# Design and Simulation of Electromagnetic Linear Actuators for Jet Dispensers
## 4 Proposal Actuators' Model  
As described in Section 2, the aim of the second part of the design framework was to modify the three proposal actuator designs in such a way as to generate the biggest thrust force. To simplify the modeling process, and reduce the computation time, the actuator models were further simplified to the forms shown in Figure 6. As shown, each model was constructed using only the most essential features of the actuator structure. For example, the SA model contains only the plunger, the iron core, and the coil. Similarly, the MCA and MMA models comprise only the permanent magnet, two iron-rings, two serial coils, the shell, and an insulated aluminum ring. Note that for all three actuators, the remaining components were simply ignored since they are either made of non-magnetic materials such as aluminum and rubber, or have no effect on the magnitude of the simulated thrust force. The modeling process comprised two steps. In the first step, the 2-D axisymmetric models of the SA, MCA, and MMA were constructed using the same basic dimensions of the coil, including the coil height and the inner and outer diameters. Note that such an approach was deliberately taken in order to facilitate an objective comparison between the actuation performance of the three models. In the second step, specific geometric variables were added to each basic model, depending on the particular actuator involved, and assigned appropriate design values. For example, variable \(d\), was added to the SA model (see Figure 6a), while variable \(e\) was added to the MCA and MMA models (see Figure 6b,c). Note that the MCA and MMA models are the same and differ only in whether the coil windings (MCA) or magnets (MMA) move during the dispensing process. Table 2 summarizes the specific geometric dimensions of three basic models prior to their modification. In constructing the actuator models, the coil windings were assumed to be fabricated of copper with a relative permeability of \(\mu_{r}\)=1, while the magnets in the MCA and MMA models were assumed to be fabricated of NdFeB with a relative permeability of \(\mu_{r}\) = 1.05 and a residual magnetism of \(B_{r}\) = 1.43 T. The core and shell of the SA model, together with the iron rings and shell of the MCA and MMA models, were assumed to be fabricated of AISI 1020 carbon steel. AISI 1020 is a nonlinear material, and hence the relative permeability, \(\mu_{r}\), has a non-constant value. Accordingly, the \(B\)-\(H\) curve was defined in the form shown in Figure 7[44]. Finally, the insulated ring in the MCA and MMA models was fabricated of Al6061 aluminum alloy with a relative permeability of \(\mu\)r = 1. Figure 8 shows the boundary conditions imposed on each of the actuator models. The goal of the simulation process was to determine the force acting on the movable parts of the actuators, namely the plunger in the SA model, the coil windings in the MCA model, and the magnet core in the MMA model. For the SA model, the actuator was assumed to comprise a single-coil winding with 900 turns. By contrast, the MCA and MMA models were assumed to have multiple coil windings with 450 turns on each winding. For all of the models, the DC current in the cross-section of the coils was assumed to be uniformly distributed, with the flow direction of the current perpendicular to the cross-section surface [42]. For the SA model, the initial \(z\) gap was set as 0.7 mm and the plunger then moved toward the core over a stroke length of 0.5 mm. For the MCA and MMA models, the mid-points of the magnets and coil windings were set to coincide initially and the magnets then moved through a distance of 0.5 mm away from this mid-point position in a direction governed by the direction of the current flow through the coil.