## 2 Materials and Methods
### Experimental Setup and Calculation Principle  
The measuring unit is a solenoid coil located around the railway track. The meter contains two windings: generating and measuring. The generating winding creates an alternating electromagnetic field with constant frequency and intensity parameters. The measuring winding induction current is induced with instantaneous parameters depending on the local properties of the moving core of the solenoid. The object being measured serves as the core, which is a moving railway car in our case. The parameters of the measuring device are shown in Table 1.  
Figure 1 shows a diagram of the setup that was used to record the measurements. The parameters of the measuring coil system and electric signals used in the measurements are listed in Table 1. The measuring coil system consists of primary and secondary windings. The inductance of a coil with a solenoid within is calculated using Equations (8) and (9):  
\[L=\frac{\Phi}{I}, \tag{8}\]  
\[\Phi=NS_{p}H, \tag{9}\]  
where \(L\) is the inductance, \(I\) is the current, \(\Phi\) is the magnetic flux, \(N\) is the turn number, \(S_{p}\) is the solenoid cross-section area, and \(H\) is the magnetic field strength. If there is no metallic core inside the solenoid frame, inductance is calculated according to Equation (1), with permeability considered as being \(I\), which is the permeability of air:  
\[L=\frac{NS_{p}H}{I}, \tag{10}\]  
\[I=\frac{Hl_{av}}{N}, \tag{11}\]  
where \(l_{av}\) is the average length of line of the magnetic force. Therefore, in the case of a coil without a core:  
\[L_{air}=\frac{NS_{p}H}{I}=\frac{NS_{p}H}{H_{\overline{N}}^{I}}=\frac{N^{2}S_{ p}H}{Hl_{av}}=\frac{N^{2}S_{p}}{l_{av}} \tag{12}\]  
When the ferrous core passes through the coil, additional inductance appears within the coil:  
\[L_{met}=\frac{N^{2}S_{met}B}{Hl_{av}}=\frac{\mu N^{2}S_{met}}{l_{av}}, \tag{13}\]  
\[B=\mu H, \tag{14}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Name** & **Value** & **Unit** \\ \hline Range of alternating current in primary winding & 2.00–8.00 \(\pm\) 0.05 & A \\ Range of alternating magnetic field strength & 100–1000 & A/m \\ Range of alternating current frequency & 20–100 \(\pm\) 1 & Hz \\ Windings real resistance (\(\Omega\)) & & \\ -magnetizing winding & 1 \(\pm\) 0.2 & \(\Omega\) \\ -measuring winding & 12.5 \(\pm\) 2 & \(\Omega\) \\ Estimated value of the measuring coil cross-section area & 43.43 \(\pm\) 0.01 & m\({}^{2}\) \\ Reference value of magnetic flux in air & 1.9-10\({}^{-2}\) & Wb \\ Range of magnetic flux measurement & 1.5 \(\times\) 10\({}^{-2}\)–6.0 \(\times\) 10\({}^{-2}\) & Wb \\ Dimensions of measuring coil & 7190 \(\times\) 6040 \(\times\) 740 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the measuring coil system and electric signals used in measurements. where \(B\) is the magnetic inductance, \(\mu\) is the magnetic permeability, and \(S_{met}\) is the cross-section area of the railway carriage with metal. The total inductance is:  
\[L_{sum}=L_{air}+L_{met}=\frac{N^{2}\Big{(}S_{p}-S_{met}\Big{)}}{l_{av}}\ +\frac{\mu_{\text{ MeT}}N^{2}*S_{met}}{l_{av}}=\frac{N^{2}}{l_{av}}\Big{(}S_{p}-S_{met}+\mu_{met}S_ {met}\Big{)}, \tag{15}\]  
where \(\mu\) is the magnetic permeability of metal. With the help of Equation (15), it is possible to calculate the quality factor \(Q\):  
\[Q=2\pi f\frac{L_{sum}}{R}=tg\Delta\varphi, \tag{16}\]  
\[\Delta\varphi=\varphi_{\text{u}}-\varphi_{\text{I}}, \tag{17}\]  
where \(\varphi_{\text{u}}\) is the voltage phase, \(\varphi_{\text{I}}\) is the current phase, \(\Delta\varphi\) is the phase difference between current and voltage in the primary winding, and \(R\) is the resistance.