## 2 Materials and Methods
Analysis of Equations (16)-(21) shows that they contain functions of the electric angle of rotation of the rotor \(\alpha\), so the system of differential equations describing electromagnetic processes in a three-phase asynchronous motor is a system of equations with variable coefficients. The solution of such systems is possible only by numerical methods. The solution result of the system of Equations (16)-(21) is in the form of currents in the rotor and stator phases of the windings. To solve the aforementioned system of differential Equations (16)-(21) numerically, it is necessary to reduce them to normal form. Such reduction is carried out by numerical method of integration of the system of differential Equations (16)-(21) by method of solution of system of linear algebraic equations relative to derivatives. In this case, the system of Equations (16)-(21) is transformed so that their left-hand sides contain derivatives of desired functions with appropriate coefficients, and their right-hand sides contain desired functions. Another way to solve the system of linear algebraic equations with respect to derivatives is to find the inverse matrix \([L]^{-1}\). In this case equations of system (6) should be presented in the form (25)  
\[\frac{d}{dt} \begin{bmatrix}[i_{1}]\\ [i_{2}]\end{bmatrix} = \begin{bmatrix}[L_{1}]&[M_{12}]\\ [M_{21}]&[L_{2}]\end{bmatrix} ^{-1}\bigg{\{} \begin{bmatrix}[u_{1}]\\ [u_{2}]\end{bmatrix} - \begin{bmatrix}[R_{1}]&[0]\\ [0]&[R_{2}]\end{bmatrix}\begin{bmatrix}[i_{1}]\\ [i_{2}]\end{bmatrix} -\omega\frac{\partial}{\partial\alpha} \begin{bmatrix}[L_{ 1}]&[M_{12}]\\ [M_{21}]&[L_{2}]\end{bmatrix}\begin{bmatrix}[i_{1}]\\ [i_{2}]\end{bmatrix} \bigg{\}}. \tag{25}\]  
The matrix \([L]\) contains variable coefficients, so the inverse matrix \([L^{-1}]\) must be determined at each integration step. The expression of the instantaneous power consumption (26):  
\[p_{1}=u_{1A}i_{1A}+u_{1B}i_{1B}+u_{1C}i_{1C}. \tag{26}\]  
Thus, the system (16)-(21) together with the expression of electromagnetic moment (23), equation of motion (23) and Equation (13) represents a mathematical model of a three-phase squirrel-cage induction electric motor in phase axes, without considering saturation of magnetic wire and losses in steel. The advantage of the considered mathematical description of electromechanical energy conversion processes in three-phase IM is that it uses instantaneous values of currents and voltages of stator and rotor winding phases as variables. This description of the dynamics gives a direct indication of the physical processes in the motor. Control in the coordinate system oriented by the calculated rotor magnetic flux vector (\(\psi_{R}\)) is called direct-field oriented control. Usually, systems with direct orientation also contain a flow stabilization loop. But a simpler approach and another class of systems have emerged in which the rotor magnetic flux vector (\(\psi_{R}\)) is not evaluated, but its angle is calculated from the slip and velocity (or position) of the rotor. Such systems are called systems with indirect field orientation. Such systems do not contain a flow stabilization loop. In this case, it is believed that if the control actions on the motor are formed based on the condition of flow constancy, then if they are accurately implemented, the flow will be constant even without stabilization. As practice has shown, such an approach has quite justified itself due to a good ratio of accuracy and simplicity of regulation. When operating at rated voltage with increasing frequency, the amplitude of the stator electric current is limited by the increasing inductive resistance of the IM. Therefore, as the frequency increases, the maximum achievable motor torque is also limited. For controller synthesis and simulation, a dynamic model of the IM developed in the system (\(d\)-\(q\)) that is synchronized with the flux of the stator was created.