# Influence of Nonstationary Processes in Drill Rigs on the Durability of Structural Elements
## 3 Mathematical Model of Electromagnetic Processes in an Asynchronous Motor  
Providing proper accuracy for the analysis of dynamic processes in a drill rig requires detailed consideration of not only the elastic and inertial properties of a mechanical system but also the driving forces and useful resistance forces time changes pattern. In particular, peculiarities for determining the electromagnetic torque of the motor based on mathematical modeling of electromagnetic phenomena in an electric machine are considered. The differential equations of electromagnetic processes in an asynchronous motor have the following form:  
\[\frac{di_{s}}{dt}=A_{s}(u+\Omega_{s}\Psi_{s}-R_{s}i_{s})+B_{s}(\Omega_{r}\Psi_ {r}-R_{r}i_{r}),\]  
\[\frac{di_{r}}{dt}=A_{r}(\Omega_{r}\Psi_{r}-R_{r}i_{r})+B_{r}(u_{s}+\Omega_{s} \Psi_{s}-R_{s}i_{s}), \tag{13}\]  
where \(i_{s}\), \(i_{r}\), \(u_{s}\)--matrix-columns of currents and voltages; \(A_{s}\), \(B_{s}\), \(A_{r}\), \(B_{r}\)--relations matrices of electromagnetic parameters; \(\Omega_{s}\), \(\Omega_{r}\)--matrices of rotation frequencies; \(\Psi_{s}\), \(\Psi_{r}\)--matrix-columns of flux linkages; \(R_{s}\), \(R_{r}\)--active resistances of windings. Index \(s\) identifies stator winging, and \(r\)--rotor winding. Matrix-columns \(i_{s}\), \(i_{r}\) and \(u_{s}\) are defined as  
\[i_{j}(j=s,r)=\text{col}(i_{jx},i_{jy});\ u_{s}=\text{col}(U_{m},0),\]  
where \(i_{jx}\), \(i_{jy}\)--projections of currents on the coordinate axes \(x,y\); \(U_{m}\)--power supply voltage amplitude. Square matrices \(A_{s}\), \(B_{s}\), \(A_{r}\), and \(B_{r}\) are defined as  
\[A_{s}=\alpha_{s}(1-\alpha_{s}G),\ B_{s}=-\alpha_{s}\alpha_{r}G,\ A_{r}=\alpha_ {r}(1-\alpha_{r}G),\ B_{r}=B_{s},\]where  
\[G=\frac{1}{i_{m}^{2}}\left[ \begin{array}{cc}Ri_{x}^{2}+T_{y}^{2}&(R-T)i_{x}i_{y} \\ (R-T)i_{x}i_{y}&T_{x}^{2}+Ri_{y}^{2}\end{array} \right],\]  
therewith:  
\[R=\frac{1}{\rho+\alpha_{s}+\alpha_{r}},\ T=\frac{1}{\tau+\alpha_{s}+\alpha_{r}}.\]  
In the given expressions, \(i_{m}\), \(i_{x}\), \(i_{y}\)--magnetizing current and its components along the axes \(x\), \(y\); \(\tau\), \(\rho\)--values determined from the magnetization curve, which represents the dependence of the working flux linkage \(\Psi_{m}\) on magnetizing current; \(\alpha_{s}\), \(\alpha_{r}\)--quantities reciprocal to the inductance of the stator and rotor. Rotation frequency matrices are  
\[\Omega_{s}=\left[ \begin{array}{cc}0&\omega_{0}\\ -\omega_{0}&0\end{array} \right],\ \Omega_{r}=\left[ \begin{array}{cc}0& \omega_{0}-\omega_{r}\\ \omega_{r}-\omega_{0}&0\end{array} \right],\]  
where \(\omega_{0}\) and \(\omega_{r}\)--synchronous angular speed of the motor and the angular speed of the rotor, expressed in radians per second. Matrix columns of complete flux linkages for stator and rotor windings are defined as  
\[\Psi_{s}=\frac{1}{\alpha_{s}}i_{s}+\frac{1}{\tau}i_{s},\ \Psi_{r}=\frac{1}{ \alpha_{r}}i_{r}+\frac{1}{\tau}i,\]  
where  
\[i=\text{col}(i_{x},i_{y}).\]  
The following values are defined as  
\[i_{x}=i_{sx}+i_{rx},\ i_{y}=i_{sy}+i_{ry},\ i_{m}=\sqrt{i_{x}^{2}+i_{y}^{2}}.\]  
Values \(\tau\) and \(\rho\) are defined by the following expressions:  
\[\tau=\frac{i_{m}}{\Psi_{m}},\ \rho=\frac{di_{m}}{d\Psi_{m}}. \tag{14}\]  
Electromagnetic torque is calculated by the following formula:  
\[M_{e}=\frac{3}{2}p_{0}\frac{1}{\tau}\left(i_{rx}i_{sy}-i_{ry}i_{sx}\right), \tag{15}\]  
where \(p_{0}\)--magnetic pole pairs number. The magnetization curve is given in the form  
\[\Psi_{m}=a_{1}i_{m}+a_{2}i_{m}^{3}+a_{3}i_{m}^{5},\ \text{if}\ i_{m}>i_{mk};\]  
\[\Psi_{m}=\alpha_{m}^{-1}i_{m},\ \text{if}\ i_{m}\leq i_{mk}. \tag{16}\]  
where \(i_{mk}\)--critical magnetizing current, above which the dependence \(\Psi_{m}(i_{m})\) is nonlinear.