# Magnetic Force Enhancement Using Air-Gap Magnetic Field Manipulation by Optimized Coil Currents
## 2 Ideal Magnetic Field Distribution for Magnetic Force Enhancement  
Before performing the optimization of coil currents, ideal air-gap magnetic field distribution for maximizing the tangential-direction magnetic force is explained in this section. An ideally manipulated magnetic field distribution between two soft ferromagnetic (i.e., iron) parts could maximize a tangential direction magnetic force exerted on the iron part. Figure 1a shows the magnetic field lines in a common operating situation, and Figure 1b shows ideally manipulated magnetic field lines for maximizing the magnetic force in a tangential direction to the air-gap line.  
The ideal magnetic field distribution for maximizing tangential direction magnetic force could be explained theoretically using [PERSON]'s stress tensor formulation [28]. [PERSON]'s stress tensor formulation describes the relation between magnetic force and magnetic field distribution along an integration path (i.e., air-gap lines):  
\[\mathbf{F}=\left[\oint\frac{1}{2\mu_{0}}\big{(}{B_{n}}^{2}-{B_{t}}^{2}\big{)} dS\right]\mathbf{n}+\left[\oint\frac{1}{\mu_{0}}B_{n}B_{t}dS\right]\mathbf{t} \tag{1}\]  
where \(S\) is any closed surface surrounding the object, \(\mu_{0}\) is the air permeability, \(\mathbf{n}\) and \(\mathbf{t}\) are normal and tangential unit vectors to the integration path, and \(B_{n}\) and \(B_{t}\) are magnetic flux densities, respectively, in normal and tangential directions with respect to the integration path. In (1), the second term of the right-hand side corresponds to the tangential direction magnetic force \(F_{t}\).  
By investigating (1), ideal magnetic field distributions, for maximizing the tangential direction magnetic force \(F_{t}\), could be found out. The tangential direction force \(F_{t}\) is calculated as the integral of the multiplication of normal and tangential magnetic flux density \(B_{n}\) and \(B_{t}\). If the amount of the magnetic flux along with the air-gap integration path, \(\Phi=\oint\sqrt{{B_{n}}^{2}+{B_{t}}^{2}}d(Area)\), stayed constant, the tangential force \(F_{t}\) might be maximized when the focused normal direction flux density, \(B_{n}\), is the same with the focused tangential direction flux density, \(B_{t}\). In other words, the magnetic field that is focused on a direction inclined 45 degrees to the integration path might be ideal for maximizing the tangential force, \(F_{t}\), as described in Figure 1b.  
The conceptual comparison of an evenly and focused distribution with 45 degrees inclination is presented in Figure 2. The tangential direction magnetic force, \(F_{t}\), is zero when the air-gap magnetic field is evenly distributed in the normal direction, as shown in Figure 2a. The magnetic force, \(F_{t}\), for focused and 45-degree inclined field in Figure 2c, is five times higher than that for evenly distributed and 45-degree inclined field in Figure 2b. In all three cases, the magnetic flux \(\Phi\) is identical. This comparison showed how the focused and 45-degree inclined magnetic field could be considered as an ideal distribution.  
Figure 1: Magnetic field lines (**a**) in a common operating situation, (**b**) manipulated ideally for maximizing tangential direction magnetic force.