## 5 Improved Tracking Differentiator (SYSTD)
### A Method for Estimating the Position of the Mover Iron Core Based on SYSTD
The calculation formula for the induced electromotive force when the left-side measuring coil cuts the magnetic field lines is as follows [20]:  
\[e_{l}=BLv=(B_{u}+B_{d})L\frac{dS_{l}}{dt} \tag{56}\]  
In the formula, \(B_{u}\) is the air-gap magnetic field strength on the upper-left side of the MRC, \(B_{d}\) is the air-gap magnetic field strength on the lower-left side of the MRC, \(L\) is the length of the measuring coil in the air-gap magnetic field, and \(S_{l}\) is the displacement of the MRC. The calculation method of the induced electromotive force \(e_{r}\) is the same as that of \(e_{l}\), so it will not be repeated here. In addition, since the measuring coil is wound along the horizontal control coil, the mutual-inductance coefficient \(k\) between the measuring coil and the horizontal control coil is infinitely close to 1. Therefore, there is mutual inductance between the measuring coil and the horizontal control coil. According to the mutual-inductance calculation formula, the mutual inductance \(M\) can be estimated as follows:  
\[M=k\sqrt{L_{c}\times L_{k}} \tag{57}\]  
In the formula, \(Lc\) is the inductance of the measuring coil; \(L_{k}\) is the inductance of the horizontal control coil. The calculation formula for the inductance is as follows:  
\[L=\frac{0.01\times D\times N^{2}}{\frac{H}{D}+0.44} \tag{58}\]  
In the formula, \(L\) represents the inductance, \(D\) represents the diameter of the coil, \(N\) represents the number of turns, and \(H\) represents the length of the coil. Figure 15: Schematic diagram of the winding of the measuring coil on the left-front side of the MRC. Thus, it can be calculated that \(L_{c}\approx\) 12,272 \(\mu\)H, \(L_{k}\approx\) 122 \(\mu\)H. Therefore, \(M\approx\) 1.2 \(\times\) 10\({}^{-3}\)\(\mu\)H. In the measuring coil, the calculation method of the induced electromotive force \(e_{c}\) generated by the mutual inductance is as follows:  
\[e_{c}=-M\frac{di_{c}}{dt} \tag{59}\]  
In the formula, \(i_{c}\) represents the current of the horizontal control coil. The MRC operates in a fixed-step-length stepping motion mode. Therefore, the current of the horizontal control coil is a square wave signal. Here, \(e_{c}\) is the induced electromotive force generated by the mutual inductance. Only at the rising edge and falling edge of the square wave signal will it cause significant fluctuations in \(e_{c}\). The duration of these fluctuations is influenced by the time constant \(\tau_{k}\) of the horizontal control coil. \[\tau_{k}=\frac{L_{k}}{R} \tag{60}\]  
\(R\) is the equivalent resistance of the horizontal control coil, which is measured to be 1.4 \(\Omega\). \(\tau_{k}\) is approximately 8.7 \(\mu\)s. To ensure the calculation accuracy, the rising edge and the falling edge of the current in the horizontal control coil are taken as six times the value of \(\tau_{k}\), that is, 52.2 \(\mu\)s. During this period, the frequency of the waveform of \(e_{c}\) is much higher than that of the induced electromotive forces \(e_{l}\) and \(e_{r}\), and its amplitude is also larger. This issue can be addressed by low-pass filtering for noise reduction. In this paper, SYSTD is used to observe the three signals \(B_{u}\), \(B_{d}\), and \(e_{l}\), respectively, so as to obtain more accurate values of the air-gap magnetic field strength and the induced electromotive force. After substituting these values into Equations (4)-(56), a relatively accurate estimated value of the position of the left end of the MRC can be calculated. The estimated value of the position of the right end of the MRC can be obtained in the same way.