## 2 Methodological Aspects
### Modeling and Simulation of the IM in Harmonics  
Harmonics are sinusoidal voltages or currents with frequencies that are integer multiples of the fundamental electrical system frequency. These are classified as positive-sequence harmonics: \(\mathrm{k_{p}}=3n+1\); and negative-sequence harmonics: \(\mathrm{k_{n}}=3n+2\), being n = 0, 1, 2. In IM, positive-sequence harmonics contribute to torque in the positive (forward) direction, and negative-sequence harmonics provide torque in the negative (backward) direction. The circulation of zero-sequence current harmonics is null because the IM is usually Delta or isolated Wye connected [15]. The model shown in Figure 1 illustrates the IM simulated in MATLAB/Simulink. The Asynchronous Machine block implements a three-phase induction machine (single squirrel-cage), operating in motor mode. We considered the following initial hypotheses for the study of the IM behavior in the harmonics:  
* The initial reference situation is that when the IM stator is supplied with a symmetrical system of sinusoidal voltages, at the network frequency, the motor is running stable at a determined constant torque and speed. * The voltage waves applied to the stator are deformed, by superposing over the fundamental of a single harmonic, with orders from the series k \(\in\)[5, 7, 11, 13, 17, 19], and having percentage levels among the values \(\mathrm{\gamma_{Vk}}\in\) {5%, 10%, 15%, 20%, 25%, 30%}, above the rated value of the voltage. * Solving the situations considered above on the physical model of the IM, the levels of the harmonics of the stator current, corresponding to those of the injected voltage, are followed. Moreover, the total harmonic distortion (THD) of the stator current, of the electromagnetic torque and its THD, the PF, and the IM efficiency are monitored as well. The voltage THD varies in the electrical networks depending on the type, power, position, and weight of the receivers and the deforming consumers. The chosen values for the voltage THD in this paper frame the real, possible situations. For supplying the IM in harmonics, a three-phase voltage source with programmable time variation of amplitude, phase, frequency, and harmonics is used. With this type of voltage source, two harmonics can be programmed and superimposed on the fundamental signal. Figure 2 shows the waveforms of the supply voltages in which 15% of the 5 th-order harmonic was superimposed on the fundamental. We used the RMS block for obtaining the values of the stator and rotor currents. This block computes the true RMS value of the input signal, which is calculated over a running average window of one cycle of the specified fundamental frequency:  
\[\text{RMS}(\text{f}(\text{t}))=\sqrt{\frac{1}{\text{T}}\int\limits_{\text{t-T} ^{\text{t}}}^{\text{t}}\text{f}(\text{t})^{2}\text{dt}}, \tag{14}\]  
where f(t) is the input signal and T is 1/(fundamental frequency). We chose, for graphical exemplification of the IM behavior running in harmonics, the case when in the supply voltage, 15% of the 5 th-order harmonic was injected over the fundamental. Figure 3  
Figure 1: Simulation of the IM in MATLAB / Simulink. Figure 2: Supply voltages, with 15% of the 5 th-order harmonic superimposed on the fundamental. shows some IM characteristic parameters when the supply voltage contains the abovementioned harmonic content. The harmonic contents of both the stator current and the electromagnetic torque were obtained by using the Powergui FFT Analysis Tool, which computes THD for the analyzed signal. We calculated the values of the rotor speed and electromagnetic torque using the Mean block, which computes the mean value of the input signal. The mean value is computed over an average running window of one cycle of the specified fundamental frequency:  
\[\text{Mean}(\text{f}(\text{t}))=\frac{1}{\text{T}}\int\limits_{\text{t}- \text{T}}^{\text{t}}\text{f}(\text{t})\text{dt}. \tag{15}\]  
For the calculation of the active (P) and reactive (Q) total powers, we used the Power Measurement block.