# Levitating Control System of Maglev Ruler Based on Active Disturbance Rejection Controller
## 2 Analysis Model of Levitating Force
### The Correlation between the Levitating Force and the Levitating Position of the MC  
From Equations (33) and (34), it can be seen that the levitating force is related to both the upward offset of the MC and the planar position of the MC.  
Let the current of the levitating coil be zero and the plane position \(y\) = 0, then Equations (33) and (34) can be transformed into:  
\[F_{left}=-(\frac{1}{0.1206-5.981\times z_{l}-1.9715\times z_{r}})^{2}+(\frac{ 1}{0.1206+5.981\times z_{l}+1.9715\times z_{r}})^{2} \tag{35}\]  
\[F_{right}=-(\frac{1}{0.1206-1.9715\times z_{l}-5.981\times z_{r}})^{2}+(\frac{ 1}{0.1206+1.9715\times z_{l}+5.981\times z_{r}})^{2} \tag{36}\]  
From Equations (34) and (35), it can be seen that there is a cross-coupling between the levitating force on both sides of the MC and its upward offset.  
When the left side of the MC moves upward, its left upward offset \(z_{l}\) increases, and the length of the upper left air gap decreases, which also decreases the length of the middle upper air gap, but the decrease is less than that of the left upper air gap. The decrease in two air gaps results in a decrease in two magnetic reluctance. Since the magnetic potentials of each magnetic loop remain unchanged, the magnetic fluxes of both loops increase, which is the increase in magnetic flux of loop 1 and loop 2, resulting in an increase in electromagnetic suction force on the upper left side. The decrease in the upper left air gap and the increase in the length of the lower left air gap result in an increase in the magnetic reluctance of the lower left air gap, which reduces the magnetic flux of loop 3, resulting in a decrease in electromagnetic suction force on the lower left side. Since the left levitating force is the sum of the upper electromagnetic suction force and the lower electromagnetic suction force on the left MC, an increase in \(z_{l}\) on the left side will result in an increase in levitating force on the left side. At the same time, an increase in magnetic flux of loop 2 results in an increase in electromagnetic suction force on the upper right side, while at this time, the length of the middle lower air gap and the left lower air gap increases, and magnetic flux of loop 3 decreases, resulting in a decrease in electromagnetic suction force on the lower right side. Therefore, the levitating force on the right side also increases upward. Since the upward offset \(z_{r}\) of the MC is less than that of the left side, \(z_{l}\) has a greater impact on the levitatingforce on the left side than on the right side. Similarly, \(z_{r}\) also has an impact on the levitating force on both sides of the MC. The influence of \(z_{r}\) on the right side levitating force is greater than that on the left side. Therefore, it can be seen that there is a strong coupling between the levitating forces on both sides of the MC and their upward offsets.  
From Equations (32) and (33), it can be seen that the levitating forces on both sides of the MC are affected by the planar position \(y\) of the MC. From Equations (10)-(13), it can be seen that changes in the planar position \(y\) of the MC cause some changes in magnetic reluctance, resulting in changes in the magnetic flux of the eight magnetic loops, which ultimately lead to changes in the levitating forces on both sides of the MC.  
The instantaneous upward offset and plane position changes of the MC can cause changes in relative magnetic permeability, but this is not included in the analytical model. To increase the precision of the model, it is necessary to use the actual curve instead of a linear approximation curve. However, this also increases the complexity of the analytical model and introduces more uncertainties.  
The permanent magnet in the MR causes the magnetic loop to be in a magnetic saturation state, so that the magnetization point is only slightly affected by the upward shift and plane position changes of the MC. Therefore, the change in the relative magnetic permeability of the MR is also very small and can be almost ignored in engineering.