## 4 Analysis of the Diagnostic Signals--Experimental Results
### Results of Stator and Rotor Fault Extraction  
The characteristic frequency components, that characterize faults in the stator winding (ITSCs) are known as [14]:  
\[f_{SH}=f_{s}\bigg{(}k\frac{(1-s)}{p_{p}}\pm m\bigg{)}, \tag{14}\]  
where:\(f_{s}\)--power supply frequency [Hz], \(s\)--slip, \(p_{p}\)--number of pole pairs, \(k=1,2,3,\dots,m=1,3,5,\dots\)  
On other hand faults in the rotor windings (broken bars) in the well-known Motor Current Signature Analysis (MSCA) are described as follows [7; 29]:  
\[f_{BB1}=f_{s}(1\pm k\cdot 2s), \tag{15}\]  
where: \(k=1,2,3,\dots\) (usually in practice \(k=1\)). Also other spectral components should be examined under rotor winding faults, according to [7; 29]:  
\[f_{BB2}=f_{s}\bigg{(}\frac{k}{p_{p}}(1-s)\pm s\bigg{)}. \tag{16}\]  
Figure 4: Illustration of the physical modelling of stator and rotor fault: (a) connection of the stator winding for inter-turn short-circuit modelling; (b) broken bars modelling in squirrel-cag rotor. These characteristic frequency components for electrical faults could be also searched for in the stator current symmetrical components. The first part of the research concerned obtaining information on the technical condition of the motor windings and isolating from the spectrum analysis those frequencies that change under the influence of the winding failures, and thus may be symptoms of a given stator or rotor fault. For this purpose, a series of tests and spectral analyses of the instantaneous symmetrical components \(i_{1}\) and \(i_{2}\) of the stator current signals obtained after the FFT were performed for various degrees of damage to the stator winding and rotor bars in accordance with the procedure shown in Figure 1. Sample spectra of the instantaneous symmetrical components are shown in Figures 5 and 6. Figure 5 shows the harmonic spectra of the positive-sequence component \(i_{1}\) for a motor loaded with rated torque \(T_{L}=T_{LN}\), under supply voltage with frequency \(f_{s}=50\) Hz and various defects of the motor windings. In the case of stator winding short-circuits (Figure 5a), only a slight increase in component \(f_{s}\) of the positive-sequence symmetrical component \(i_{1}\) can be seen as a result of the increasing stator winding degradation. However, these changes are too small to be taken into account when diagnosing the stator winding. In addition, the value of the amplitude of component \(f_{s}\) strongly depends on the changes of the load torque. The influence of the ITSCs of the stator winding is also visible for the \(3f_{s}\) frequency but only when the damage is relatively high (as e.g., \(N_{sh}=5\)). This shows practically no changes in spectrum \(i_{1}\) due to the stator winding incipient faults. Thus positive-sequence component \(i_{1}\) does not carry important information for the case of stator winding fault or changes in the initial phase of the failure (for \(N_{sh}=1\), \(2\)) are negligible. In the case of broken rotor bars (Figure 5b), the so-called slip frequencies around the fundamental frequency \(f_{s}\) appear, which are characteristic of this type of damage (see Equation (15)). Although they can also be extracted for the undamaged motor, the amplitude level of these components is very small comparing to a faulted rotor case and can be negligible, indicating a non-ideal casting of the rotor cage. However, along with the increase in the degree of damage, the amplitude values of these slip frequencies increase rapidly (for one broken rotor bar it is 10 times more than in the case of an undamaged rotor). On the contrary, the components calculated in accordance with Equation (16) are too low to be extracted directly from the spectrum shown. Figure 5: FFT analysis of the positive-sequence instantaneous symmetrical component \(i_{1}\) of the stator current for nominal load torque and \(f_{s}=50\) Hz with (**a**) different number of shorted turns \(N_{sh}\); (**b**) different number of broken bars \(N_{hb}\).