# 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  
The LMA (Loss of Metallic Area) signal generated by Hall sensors has been used to indirectly detect changes in the average magnetic properties of the rope fragment material, including magnetic permeability [43], coercivity, and remanence \(\mu_{m}H_{c}B_{r}\). Changes in the magnetic properties of the rope material can be caused by, among others, the current state of stress, change in the diameter of the rope, local plastic deformation of the wires or the rope, corrosion of the wires, fatigue of the material, and changes in the temperature of the material [44]. A higher number of broken or corroded wires corresponds to a greater reluctance of the tested rope fragment. In the MRT method, the Hall sensor can be mounted as follows:  
* In the air gap under the pole piece of the magnetization circuit in order to record the induction of the magnetizing field \(\mathcal{F}\) of the rope \(B_{m}\). With the known parameters of the magnetizing circuit (geometry, magnetomotive force, and magnetic flux \(\Phi_{m}\)), the signal indirectly reflects the magnetic reluctance \(\mathcal{R}=\mathcal{F}/\Phi_{m}\cong 1/(\mu_{m}\cdot S_{ur})S_{ur}\) of a fragment of the tested rope with a cross-section of the ferromagnetic part (the rope fragment is a component of the main magnetic circuit) and the existing state of magnetization of the rope. * Inside the measuring head, in which the rope is fully demagnetized \(\langle B_{m},-B_{m}\rangle\), in order to record the scattered magnetic field inside the head \(\Phi_{int}\), depending on the existing magnetization of the rope, the magnetic parameters prevailing in the air gaps of both pole pieces of the measuring head, local magnetic anomalies of the rope, and broadband Barkhausen noise. * Outside near the probe in order to record the external scattered magnetic field \(\Phi_{out}\) of the probe and the level of magnetization of the rope when magnetizing the rope along the \"original\" magnetization curve, taking into account the position of the Hall sensors relative to the external magnetic neutral axis. In contrast to the coils' Hall effect, sensors can measure magnetic field strength in a static mode, as they are based on the Lorentz force acting across the sensor (Figure 5). The response of a Hall effect sensor is given by the formula in Equation (17) up to the point of saturation determined by the constraints of the power supply:  
\[V_{H}=R_{H}\frac{i\cdot B}{s} \tag{17}\]  
where  
* \(V  
* \(R_{H}\) is the resistance of the Hall sensor;
* i is the electric current;
* \(s\) is the thickness of the conductor;
* \(B\) is the magnetic flux density (the only changing parameter in the input). Hall sensors have many advantages, including being cheap, small, and having the ability to detect both the presence and direction of a magnetic field at a given point in space. They are highly resistant to dust, dirt, and moisture, making them ideal for use in harsh environmental conditions and in typical MRT conditions. Fast response times allow them to be used effectively in dynamic applications. Hall effect sensors are available with different sensitivities, which facilitates their proper selection at the stage of designing the measuring head for a given range of wire rope diameters. The signals of several Hall effect sensors evenly arranged around the rope can be converted to a conventional 2D curve that shows the same typical defect signals as the search coil (Figure 6). The approximate relationship between the LF signal and the LMA is described in Formula (18). The measuring head for testing wire ropes with multiple Hall sensors instead of search coils was developed at the end of the twentieth century at the Technical University of Stuttgart and is known as the \"STUTTGART High Resolution Magnetic Rope Testing Method\" [46].