Measurement of Electromagnetic Force of Pilot-Operated Type Solenoid Valve and Analysis of its Driving Characteristics

2.1 Structure of a Solenoid Valve

The solenoid valve consists of a main plunger, a pilot plunger (movable core), a sol guide core (fixed core), a coil, and a sol housing, as shown in Fig. 1. When current is applied to the coil, a magnetic field is generated and an attractive force develops across the air gap, causing the plunger to be drawn toward and adhere to the core. The air gap region changes depending on whether current is applied to the solenoid, as indicated by the red box in Fig. 1 and illustrated in Fig. 2. When the electromagnetic force exceeds the spring force, the plunger moves by the air gap distance, resulting in a change in stroke.

Fig. 1.

Structure for a Pilot-Operated Solenoid Valve with corresponding materials

Fig. 2.

Pilot-Operated Solenoid Valve Operation

2.2 Electromagnetic Model of a Solenoid Valve

The resistance of a solenoid varies with temperature. In this study, the solenoid temperature and resistance are assumed to be constant [6]. Under DC voltage excitation, the solenoid coil can be represented by the circuit shown in Fig. 3, where the resistance is constant; in steady-state, it can be modeled as an electrical circuit with constant inductance and current. According to Kirchhoff’s voltage law, the voltage applied to the coil is equal to the sum of the voltage drop across the equivalent coil resistance and the electromotive force induced by the change in magnetic flux.

Fig. 3.

RL circuit diagram

Where, R is the ohmic resistance of the coil, i is the coil current, V_e is the applied voltage, and φ the total magnetic flux. The total flux linkage λ is defined based on the number of coil turns (N) and the total magnetic flux (φ), and is expressed as in Eq. (2).

By substituting λ from Eq. (2) into Eq. (1), the applied voltage can be obtained as follows

The electromagnetic force F of a solenoid can be expressed, as shown in Eq. (4), as a function of the air gap flux density B_g, the absolute permeability μ₀ of air, and the cross-sectional area of the air gap region A_g. Where, the air gap flux density B_g can be written, as in Eq. (5), in terms of the relationship between the total flux linkage λ and the air gap cross-sectional area A_g. Substituting this expression into Eq. (4), the electromagnetic force can be reformulated as a relationship between the total flux linkage λ and the air gap cross-sectional area A_g [7,8].

Meanwhile, in this study, the electromagnetic force of the solenoid valve was described using a simplified quasistatic model in which the force is expressed as a function of the air gap flux density and the total magnetic flux. In actual valve actuation, however, Faraday’s law implies that variations in coil current and changes in the air gap due to plunger motion cause the magnetic flux inside the core to vary with time. This time-varying flux induces an electric field within the core, generating eddy currents in closed loops. According to Lenz’s law, these eddy currents produce a magnetic field that opposes the flux change, and the effect appears as a braking force acting opposite to the plunger’s direction of motion. Accurately modeling eddy currents inside the core and plunger is difficult because many variables must be considered and their interactions are complex; thus, a rigorous analysis is not straightforward.

Therefore, this study adopts a model with simplifying assumptions and approximations, and the braking force is estimated by introducing appropriate approximations based on electromagnetic laws.

First, the expression is simplified by assuming that the coil has a single turn. Without this assumption, the expression for the total flux passing through the coil becomes a summation over the number of turns, which becomes highly complicated.

As a second simplification, the coil self-inductance is neglected. If it is not neglected, coupled differential equations containing two unknown functions the current I(t) and the plunger velocity v(t) must be solved, which significantly increases the complexity of the analysis. If z(t) is defined as the distance from the coil center to the plunger, a simplified expression for the z-component of the magnetic field normal to the coil cross section can be written as in Eq. (6).

Where, B₀/C denotes the magnetic field at the plunger surface. The magnetic flux passing through the coil can then be expressed as in Eq. (7).

