# Design and Simulation of Electromagnetic Linear Actuators for Jet Dispensers
## 5 Results and Discussion  
### Models, Convergence, and Thrust Force after Modification  
As described in the previous section, two specific geometry variables (\(d\) and \(e\)) were assigned to the SA and MCA/MMA actuators, respectively, and simulations were then performed to determine the geometry settings which maximized the actuator thrust with the minimal current. Table 3 shows the considered design range for each variable and the value found to maximize the actuator thrust force. Figure 9 shows the models of the SA, MCA, and MMA before and after the modification process, respectively.  
Figure 8: Boundary conditions imposed on 2-D symmetric models. (**a**) Balloon boundary; (**b**) movable part; and (**c**) symmetric axis.  
Figure 7: B-H curve of AISI 1020 carbon steel.  
Figure 10 shows the convergence plot of the thrust force by varying the mesh density in each case given an excitation current of 2A at the initial position of \(z\) (0.2 mm for SA and 0 mm for both MCA and MMA). The thrust forces generated by SA, MCA, and MMA models start to reach a good convergence when mesh element numbers are higher than 5000, 8000, 8000 elements, respectively. Accordingly, the thrust forces approximately converge at 63.42, 15.65, and 14.72 N for SA, MCA, and MMA, respectively. Although a higher mesh density enhances the accuracy of predicted thrust forces, it significantly increases computation time. Consequently, the number of mesh elements mentioned above for each model is reliable enough and is used for setting mesh operations in the next simulation steps.  
Figure 11 shows the meshed geometric models and magnetic field distribution at \(I\) = 2A of SA, MCA and MMA after modification, respectively. For SA model, the biggest values of the magnetic field at the position of corners on the coil holder exceeds saturation value because of the edge and thin wall effects. As described above, the MCA and MMA structures are essentially similar, with the only difference between them being whether the coil windings (MCA) or magnet (MMA) move during operation. As a result, the two actuators have similar magnetic flux distribution. The highest value of the magnetic field concentrates on iron-rings where are close to the coil.  
Figure 12 shows the simulated thrust force for the three models before and after the modification process. It is seen that for every model, the re-design process increases the thrust force produced by the actuator. From inspection, the average percentage improvements in the thrust force of the SA, MCA, and MMA models are found to be about 27.5%, 36%, and 48%, respectively, over the considered actuation current range of 0.6-2.6 A with increments of 0.2 A.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Symbol**} & **Variable** & **After** & \multirow{2}{*}{**Symbol**} & **Variable** & **After** \\  & **Range** & **Modification** & & **Range** & **Modification** \\ \hline \(d_{1}\) & 2.5–4 & 2.5 & \(e_{3}\) & 1–2 & 1.25 \\ \(d_{2}\) & 0.3–1 & 0.4 & \(e_{4}\) & 0.3–1 & 0.6 \\ \(d_{3}\) & 1–2.5 & 1.7 & \(e_{5}\) & 1–2.5 & 1.5 \\ \(d_{4}\) & 1–3 & 3 & \(e_{6}\) & 5–23 & 13 \\ \(d_Figure 11: Meshed geometry and magnetic field distribution of models after modification. (**a**) SA model with 5118 mesh elements, and (**b**) MCA and MMA models with 8612 mesh elements.  
Figure 10: Thrust force convergence of modified models.