## 2 Materials and Methods
### Measurement Result Calculation  
The first stage of signal mathematic processing, which includes measurement process control and displayed information processing, is performed via a hardware-software system. The system software consists of a built-in part (embedded write-protected microcontrollers) and an external part, controlled by the server's operational systems and the operator's automated workstation.  
Calculations are performed according to the following algorithms of signal mathematical processing:  
1. The signal (voltage and current) from the coil windings is digitized at a 62,500 Hz frequency;
2. The digitized signal is processed using Fourier transformation; and
3. The algorithm input is the interval of a signal that equals three whole cycles at the generator's actual frequency.  
The following equations are used to calculate the Fourier transformation coefficients:  
\[a_{i}=\frac{2}{N}\sum_{n=0}^{N-1}F_{n}\cos\!\left(\frac{2\pi in}{N}\right)\,\ i=1,2,\ldots,11 \tag{30}\]  
\[b_{i}=-\frac{2}{N}\sum_{n=0}^{N-1}F_{n}\sin\!\left(\frac{2\pi in}{N}\right)\,\ i=1,2,3,\ldots,11 \tag{31}\]  
where \(F_{n}\) is the value of the signal (current, voltage) and \(N\) is the number of points. The third harmonic provides the main contribution against which other harmonics are insignificant. Consequently, the phase of signal \(\varphi\) is calculated by  
\[\varphi=\arctan\!\left(\frac{a_{3}}{b_{3}}\right). \tag{32}\]  
As a result, instantaneous values of the phase difference between the current and voltage are obtained. The instantaneous values of the loss angle are calculated using Equation (18).  
Since the values of the loss angle can be considered small, we can use an approximation for a small angle, so it is advisable to integrate the envelope curve. The envelope curve is obtained as follows: each subsequent instantaneous value is calculated after the previous one after the appearance of the 512 new instantaneous values of the digital signal.  
A ferromagnetic object that moves through a coil is characterized by a set of instantaneous 219 loss angle values, each of which corresponds to its own axial coordinate A, of the cross-section of the core (car with load). In the case of 10 series-connected railway cars, the graph of the loss angle versus the longitudinal coordinate takes the form shown in Figure 2.  
Every envelope in the interval between two adjacent absolute minima corresponds to the magnetic characteristics of a single railcar. The envelope curve can be plotted using any of the coil parameters: coil \(Q\), the inductance, or the loss angle. Such an experiment was conducted. One car with the same load repeatedly traveled through the coil. All three parameters were measured during each passage. After that, the variances of the random values for each parameter were compared. The experiments showed that the smallest error and best reproducibility were achieved using the loss angle function using Equation (18).  
The second stage of signal processing is based on the fact that integrating the signal in any chosen interval yields the mass characteristics of the magnetic component of a single fre  
Figure 2: Experimental envelope function of a core consisting of 10 consecutively connected railcars with the ferromagnetic freight.