Footnote †: journal: Applied sciences  
This allows them to generate a continuous range of outputs rather than just binary states, making them indispensable in systems requiring high precision and flexibility. However, designing a proportional actuator is complex and depends on different parameters. To create a proportional solenoid from an on-off design, modifications are needed [4], especially in the geometry and magnetic circuit [5], to achieve the desired range of control. Traditional methods often struggle with the nonlinear behavior of magnetic systems [6], leading to complex equations and longer computation times. Although research has explored the impact of geometry on force characteristics [4; 5] and dynamic system models [7; 8], these approaches are often limited by their computational inefficiency. Finite element method (FEM) analysis is a powerful tool for designing electromechanical systems by predicting thrust force characteristics under various conditions. In their study, [PERSON] and [PERSON] [9] used the FEM to optimize three electromagnetic actuators--a solenoid actuator, a moving coil actuator (MCA), and a moving magnet actuator (MMA)--for jet valve applications. The results showed that the MCA and MMA required lower excitation currents (1.8 A and 1.9 A) and maintained consistent thrust forces with minimal drop-offs (0.7% and 1.8%) compared to the solenoid actuator, which showed significant force reduction over longer strokes. These findings highlight the efficiency and reliability of MCA and MMA models for jet dispensers. Current research suggests that the constant force characteristic of proportional solenoids is primarily influenced by the shape of the pole shoe. Numerous researchers have analyzed the structural properties and impact mechanisms of pole shoes. [PERSON] et al. [10] proposed an accurate model for a proportional solenoid valve, incorporating the effects of magnetic saturation in soft magnetic materials and the flow force acting on the valve spool. This model provided a valuable framework for optimizing system design. [PERSON] et al. [11] enhanced the pole shoe design and working air-gap dimensions of a proportional solenoid using a multi-objective optimization approach that combined traditional surrogate models with machine learning techniques. This led to an improvement in the constant force characteristic. [PERSON] et al. [12] demonstrated a robust framework for the optimization of high-speed on-off valves (HSVs), leveraging machine learning and metaheuristic algorithms to handle complex, nonlinear design interactions. [PERSON] et al. [13] conducted a detailed simulation study on a direct-drive compressor powered by an electromagnetic linear actuator.