# Levitating Control System of Maglev Ruler Based on Active Disturbance Rejection Controller
## 2 Analysis Model of Levitating Force
### Analysis Model of Buoyancy Force  
The levitating force of the MC comes from three parts: the electromagnetic suction force on the left support, the right support, and the center point. The schematic diagram of the electromagnetic suction force on the magnetic loop and the MC is shown in Figure 1.  
The MR is divided into three parts, the left, right, and middle, as shown in Figure 1b. The levitating force in the middle is affected by the levitating forces on the left and right, and these levitating forces are similar. Therefore, this paper first studies the levitating force using the left levitating force as an example. Figure 1b also indicates the electromagnetic suction force acting on the MC. The resultant force of the upper and lower electromagnetic suction forces on the MC is the left levitating force:  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]  
where \(F_{left}\) is the levitating force on the left end of MC, \(F_{l\_air\_up}\) is the upward electromagnetic suction force on the left end of MC, and \(F_{l\_air\_down}\) is the downward electromagnetic suction force on the left end of MC. According to the electromagnetic suction force calculation equation [15]:  
\[F_{R}=-\frac{1}{2}\frac{\phi^{2}}{\mu_{0}\cdot A} \tag{2}\]  
where \(\mu_{0}\) is the permeability of the air gap; \(A\) is the area of the pole surface, which is the vertical intersection of the U-shaped conductor and the MC, i.e., the black area on the MC in Figure 1b. Therefore, the electromagnetic suction force \(F_{R}\) is proportional to the square of the magnetic flux \(\phi\)[1]. There are eight magnetic loops on the MR, which are numbered as shown in Figure 1a. The magnetic fluxes of the loop 1 to loop 8 magnetic loops are respectively: \(\phi\)1, \(\phi\)2, \(\phi\)3, \(\phi\)4, \(\phi\)5, \(\phi\)6, \(\phi\)7, and \(\phi\)8. Among them, \(\phi\)1, \(\phi\)2, \(\phi\)3, and \(\phi\)4 are the magnetic fluxes of the basic levitating loop; \(\phi\)5, \(\phi\)6, \(\phi\)7, and \(\phi\)8 are the magnetic fluxes of the control levitating loop. As shown in Figure 1a, the magnetic flux in the upper air gap on the left support is the sum of the magnetic flux of the base levitating loop and the control levitating loop, and both directions are downward. Therefore, the calculation equation for the magnetic flux in the upper air gap on the left support is [16]:  
\[\phi_{l\_up\_air}=\phi_{1}+\phi_{5} \tag{3}\]  
Similarly, the magnetic flux in the lower air gap of the left support is also the sum of the magnetic flux of the base levitating loop and the control levitating loop, and both directions are upward. Therefore, the calculation equation for the magnetic flux in the lower air gap of the left support is [16]:  
\[\phi_{l\_down\_air}=\phi_{3}+\phi_{7} \tag{4}\]  
Figure 1: (**a**) Magnetic loops of MR. (**b**) The electromagnetic suction force on the MC. The total magnetic flux of the air gap in the upper part of the left support starts from the left upper permanent magnet and the left upper levitating coil. According to [PERSON]'s law [16]:  
\[\phi=\frac{V}{R} \tag{5}\]  
where \(V\) represents the magnetic potential and \(R\) represents the magnetic reluctance. As can be seen from Figure 1, the structures of the eight levitation magnetic loops are similar, consisting of permanent magnets or levitating coils and six magnetic conductors. For ease of calculation, a label is assigned to the magnetic reluctance of each magnetic conductor, as shown in Figure 1a. As shown in Figure 1a, the MR has 28 magnetic resistors, including \(R_{11}\), \(R_{12}\), \(R_{21}\), \(R_{22}\), \(R_{31}\), \(R_{41}\), \(R_{42}\), \(R_{51}\), \(R_{52}\), \(R_{61}\), \(R_{62}\), \(R_{71}\), \(R_{72}\), \(R_{81}\), \(R_{82}\), \(R_{M\_U}\), \(R_{M\_D}\), \(R_{F\_U}\), \(R_{F\_D}\), \(R_{B\_U}\), \(R_{B\_D}\), \(R_{L\_M}\), \(R_{R\_M}\), \(R_{R\_U\_A}\), \(R_{L\_D\_A}\), \(R_{R\_U\_A}\), \(R_{R\_D\_A}\). The air gap length of the MR can also be divided into two parts: the nominal air gap length and the coil air gap length occupied by the coil.