# Magnetic Levitation Control Based on Flux Density and Current Measurement
## 2 Modeling
Soft iron domains, on the other hand, are represented with a nonlinear relation to account for saturation:  
\[\mathbf{H}=f_{\text{Fe}}\left(\left|\mathbf{B}\right|\right)\frac{\mathbf{B}}{ \left|\mathbf{B}\right|} \tag{5}\]  
The coil domain has the same magnetic properties as air. However, a specific feature of the software allows to define a homogenized multi-turn coil with external current density  
\[\mathbf{J}_{e}=\frac{N_{c}i}{A_{c}}\mathbf{e}_{c}, \tag{6}\]  
where \(N_{c}\) is the number of turns of the coil, \(A_{c}\) is its total cross section and \(\mathbf{e}_{c}\) is a vector field representing the local direction of the coil wires. This latter term is determined by a coil geometry analysis study step embedded in COMSOL Multiphysics. The symmetry of the model allows reducing the geometry size to one quarter, provided that the mover is perfectly centered with respect to the electromagnet. To decrease the computational overhead, this symmetry is exploited by imposing a Dirichlet's boundary condition on all the surfaces of the model:  
\[\mathbf{n}\times\mathbf{A}=\mathbf{0} \tag{7}\]  
where \(\mathbf{n}\) is the unit vector normal to the surface where the condition is applied. The resulting FE model is shown in Figure 2. It is simulated for a matrix of current and vertical displacement input values ranging from 0.1 to 1 A and 1 to 10 mm, respectively. The coordinate \(q\) is assumed null when the mover is in contact with the central limb of the electromagnet and positive as the mover goes downwards. Figures 3-5 illustrate the behavior of the analyzed terms for the working set of \((i,q)\) input pairs. It is observed that the variables exhibit nonlinear behavior with respect to the inputs. This is particularly emphasized for the magnetic force developed by the actuator. In addition, the coil flux linkage \(\lambda\) and the measured magnetic flux density \(B_{s}\) exhibit similar behaviors. Discrepancies are found when the mover is very close to the electromagnet because the flux on the side limb tends to change direction, while \(B_{s}\) takes only the flux density component orthogonal to the cross section of the  
sensing element. Please note that the behavior of \(B_{s}(i,q)\) is numerically defined in the examined range. Therefore, the measurements of \(B_{s}\) and \(i\) can be used to determine indirectly the vertical position of the mover. Figure 4: Numerical behavior of the magnetic force \(f_{m}\) as a function of current \(i\) and displacement \(q\). Figure 3: Numerical behavior of the coil flux linkage \(\lambda\) as a function of current \(i\) and displacement \(q\). Figure 1:0 Finite-element 3D model of the levitation system: (**a**) geometry and domains; (**b**) mesh consisting of 109,210 elements; (**c**) magnetic flux density norm distribution for \(i=650\) mA and \(q=5\) mm, where the geometry is mirrored and the air domains are omitted for display purposes. As an alternative to the numerical approach proposed here, the relation \(B_{s}(i,q)\) (see Figure 5) could be characterized experimentally by adding a position sensor. However, a suitable position control is needed _a priori_ to stabilize the plant and obtain reliable measurements. When applying a feedback control law, the flux density and the coil current values are prone to corruption due to feedback noise propagation.