## 2 Materials and Methods  
The rotor coordinate system is rotating relatively to the coordinate system of the stator with angular velocity \(\omega\), their mutual arrangement is characterized by the electric angle \(\alpha\) between the same-name axes. The initial system of differential equations of three-phase asynchronous electric motor can be written in vector-matrix form Equation (1) based on the second Kirchhoff's law with regard to [PERSON]'s equation:  
\[[u]=[R][i]+\frac{d}{dt}[\Psi]=[R][i]+\frac{d}{dt}([L][i])=[R][i]+[L]\bigg{(} \frac{d}{dt}[i]\bigg{)}+\bigg{(}\frac{d}{dt}[L]\bigg{)}[i], \tag{1}\]  
where \([R]\) is the matrix of active resistances; \([u]\) is the voltage vector; \([\Psi]\) is the flux vector; \([i]\) is the vector of electric currents; and \([L]\) is the matrix of inductances. Matrices and vectors of three-phase IM enter the Equation (1), and have the form Equations (2)-(4):  
\[[u]= \begin{bmatrix}[u_{1}]\\ [u_{2}]\end{bmatrix} = \begin{bmatrix}u_{1A}\\ u_{1B}\\ u_{1C}\\ u_{2A}\\ u_{2C}\end{bmatrix} ;[i]= \begin{bmatrix}[i_{1}]\\ [i_{2}]\end{bmatrix} = \begin{bmatrix}i_{1C}\\ i_{1B}\\ i_{2A}\\ i_{2B}\\ i_{2C}\end{bmatrix} ;[\Psi]= \begin{bmatrix}[\Psi_{1}]\\ [\Psi_{2}]\end{bmatrix} = \begin{bmatrix}\[PERSON]_{1C}\\ \[PERSON]_{2A}\\ \[PERSON]_{2B}\\ \[PERSON]_{2C}\end{bmatrix} , \tag{2}\]  
\[[R]= \begin{bmatrix}[R_{1}]&[0]\\ [0]&[R_{2}]\end{bmatrix} = \begin{bmatrix}R_{1o}&0&0&0&0&0\\ 0&R_{1o}&0&0&0&0\\ 0&0&R_{1o}&0&0&0\\ 0&0&0&R_{2o}&0&0\\ 0&0&0&0&R_{2o}&0\\ 0&0&0&0&0&R_{2o}\end{bmatrix} ; \tag{3}\]  
\[[L]= \begin{bmatrix}[L_{1}]&[M_{12}]\\ [M_{21}]&[L_{2}]\end{bmatrix} = \begin{bmatrix}L_{1o}&M_{1A1C}&M_{1A2A}&M_{1A2B }&M_{1A2C}\\ M_{1B1A}&L_{1o}&M_{1B1C}&M_{1B2C}&M_{1B2B}&M_{1B2C}\\ M_{1C1A}&M_{1C1B}&L_{1o}&M_{1C2A}&M_{1C2B}&M_{1C2C}\\ M_{2A1A}&M_{2A1B}&M_{2A1C}&L_{2o}&M_{2A2B}&M_{2A2C}\\ M_{2B1A}&M_{2B1B}&M_{2B1C}&M_{2B2A}&L_{2o}&M_{2B2C}\\ M_{2C1A}&M_{2C1B}&M_{2C1C}&M_{2C2A}&M_{2C2B}&L_{2o}\end{bmatrix} , \tag{4}\]  
where \([i_{1}],\ [\Psi_{1}],\ [u_{1}]\) are vectors of stator electric currents, stator flux vector and voltages of stator winding phases; \([L_{1}],[R_{1}]\) are matrices of active inductances and stator winding resistances; \([i_{2}],\ [\Psi_{2}],\ [u_{2}]\) are vectors of rotor electric currents, rotor flux vector and voltages of rotor winding phases; \([L_{2}],\ [R_{2}]\) are matrices of inductances and active resistances of the rotor winding, reduced to the stator winding; \([0]\) is zero matrix; \([M_{21}]\) is matrix of mutual inductances of the rotor and stator; \([M_{12}]\) is matrix of mutual inductances of stator and rotor. The indices 1A, 1B, 1C indicate that the parameter belongs to the corresponding stator phase and the indices 2A, 2B, 2C indicate that the parameter belongs to the corresponding rotor phase. The index \(o\) indicates that the parameter belongs to a three-phase asynchronous electric machine with the parameters of the rotor winding connected to the stator. The differentiation of the matrix \([L]\), is written in the following form (5):  
\[\frac{d[L]}{dt}=\frac{d\alpha}{dt}\frac{\partial[L]}{\partial\alpha}=\omega \frac{\partial[L]}{\partial\alpha}. \tag{5}\]  
Taking into account the adopted notations, system (5) is written in the form (6):  
\[ \begin{bmatrix}[u_{1}]\\ [u_{2}]\end{bmatrix} = \begin{bmatrix}[R_{1}]&[0]\\ [0]&[R_{2}]\end{bmatrix}\begin{bmatrix}[i_{1}]\\ [i_{2}]\end{bmatrix} + \begin{bmatrix}[L_{1}]&[M_{12}]\\ [M_{21}]&[L_{2}]\end{bmatrix} \frac{d}{dt}\bigg{(} \begin{bmatrix}[i_{1}]\\ [i_{2}]\end{bmatrix} \bigg{)}+\frac{d}{dt}\bigg{(} \begin{bmatrix}[L_{1}]&[M_{12 }]\\ [M_{21}]&[L_{2}]\end{bmatrix} \bigg{)} \begin{bmatrix}[i_{1}]\\ [i_{2}]\end{bmatrix} , \tag{6}\]The stator and rotor inductance matrices included in Equation (6) are written as Equations (7) and (8):  
\[[L_{1}]= \begin{bmatrix}L_{1o}&M_{1A1B}&M_{1A1C}\\ M_{1B1A}&L_{1o}&M_{1B1C}\\ M_{1C1A}&M_{1C1B}&L_{1o}\end{bmatrix} =