## 5 Improved Tracking Differentiator (SYSTD)
### A Method for Estimating the Position of the Mover Iron Core Based on SYSTD  
In order to simplify the system structure, the horizontal position of the MRC in this paper is not obtained by survey with a position sensor. Instead, a measuring coil can be wound along the horizontal control coil, and the position feedback signal of the mover core can be estimated from the back electromotive force obtained from the measuring coil. During the operation process, the MRC is only affected by the electromagnetic suction force, the horizontal thrust force, and the gravitational force. Among them, the magnitude and direction of the electromagnetic suction force are related to the air-gap magnetic field of the magnetic levitation ruler system. The air-gap magnetic field is mainly distributed between the stator yoke and the mover iron core [2]. However, there are still residual air-gap magnetic fields on the front and rear sides of the MRC. When the suspension position of the mover is 1.1 mm, the upper and lower residual air-gap magnetic fields are symmetrical. Under the action of this part of the air-gap magnetic field, there are horizontal components in the upward and downward electromagnetic suction forces of the MRC. In this paper, the force analysis of the MRC is carried out by means of MAXWELL. Figure 13 shows the force analysis diagram of the MRC at the left-hand end of the MRC. As can be seen from the figure, the electromagnetic suction force generated by the air-gap magnetic field is mainly in the Z-direction. However, there are also horizontal components. The horizontal component of the electromagnetic suction force on the front side of the MRC is in the backward direction; the horizontal component of the electromagnetic suction force on the rear side of the MRC is in the forward direction. Most of the horizontal components of the electromagnetic suction force cancel each other out. By means of the MAXWELL field calculator, it is concluded that the total horizontal electromagnetic suction force component of the MRC is approximately 0.04 mN, which is less than 0.1% of the horizontal thrust. Therefore, the horizontal component of the electromagnetic suction force can be neglected. Figure 12: Comparison of filtering characteristics of three TDs. The horizontal control coil is in contact with the upper and lower surfaces of the MRC. As can be seen from Figure 14, the magnetic field lines of the air-gap magnetic field are perpendicular to the upper and lower surfaces of the MRC and the horizontal control coil in contact with these surfaces. Figure 14: Distribution diagram of the magnetic field lines of the upper and lower air-gap magnetic fields of the mover iron core. (**a**) Upper air-gap. (**b**) Lower air-gap. Figure 13: Force analysis diagram of the MRC at the left end. Thus, an induced-electromotive-force measuring coil can be wound along the horizontal control coil, and the position change of the MRC can be estimated by detecting the induced electromotive force in the measuring coil. The design objective of the measuring coil is to detect the induced electromotive force, and there is no requirement for the coil current. Therefore, in this paper, enameled wire with a wire diameter of 0.05 mm is chosen to wind the induced-electromotive-force measuring coil along the gaps between the wires of the horizontal control coil, and the number of turns of the coil can reach around 300 as shown in Figure 15.  
Figure 15 is a schematic diagram of the winding method of the coil on the left-front side of the mover core. The installation method of the measuring coils on the left-rear side, right-front side, and right-rear side of the mover iron core is the same as this. Among them, the port AB is used to measure the induced electromotive force. By connecting the two groups of measuring coils on the left side in series, the left-side induced electromotive force \(e_{l}\) can be obtained; by connecting the two groups of measuring coils on the right side in series, the right-side induced electromotive force \(e_{r}\) can be obtained.