Where, L is the radius of the coil. Therefore, the induced EMF is given by Eq. (8).

Where, v(t) = dz/dt denotes the translational velocity of the plunger. Using another rough approximation, the expression can be written as in Eq. (9).

In other words, this expression is written by treating it as a constant C'. However, in practice, it is not a constant; it varies with time t and can be interpreted as the time-averaged value of dB_z/dz. In addition, if the coil self-inductance is neglected, the current flowing through the coil can be expressed as in Eq. (10).

According to the Lorentz force, the force exerted by the plunger on the coil is given by Eq. (11).

Where, $$ \overrightarrow{B}$$ is the total magnetic field generated by the plunger at each coil winding position, and $$ \overrightarrow{dl}$$ is an infinitesimal length vector along the current-carrying path of the coil. The magnetic field can be expressed as $$ \overrightarrow{B}=B_x\tilde{X}+B_y\tilde{Y}+B_z\tilde{Z}$$, which has x-, y-, and z- components. Evaluating the integral shows that the net force due to the z- component B_z(z) is always zero; therefore, it is sufficient to compute only the integrals associated with the x - and y - components of $$ \overrightarrow{B}$$. We further make the simplifying assumption that those x - and y - components are such that their sum $$ {\overrightarrow{B}}_N=B_x\tilde{X}+B_y\tilde{Y}$$ is normal to the $$ \overrightarrow{dl}$$. With this assumption, the expression can be rearranged as in Eq. (12).

Therefore, the braking force acting on the plunger can be expressed, according to Newton’s third law, as −F.

3.1 Electromagnetic Force Test Setup

Fig. 4 illustrates the configuration of the electromagnetic force test setup. A digital push-pull gauge is connected to a tensile-compression testing machine, and the main plunger is fastened to a threaded attachment of the push-pull gauge. The tensile force measured by the load cell is converted to electromagnetic force (N) and displayed as a digital value. The data measured by the load cell are transmitted by connecting the USB port of the tensile-compression testing machine to a desktop computer, and the results can be monitored as graphs via the measurement software (Force Recorder Professional) synchronized with the load cell digital output. The stroke between the plunger and the core is adjusted using a ball-screw-driven vertical translation stage that moves the load cell up and down. In addition, a fixing jig and an angle adjuster capable of tilting and rotating were used to secure the solenoid valve and to ensure coaxial alignment between the core and the plunger.

Fig. 4.

Experimental setup for electromagnetic force measurement

3.2 Experimental Conditions and Methods

Using a DC power supply, constant voltage (CV) levels of 8 V, 10 V, 12 V, and 14 V were applied to the solenoid valve coil. The experiment was initiated at an air gap distance of 1.20 mm, and the translation stage was driven at a feed rate of 5.00 mm/min under a vertical reciprocating motion of 2.40 mm. The sampling frequency of the measurement program was set to 10 Hz. All measurements were repeated three times for each condition, and the average values were reported. To measure the intrinsic electromagnetic force, the ball, spring guide, and spring were removed during the experiments.

4.1 Experimental Results of Electromagnetic Force Measurements with Varying Air Gap Distance

In this experiment, the electromagnetic force of a solenoid actuator was measured as a function of air gap variation. The results showed an asymmetric behavior: compared with the air-gap-decreasing interval (1.20 mm → 0.00 mm) shown in Table 1. and Fig. 5, the electromagnetic force at the same air gap was higher in the subsequent air-gap-increasing interval (0.03 mm → 1.20 mm) shown in Table 2. and Fig. 6. This phenomenon can be interpreted as a dynamic hysteresis effect induced by eddy currents generated during the air gap transition process.

Table 1.

Electromagnetic Force Analysis Results for Downward Motion

Fig. 5.

Electromagnetic Force-Air Gap Curves for Downward Motion

Table 2.

Electromagnetic Force Analysis Results for Upward Motion

Fig. 6.

