The experimental bench consists of a squirrel cage induction motor of 5.5 kW (Figure 1). The engine is Leroy Somer LS 132S, IP 55, class F, {T}^{\circ}\text{C}={40}^{\circ}\text{C}. The nominal voltage between phases 400 V, supply frequency 50 Hz, speed rate 1440 r/min, the number of slots in the rotor \mathit{Nr}=28. The number of slots in the stator \mathit{Ns}=48. The stator windings are star-coupled [5]. The motor is loaded with a powder brake. Its maximum torque (100 Nm) is reached at the nominal speed.

### 4.1 Rotor fault influence on the stator current spectrum phase

The spectrum modulus and phase of the stator current of a rotor cage with four broken bars (4b-C100) (Figure 2) (connection to a 3-phase supply) are shown in Figures 3 and 4. It is clear that the frequency components (1\pm 2kg)fs are present in the amplitude spectrum of the stator current as it is shown in Figure 3.

To be sure that the phase jumps of frequencies (1\pm kg)fs present in this spectrum are due to the presence of a damaged rotor bar, we compared with the spectrum of the stator current when the induction machine operates with a healthy rotor. This analysis helps to reinforce the fact that the appearance of a broken bar in the machine rotor leads to picks in the spectrum at frequencies (1\pm 2kg)fs [1].

We have shown that the analysis of the stator current spectrum tells us about the rotor state of the induction machine.

We notice that the jumps of the spectrum present at frequencies (1\pm 2kg)fs were clearly due to the presence of one or more damaged rotor bars. Therefore, based on this information, it is possible to establish a diagnosis of squirrel cage by analyzing the spectrum of particular picks.

To undertake a rotor fault diagnosis without need for comparison with a reference (reference obtained from a healthy functioning) [6], the final decision, that is, ‘Is the rotor healthy or not?’, must be made exclusively from the analyzed signal. This will allow us to apply the method to low or high power machines. We know that all induction machines have a slight asymmetry of construction that induces, in the stator current spectrum, a frequency component (1-2g)fs. Sometimes, the oscillation speed created by this component is large enough to make an additional component of frequency (1+2g)fs appear in the same frequency spectrum. However, induction motors manufacturers ensure that the machines present asymmetry as small as possible because it could be the main cause of faults [7]. For instance, a static eccentricity causes a homopolar current enclosed in the bearings reduce their lifetime significantly [8]. It is in this light that the diagnostic method will be developed. We study the stator current spectrum and especially the frequency jump at (1+2g)fs. Normally, this jump is very low or even zero for a healthy induction machine, and this is true whatever the charge is.

### 4.2 Hilbert transform for rotor faults diagnosis

This section develops the diagnosis method based on the calculation of the phase of the analytical signal obtained by a Hilbert transform of the spectrum amplitude of the current absorbed by the induction machine. In other words, instead of working directly on the stator current (time signal), we suggest working with the module of its Fourier transform. As we previously mentioned, the Hilbert transform of a signal returns a representation of this signal in the same domain. Thus, if we apply the Hilbert transform of the modulus of the Fourier transform of the stator current, the resulting signal will therefore be expressed in the frequency domain.

This approach uses the Hilbert transform calculated from the spectrum module of the stator current, its phase has no importance here. Figure 4 represents the analytic signal phase obtained by calculating the Hilbert transform of the spectrum module of the stator current, when the machine operates with a healthy rotor Figure 4(a) and a failing rotor Figure 4(b). These figures reveal the presence of ‘phase jumps’ at fault frequencies (1\pm 2kg)fs. Moreover, we can notice that the appearance of the rotor fault increases the amplitude of the jumps present at phase \phi \mathrm{HT}(f).

We can notice the presence of a rapid change in the phase at 50 Hz. As the phase of the FT of the current, having a clear phase change at 50 Hz, allows the evaluation of the amplitude of the phase jump at (1-2g)fs more easily than the amplitude of component of the same frequency present in the stator current spectrum module Figure 4(b).

For our machine, there is no problem in the detection of this frequency, either in the spectrum amplitude or in the phase \mathrm{HT}(f), but in the case of high power motors, this detection may be difficult due to the low slip value (about 1%) because of the dominance of the fundamental harmonic frequency 50 Hz.

The difference between the phase of the Fourier transform and the phase of the analytic signal lies in the fact that the latter is calculated from the spectrum amplitude of the stator current. This means that, as soon as the frequency component (1-2g)fs appears in the spectrum module, it will also appear in the phase \phi \mathrm{HT}(f). Even the component created by the rotor fault has a relatively low amplitude in the module of the stator current frequency spectrum, it appears in the phase of the analytic signal \phi \mathrm{HT}(f) because the modulus of the spectrum contains this information. Moreover, it should be noted that the amplitude of the phase jumps located at frequencies (1\pm 2kg)fs of the phase \phi \mathrm{HT}(f) is directly related to the amplitude of the components located at the same frequencies in the modulus of the spectrum of the stator current.