# Levitating Control System of Maglev Ruler Based on Active Disturbance Rejection Controller
## 2 Analysis Model of Levitating Force
### Analysis Model of Buoyancy Force
In addition, due to the different leakage coefficients of the basic levitating loop and the control levitating loop, \(R_{M\_U}\), \(R_{M\_D}\), \(R_{L\_U\_A}\), \(R_{L\_D\_A}\), \(R_{R\_U\_A}\), \(R_{R\_D\_A}\) can also be decomposed into \(R_{M\_U\_B}\), \(R_{M\_U\_C}\), \(R_{M\_D\_B}\), \(R_{M\_D\_C}\), \(R_{L\_U\_A\_B}\), \(R_{L\_U\_A\_C}\), \(R_{ICU\_U}\), \(R_{L\_D\_A\_B}\), \(R_{L\_D\_A\_C}\), \(R_{ICU\_D}\). The known equation for calculating the magnetic reluctance is as follows [17]:  
\[R=\frac{l}{\mu A} \tag{6}\]  
According to the dimensions of the components of the MR [1], combined with the magnetic reluctance calculation Equation (6) and the finite element analysis results, the following magnetic reluctance calculation equations can be obtained:  
\[R_{11}=R_{21}=R_{31}=R_{41}=\frac{l_{h}}{\mu_{h\_\_\prime}A} \tag{7}\]  
\[R_{51}=R_{61}=R_{71}=R_{81}=\frac{l_{h}}{\mu_{h\_\_\prime}A} \tag{8}\]  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]0  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]1  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]2  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]3  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]4  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]5  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]6  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]7  
\[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]8  
where \(l_{h}\) is the length of loops 1, 2, 3, and 4 on the T-shaped conductor arm and the length of loops 5, 6, 7, and 8 on the bottom of the U-shaped conductor. \(l_{s\_y0}\) is the length of the eight magnetic loops on the U-shaped conductor side arm when the MC is in its center position. \(l_{m\_y0}\) is the length of the eight magnetic loops on the T-shaped conductor vertical arm. \(l_{A}\) is the length of each air gap when the MC is in the center position. \(l_{CU}\) is the diameter of copper wire. \(y\) is the offset of the MC on the y-axis. \(z_{l}\) is the upward offset of the left end of the MC. \(z_{r}\) is the upward offset of the left support of the MC [1]. After setting the magnetic leakage coefficients \(\sigma_{BM}=6\) and \(\sigma_{CM}=3\), the equation calculates the magnetic flux of eight magnetic loops from Equations (7)-(27). The results of Equations (7)-(27) are presented in \[F_{left}=F_{l\_air\_up}-F_{l\_air\_down} \tag{1}\]9  
This paper takes magnetic loop 1 and magnetic loop 5 as examples to analyze the calculation method of the magnetic flux of each magnetic loop [1]. As shown in Figure 1a, the magnetic flux of loop 1 starts from the permanent magnet, travels along the left arm of the U-shaped magnetic conductor, the MC, the vertical arm of the T-shaped magnetic conductor, and then returns to the permanent magnet. According to Equation (5), the magnetic flux of the 1 magnetic loop is the ratio of the average magnetic potential of the permanent magnet to the total magnetic reluctance of loop 1. Let the total magnetic reluctance of the loop 1 be \(R_{1C}\). The equation for calculating the total magnetic reluctance is as follows:  
\[F_{R}=-\frac{1}{2}\frac{\phi^{2}}{\mu_{0}\cdot A} \tag{2}\]0  
Substituting the average magnetic potential \(V_{m}\) and \(R_{1C}\) into Equation (5), we can obtain the magnetic flux of the loop 1 \(\phi_{1}\):  
\[F_{R}=-\frac{1}{2}\frac{\phi^{2}}{\mu_{0}\cdot A} \tag{2}\]1The magnetic flux of loop 5 starts from the levitating coil, follows the left arm of the U-shaped magnetic conductor, the MC, the vertical arm of the T-shaped magnetic conductor, and then reaches the bottom of the U-shaped magnetic conductor and finally returns to the levitating coil. According to Equation (5), the magnetic flux of loop 5 is the ratio of the magnetic potential of the levitating coil to the total magnetic reluctance of the loop 5. Let the total magnetic reluctance of loop 5 be \(R_{\rm 5C}\).