# Application of Dynamic Time Warping (DTW) in Comparing MRT Signals of Steel Ropes
## 2 Materials and Methods
### MRT Methods
#### 2.1.1 Numerical Model of Search Coils
\[V_{c}=-\frac{d\Phi}{dt}=-\left(A_{c}\frac{dB}{dt}+A_{c}B\omega+A_{c}\
abla Bv\right) \tag{6}\]  
The frequency characteristics of the coil--a series circuit \(R_{c}L_{c}\)--are described by the impedance expressed by Formula (7) and the goodness of the coil described by Formula (8). The time constant \(\tau\) of the coil without the influence of the winding capacitance is given by Formula (9), and the change in voltage and current in transients is described by Formulas (10) and (11):  
\[Z_{c}(\omega_{s})=|Z_{c}|(\omega_{s})e^{i\varphi(\omega_{s})}=R(\omega_{s})+jX( \omega_{s})=R_{c}+j\omega_{s}L_{c}-j\frac{1}{\omega_{s}C_{c}} \tag{7}\]  
\[Q=\frac{|X_{L}|}{R} \tag{8}\]  
\[\tau=\frac{R_{c}}{L_{c}} \tag{9}\]  
\[u(t)=I\cdot R_{c}+L_{c}\frac{di(t)}{dt}=V\cdot e^{-\tau\cdot t} \tag{10}\]  
\[i(t)=\frac{V}{R}\big{(}1-e^{-\tau\cdot t}\big{)} \tag{11}\]  
where  
* e is the base of the Natural Logarithm = 2.71828,
* \(R=R_{c}\) is the actual part of the coil impedance,
* \(X=X_{L}-X_{c}\) is the imaginary part of the coil impedance (reactance),
* \(X_{L}=\omega_{s}L_{c}\) is the inductive reactance of the coil,
* \(X_{C}=1/\omega_{s}C_{c}\) is the capacitive reactance of the coil related to the winding-to-turn capacitance,* \(\omega_{s}=2\pi f_{s}\) is the circular frequency of a signal (magnetic field) with a frequency of \(f_{s}\),
* \(|Z_{c}|=\sqrt{R^{2}+X^{2}}\) is an impedance module,
* \(\varphi\) is the phase angle between the voltage and electric current. The study used a GP-type measuring head developed at the AGH University of Science and Technology, in which the LF1 and LF2 coils are series \(N_{LF1}\) circuits and cylindrical coils evenly distributed around the circumference and \(N_{LF2}\) wound on an E-type ferromagnetic core in order to  
* Reduce inter-turn capacitances,
* Increase the inductance of the coil and the amplitude of the induced voltage, \(\mu=\mu_{m}\mu_{0}\),
* Differentiate the induced signal on the basis of known core design features. The LF1 and LF2 signals recorded in the defectograph reproduce the resultant signals of the cylindrical coils, taking into account the following:  
* Coil nonlinearity resulting from the influence of the magnetization characteristics of the ferromagnetic core \(B(H)\) on the instantaneous values of current \(i(t)\) in the coil winding and hysteresis and eddy current losses in the core. For low-magnetic-field frequencies, the unit power of the losses \(P_{Fe,h}\) in the ferromagnetic and hysteresis cores is proportional to the frequency (12), and the unit power of eddy current losses \(P_{cc}\) is proportional to the square of the frequency (13);
* Local heterogeneity of the magnetic permeability of rope wires (influence of different degrees of their degradation and current material strain, abrasion, and cracking of wires);
* Actual characteristics of the point sensitivity of the LF1 and LF2 coils, taking into account the heterogeneity of their sensitivity along the circuit, depending on the number of cylindrical coils distributed along the circuit on a given radius and the parameters of the magnetic field concentrator;
* E-core parameters;
* The impact of transmission line parameters;
* Parameters of the defectograph input circuit;
* The impact of the scanning speed compensation system;
* Continuous quantization parameters in Analog-to-Digital convectors (ADCs). \[P_{Fe,h}=\sigma_{h}\cdot f\cdot B_{m}^{n} \tag{12}\]  
\[P_{cc}=\sigma_{cc}\cdot f^{2}\cdot B_{m}^{2} \tag{13}\]  
where  
* \(\sigma_{h}\) is the coefficient depending on the grade of core material;
* \(\sigma_{w}\) is the coefficient depending on the grade of core material, its geometry and construction;
* \(f\) is the frequency;
* \(B_{m}\) is the amplitude of magnetic induction in the core;
* \(n\) is the power exponent equal to 1.6 for \(B_{m}\in(0,1\;T;1,0\;T)\) and 2.0 for \(B_{m}\in(1,0\;T;1,6\;T)\).