# Magnetic Levitation Control Based on Flux Density and Current Measurement
## 2 Modeling  
The magnetic levitation demonstrator consists of an electromagnet fixed to a frame and a levitated spherical body (mover), as illustrated in Figure 1. The electromagnet is an E-shaped core made of laminated soft iron with a coil wound on its central limb. The mover is devised with a thin soft iron yoke to reduce its mass. Table 1 summarizes the main geometric and layout features of the described setup. The system is governed by the following dynamic equations:  
\[u =\dot{\lambda}(i,q)+Ri, \tag{1}\] \[m\ddot{q} =f_{m}(i,q)-f_{d}. \tag{2}\]  
Equation (1) describes the behavior of the coil circuit, where the input voltage \(u\) is contrasted by [PERSON]'s law through the coil resistance \(R\) and by [PERSON]'s law with the variation of the coil flux linkage \(\lambda\) in time. The flux linkage \(\lambda\) depends on the magnetomotive force applied to the magnetic circuit and the reluctance, which decreases with the air gap. Hence, \(\lambda\) is a function of the vertical displacement \(q\) and the current \(i\). Equation (2) defines the dynamic equilibrium of the mover body. The resultant force includes a magnetic component \(f_{m}\) developed by the electromagnet and an external disturbance \(f_{d}\) (usually the weight of the sphere). The force \(f_{m}\) is a function of the vertical displacement coordinate \(q\) and the supply current \(i\). \begin{table}
\begin{tabular}{l l c c} \hline \hline
**Subsystem** & **Property** & **Value** & **Unit** \\ \hline \multirow{4}{*}{Electromagnet} & Height & 64 & mm \\  & Depth & 17 & mm \\  & Limb width & 16 & mm \\  & Slot width & 16 & mm \\  & Total width & 80 & mm \\ \hline \multirow{4}{*}{Coil} & Wire gauge & 0.5 & mm \\  & Number of turns & 684 & - \\ \hline \multirow{4}{*}{Mover} & Outside diameter & 92 & mm \\  & Yoke thickness & 0.5 & mm \\  & Mass & 67 & g \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric and layout features of the magnetic levitation demonstrator. Figure 1: Model scheme of the magnetic levitation demonstrator. From the control point of view, the voltage \(u\) is the command input and the current \(i\) and flux density \(B_{s}\) are the measured variables. This latter quantity is determined using an analog sensor installed on one of the side limbs of the electromagnet. Its role is to measure the flux density component \(B_{s}\) which is orthogonal to the cross section of the sensing element (see Figure 1). The relation between \(\lambda\), the current \(i\) and the vertical displacement \(q\) can be known either experimentally or from simulation. Thus, the measurement of \(B_{s}\) and \(i\) allows to estimate \(q\), provided that the magnetic hysteresis of the ferromagnetic parts is negligible. The behavior of \(\lambda\), \(f_{m}\) and \(B_{s}\) is necessary to implement any model-based control strategy on the described system. However, the analytical determination of these terms is not trivial due to magnetic saturation on the sphere yoke, irregular or unknown flux paths, non-homogeneous air gap, leakage and stray flux effects. To simplify the analysis, the model is evaluated with the aid of a 3D finite element model in COMSOL Multiphysics. If losses in the iron domains are neglected, the electromagnetic problem formulation in terms of the magnetic vector potential \(\mathbf{A}\) is given by  
\[\mathbf{B}=\mathbf{\
abla}\times\mathbf{A}, \tag{3}\] \[\mathbf{\
abla}\times\mathbf{H}=\mathbf{J}_{e}, \tag{4}\]  
where \(\mathbf{B}\) and \(\mathbf{H}\) are the magnetic flux density and field vectors, respectively, and \(\mathbf{J}_{e}\) is the external current density vector. Neglecting iron losses implies that the stated problem can be solved in a magnetostatic formulation, i.e., without the need for a computationally intensive time-stepping solution. Air domains surrounding the actuator and inside the sphere are characterized by a unitary relative permeability with respect to vacuum.