## 2 Methodological Aspects  
### Mathematical Model of the IM  
The IM chosen for the study of its operation in the harmonics has the nominal data and other characteristic sizes indicated in Table 1. A nominal power was chosen from the medium power range, i.e., P\({}_{\rm n}\) = 4 kW, considering that these types of machines are more common in the laboratories of technical universities where they are used for experimental verifications. The machine windings are connected in a three-wire Y-configuration with isolated neutral.  
The electrical part of the IM is represented by a fourth-order state-space model. All electrical variables and parameters are referred to the stator, indicated by the prime signs in the following machine equations [24]:  
\[\rm V_{ds}=R_{s}i_{ds}+d\varphi_{ds}/dt-\omega\varphi_{qs}; \tag{1}\]  
\[\rm V_{qs}=R_{s}i_{qs}+d\varphi_{qs}/dt+\omega\varphi_{ds}; \tag{2}\]  
\[\rm V^{\prime}_{dr}=R^{\prime}r_{i}^{i}_{dr}+d\varphi^{\prime}_{dr}/dt-(\omega -\omega_{r})\varphi^{\prime}_{qr}; \tag{3}\]  
\[\rm V^{\prime}_{qr}=R^{\prime}r_{i}^{i}_{qr}+d\varphi^{\prime}_{qr}/dt+( \omega-\omega_{r})\varphi^{\prime}_{dr}; \tag{4}\]  
\[\rm T_{e}=1.5p(\varphi_{ds}i_{qs}-\varphi_{qs}i_{ds}), \tag{5}\]  
in which  
\[\rm\varphi_{ds}=L_{s}i_{ds}+L_{m}i^{\prime}_{dr}; \tag{6}\]  
\[\rm\varphi_{qs}=L_{s}i_{qs}+L_{m}i^{\prime}_{qr}; \tag{7}\]  
\[\rm\Phi^{\prime}_{dr}=L^{\prime}r_{i}^{i}_{dr}+L_{m}i_{ds}; \tag{8}\]  
\[\rm\[PERSON]^{\prime}_{qr}=L^{\prime}r_{i}^{i}_{qr}+L_{m}i_{qs}; \tag{9}\]  
\[\rm Ls=L_{ls}+L_{m}; \tag{10}\]  
\[\rm L^{\prime}_{r}=L^{\prime}_{lr}+L_{m}. \tag{11}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline P\({}_{\rm n}\) & Rated power & 4 kW \\ V\({}_{\rm n,RMS}\) & Rated line-to-line voltage & 400 V \\ f & Frequency & 50 Hz \\ N\({}_{\rm n}\) & Rated speed & 1430 rpm \\ R\({}_{\rm s}\) & stator resistance & 1.405 \(\Omega\) \\ L\({}_{ls}\) & stator leak inductance & 0.005839 H \\ R\({}_{r}\) & Rotor resistance & 1.395 \(\Omega\) \\ L\({}_{lr}\) & Rotor leak inductance & 0.005839 H \\ L\({}_{m}\) & Mutual inductance & 0.1722 \\ J & Inertia & 0.0131 kgm\({}^{2}\) \\ F & Friction factor & 0.002985 Nms \\ P & Pole pairs & 2 \\ PF\({}_{\rm n}\) & Rated power factor & 0.82 \\ n\({}_{\rm n}\) & Rated efficiency & 88.5\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the induction motor (IM).  
where \(\mathrm{R_{s}}\), \(\mathrm{L_{ls}}\) represent the stator resistance and leakage inductance; \(\mathrm{L_{m}}\)--magnetizing inductance; \(\mathrm{L_{s}}\)--total stator inductance; \(\mathrm{V_{ds}}\), \(\mathrm{i_{ds}}\)--d-axis stator voltage and current; \(\mathrm{V_{qs}}\), \(\mathrm{i_{qs}}\)--q-axis stator voltage and current; \(\mathrm{\varphi_{ds}}\), \(\mathrm{\varphi_{qs}}\)--stator d- and q-axis fluxes; p--number of pole pairs; \(\omega\)--reference frame angular velocity; \(\mathrm{\omega_{r}}\)--electrical angular velocity; \(\mathrm{T_{e}}\)--electromagnetic torque.  
All stator and rotor quantities are in the arbitrary two-axis reference frame (dq frame). The subscripts used are defined as follows: d--d-axis quantity; q--q-axis quantity; r--rotor quantity; s--stator quantity; l--leakage inductance; m--magnetizing inductance.  
The mechanical part of the machine is represented by a second-order system as follow:  
\[\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{\omega_{m}}=\frac{1}{J}(\mathrm{T_{e}}- \mathrm{F\omega_{m}}-\mathrm{T_{m}}); \tag{12}\]  
\[\frac{\mathrm{d}}{\mathrm{dt}}\mathrm{\theta_{m}}=\mathrm{\omega_{m}}, \tag{13}\]  
where \(\mathrm{\omega_{m}}\) is the angular velocity of the rotor; J--combined rotor and load inertia coefficient; \(F\)--combined rotor and load viscous friction coefficient; \(\mathrm{T_{m}}\)--shaft mechanical torque; \(\mathrm{\theta_{m}}\)--rotor angular position.  
The physical model considered is linear, the inductivities are considered constant, so here the phenomenon of saturation of the magnetic circuit is not considered.