Electromagnetic Force-Air Gap Curves for Upward Motion

Therefore, the underlying mechanism is discussed theoretically below. As the air gap changes, the magnetic flux distribution inside the plunger and the core varies with time. In electrically conductive metallic cores and plungers, a flux-change-rate-dependent induced electromotive force is generated, which leads to the formation of eddy currents. According to Lenz’s law, these eddy currents produce a magnetic field that opposes the flux change, thereby tending to delay the rate of increase or decrease of the actual air gap flux. Consequently, during the air-gap-decreasing interval, eddy currents suppress the increase in magnetic flux, resulting in a relatively lower effective flux and electromagnetic force at a given air gap.

Conversely, during the air-gap-increasing interval, eddy currents suppress the decrease in magnetic flux, allowing the flux to persist for a longer time and thus producing a larger electromagnetic force at the same air gap. This behavior is also consistent with the fact that a time-varying magnetic field penetrates a conductor via magnetic diffusion, and an effective time constant is determined by the material permeability and electrical conductivity.

In other words, even at the same air gap, the force can differ depending not only on the instantaneous air gap value but also on the direction (decreasing/increasing) and the rate at which the air gap is reached. The hysteresis error shown in Fig. 7 is summarized in Table 3. The error varied with both air gap and applied voltage; for example, at an air gap of 0.10 mm, the hysteresis error under the 8 V condition was the largest, reaching 9.00%. In contrast, at an air gap of 0.70 mm under 14 V, the hysteresis error was 0.00%, indicating that the difference between the increasing and decreasing intervals was negligible. Overall, relatively larger errors tended to appear in the intermediate air gap range of approximately 0.10-0.60 mm. This can be explained by the fact that the flux change rate and the degree of magnetic diffusion within the conductor differ across air gap ranges.

Fig. 7.

Hysteresis Analysis Results for Upward Motion and Downward Motion

Table 3.

Hysteresis Analysis Results for Upward Motion and Downward Motion

Specifically, at large air gaps, the flux magnitude is small, and the change rate is limited, whereas at very small air gaps, even if the flux increases, the force sensitivity to further changes may become relatively mild; as a result, the eddy current effect may manifest more prominently as a force difference in the intermediate air gap region [9,10].

In addition, regression analysis of the electromagnetic force data in the two intervals confirmed that the force can be approximated by a second-order polynomial function of the air gap, as shown in Fig. 5 and Fig. 6. This quantitatively demonstrates the nonlinear characteristics of the electromagnetic force curve with respect to air gap variation, and the resulting approximation can be used as a surrogate model for subsequent comparison with FEM results and for predicting actuation characteristics.

4.2 Results of Magnetostatic FEM Analysis

Although the electromagnetic force of a solenoid can be calculated using Eq. (4), accurately predicting its exact value is difficult. This is because it is challenging to fully account for the leakage flux generated in the solenoid, and it is not straightforward to quantify the loss of electromagnetic force caused by eddy currents. Therefore, it is common to estimate the electromagnetic force through electromagnetic field analysis using FEM software.

In this study, the analysis was performed under steady-state conditions with no time variation of magnetic flux, using a two-dimensional axisymmetric model. The model consisted of the core, plunger, coil, and surrounding air region, and the coil was assigned a current of 0.8 A and 1148 turns. The core material was set to STS430FR, and magnetic saturation was taken into account by inputting the nonlinear B–H curve of the material, as shown in Fig. 8. In a solenoid, the magnetic flux path changes with the air gap distance, and the leakage flux distribution changes accordingly.

Fig. 8.

Magnetic flux density versus magnetic field intensity for (a) STS430FR and (b) SPCC.

In this section, FEM analyses were conducted for each valve stroke (air gap distance) to obtain the magnetic flux density and electromagnetic force, and the air-gap-dependent electromagnetic force characteristics were compared with and analyzed against the experimental results. Fig. 9 shows the air-gap-dependent magnetic flux density distributions obtained from the FEMM analysis. It can be confirmed that the flux is concentrated around the air gap region between the core and the plunger, and the air gap flux density exhibits a nonlinear decrease as the air gap distance increases, as shown in Fig. 10.

