# Application of Dynamic Time Warping (DTW) in Comparing MRT Signals of Steel Ropes
## 2 Materials and Methods
### MRT Methods
#### 2.1.2 Numerical Model of the LMA Signal
\[\frac{V_{H}(t)}{dt}=R_{H}\frac{i}{s}.\frac{dB(t)}{dt}\propto V_{LF}(t) \tag{18}\]  
Figure 6: (**a**) Sensor arrangement for a high-resolution testing system [47], (**b**) theoretical leakage field for a single wire break [46]. The existing differences between the signal from the search coil and the derivative of the signal from the Hall sensor result from the size of the cross-sectional area of the magnetosensitive element in which the measurement of the surrounding magnetic field is performed (it is larger for the coil) and the nonlinearity introduced by the ferromagnetic core of the coil. Using these methods, it is possible to access more information about a defect from the shape of the magnetic field:  
* The location around the rope circumference (e.g., upper or lower side of a track rope),
* The shape of the defect,
* The arrangement of clusters of wire breaks,
* The depth within the rope (for stranded ropes) and the distribution of wire breaks, e.g., in a single strand. The theoretical signature of the magnetic anomaly from a broken single wire of a wire rope is described by the dipole model and the Nussbaum formulas in Equations (19) and (20). The actual magnetic signature of rope defects changes as the damage density and rope type increase. The signatures of the internal defects of the rope are weakened in multi-layer polymer-impregnated steel ropes, which results from the lack of mechanical contact of individual layers of the strands. \[B_{axial}(z_{n})=I\cdot\frac{\pi d^{2}}{16\pi}\cdot\sum_{k}\Delta B(z_{k}) \cdot\frac{2\cdot(z_{k}-z_{n})^{2}-a^{2}}{\left((z_{k}-z_{n})^{2}+a^{2}\right)^ {5/2}} \tag{19}\]  
\[B_{radial}(z_{n})=I\cdot\cos\alpha\cdot\frac{\pi d^{2}}{16\pi}\cdot\sum_{k} \Delta B(z_{k})\cdot\frac{3\cdot(z_{k}-z_{n})\cdot a}{\left((z_{k}-z_{n})^{2}+ a^{2}\right)^{5/2}} \tag{20}\]  
where  
* \(B_{axial}\) is the axial component of magnetic flux density of the defect field;
* \(B_{radial}\) is the radial component of magnetic flux density of the defect field;
* \(a\) is the distance from a measuring point on the surface to the rope axis;
* \(d\) is the wire diameter;
* \(I\) is the current;
* \(z_{k},\ z_{n}\) is the longitudinal coordinate in relation to the wire break end;
* \(\alpha\) is the angle;
* \(\Delta B\) is the difference in flux density. In addition to their advantages, Hall sensors also have some weaknesses. Temperature fluctuations can affect their accuracy, so they require the implementation of temperature compensation mechanisms. Precise calibration is often crucial to guarantee accurate magnetic field measurements. Hall sensors typically exhibit higher noise and lower sensitivity compared with other magnetic field sensors, and they require a current-stabilized power supply to operate. The current consumption of a Hall effect sensor is much higher than that of magnetoresistive (AMR, GMR, TMR) and magnetoimpedance (MI) sensors. The LMA signal is much more sensitive to the external environment (electromagnetic interference, moving ferromagnetic objects) than the LF signal of the search coil. Before the MRT results are graphically displayed, the LMA signal is usually highly filtered by software procedures that can significantly alter it (Figure 7). If the defect area is longer than the distance between the pole pieces of the measuring head, then the reduction in the volume and cross-section of the rope are proportional--the LMA signal correctly reproduces the LMV volume changes in the metallic part of the steel rope not impregnated with polymer. If the fault is shorter than the pole piece spacing, the LMA signal has a lower amplitude than expected and does not correctly reproduce the volumetric changes in the metallic part. In the case of polymer-impregnated steel rope, the LMA signal will also change as the gap between successive layers of strands changes. A filtered LMA signal provides better trend tracking and interpretation, but the end user should remain in control of all processes.