# Magnetic Force Enhancement Using Air-Gap Magnetic Field Manipulation by Optimized Coil Currents
## 5 Optimization Result  
The proposed optimization strategy and problem formulation are applied for finding the optimal coil current distribution in the benchmark actuator model in Figure 3. Specifically, the number of coil turns (i.e., design variable \(\beta_{i}\)) at the design domain located at the yoke near the air-gap is determined for maximizing the average force during the plunger movement in a tangential direction to the air-gap line. In this work, the number of design variable \(N\) is set to 20, which means the design domain is divided into 20 small areas. The total number of coil turns, \(\beta_{total}^{*}\) in (10), is set to 20, which is empirically determined as a value that could reveal the magnetic field manipulation effect. When \(\beta_{total}^{*}\) is too large or small compared to winding turns, the manipulation effect is reduced. The maximum number of coil turns \(\beta_{max}\) in (11) is also set to 20, and the coil current coil \(I_{0}\) in (5) is set to 10 A. Figure 6a-d shows the distribution of design variables \(\beta_{i}\) during the optimization iteration. As shown in Figure 6a, the design variables are evenly distributed as 0.1 at the initial iteration. As the optimization iteration proceeded, the design variables moved to find the optimal distribution in Figure 6b,c. Then, the design variables \(\beta_{i}\) tended to converge into discrete integer numbers, as shown in Figure 6d, due to the proposed continuous and discrete optimization scheme in (5) and (6). Figure 6: Distribution of the design variables \(\beta_{i}\) during the optimization iteration: (**a**) iteration step 1, (**b**) iteration step 51, (**c**) iteration step 102, and (**d**) iteration step 192 (final step). The final optimization results are described in Figure 7. Figure 7a,b shows the distribution of current density magnitude \(\mu_{s}^{i}\) (\(i=1\dots 20\)) and relative permeability \(\mu_{r}^{i}\) at 20 small design areas, respectively. In Figure 7b, the area where the relative permeability \(\mu_{r}^{i}\) is 1200 represents the iron material, and the area where \(\mu_{r}^{i}\) is 1 represents the coil. Using the proposed formulation, both the current density and material state at the design domain are successfully determined simultaneously. In the optimization result, it is observed that the left part of the design domain is optimized to have a strong current density, while the right part is designed as iron material. To validate the effectiveness of the design result, the force profile of the design with magnetic field manipulation by optimized current distribution is compared with that of an original design without manipulation (i.e., no coil currents near the air-gap) in Figure 8. Here, the original design is built by setting all design variables \(\beta_{i}\) as zero, which means that every design domain is simply set as the iron material. In the original design, the number of winding turns is set as 120, which is 20 (i.e., the total number of coil turns in design domain) higher than the optimized coil current case, for a fair comparison. As shown in Figure 8, the design with optimized coil current successfully enhances the average magnetic force compared to the original design. The average force \(F_{avg}\) (i.e., objective function in (8)) of the design with optimized coil current is calculated as 128 N, which is 11.12% higher than that of the original design. This force enhancement might come from the magnetic field manipulation by the optimized current distribution. In order to confirm that the force enhancement results from the manipulated magnetic field, the magnetic flux density along the air-gap line is investigated. Figure 9a,b compares the magnetic flux density distribution along the air-gap line of the original design without magnetic field manipulation and the design with the manipulation by optimized coil currents. As expected, it is observed that the optimized coil currents manipulated the air-gap magnetic field toward the ideal field distribution.