Fig. 9.

Magnetic Flux Flow Comparison for Different Stroke Positions. (a) Air gap is 0.10 mm, (b) Air gap is 0.30 mm, (c) Air gap is 0.60 mm, (d) Air gap is 0.90 mm and (e) Air gap is 1.20 mm

Fig. 10.

Variation of air gap magnetic flux density with air-gap distance

As presented in Table 4. and Fig. 11, when the electromagnetic force acting on the plunger was measured separately during the air-gap-decreasing (descending) and air-gap-increasing (ascending) interval, and the average value was used, the FEM-predicted forces were generally higher than the experimental values. This is because the 2D axisymmetric FEM model, defined about the model axis, cannot adequately capture (i) the leakage flux occurring on the side of the housing near the power supply unit and (ii) the leakage flux associated with the fluid flow passage machined on one side of the plunger. As a result, the experimentally measured force tends to be lower than the simulated value [11].

Table 4.

Comparison of electromagnetic Experimental Values and Analysis Values

Fig. 11.

Comparison of Electromagnetic Force-Air Gap Curve Experimental Values and Analysis Values

As shown in Fig. 9(a), when the air gap distance is around 0.10 mm, the flux density in the core-plunger air gap region increases to approximately 1.32 T, entering the pre-saturation region on the B–H curve of STS430FR. In this pre-saturation region, the slope of the B–H curve changes rapidly due to permeability variations, and local flux concentration, and three-dimensional changes in flux distribution can occur near the plunger edge. Therefore, even with a nonlinear B–H curve, a 2D axisymmetric model cannot fully reproduce the flux distribution and saturation behavior of the actual three-dimensional structure [12]. Consequently, at an air gap distance of (a) 0.10 mm, the experimental force was 34.9 N while the FEM result was 39.8 N, yielding an absolute error of 4.90 N (relative error of 14.0%), which was the largest discrepancy among the tested points.

In contrast, in regions where the air gap increases (e.g., (b) 0.30 mm~(e) 1.20 mm), most of the magnetic flux forms a closed magnetic circuit through the core-plunger path, so the influence of material nonlinearity becomes relatively smaller. As the air gap length increases, the magnetic reluctance of the air gap increases and the proportion of fringing and leakage flux increases; however, the overall flux level and the electromagnetic force decrease. Moreover, in a 2D axisymmetric model, if a sufficiently large air region and appropriate boundary conditions are applied, the fringing flux distribution around the air gap can be reasonably reproduced. Thus, when the geometry is close to axisymmetric, and the leakage is dominated by axisymmetric components, the discrepancy between FEM and experiment tends to decrease as the air gap increases [13].

Meanwhile, in the air gap region around (d) 0.90 mm, the absolute magnitude of the electromagnetic force is small; therefore, even an absolute error on the order of 1.30 N corresponds to a relatively large relative error of 12.6%, showing a tendency to increase compared with other air gap points except at (a) 0.1 mm and (b) 0.3 mm. However, this absolute error corresponds to about 3.2% of the maximum FEM-predicted force (approximately 39.8 N), and overall, the FEM results and experimental measurements show good qualitative agreement across the entire air gap range.

Accordingly, the FEM program used for electromagnetic field analysis and the analysis procedure employed in this study can be regarded as reliable, given that the difference between the simulated and measured forces remains within approximately 5 N. This trend is also consistent with previous reports indicating that modeling leakage flux as a separate magnetic reluctance element can significantly reduce discrepancies between magnetic circuit analysis and FEM results. Likewise, the FEM-experiment differences observed in this study support the conclusion that the primary sources of error are leakage flux and differences in three-dimensional flux distribution [14].