## 2 Modeling Electrical Generator Faults Using the Current Signal
As described previously [19], the different stresses that cause a torque on the rotor include coupled loads; unbalanced dynamic forces; torsional vibration; transient torques; magnetic forces caused by leakage flux over the slots, making them vibrate at twice the frequency of the rotor; air gap eccentricity; centrifugal forces; thermal stresses caused by heat in the short-circuit ring and heat in the bars during starting (skin effect); residual forces due to casting; machining and welding. Under normal operating conditions, the spectrum of the signal has defined components. However, the asymmetries of the generator and the loads coupled to it (gearbox, blades) transmit torsional vibrations that act on the rotor, causing variations in the speed, torque, air gap magnetic flux and current bars. In this way, both mechanical and electrical faults manifest as lateral components of the fundamental wave. The number of harmonics and their amplitude depend on the magnitude of the fault [20; 21]. \[i_{sM}=i_{sM_{0}}+\sum A_{sM_{i}}sin(2\pi f_{v}t+\varphi_{M_{i}}) \tag{5}\]  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]  
\[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]  
\[+\frac{1}{2}(A_{sMi}\cos[2\pi(f_{s}-f_{v})t-\varphi_{M}]\]  
\[+A_{sT_{i}}\cos[2\pi(f_{s}-f_{v})t-\varphi_{T}]\} \tag{7}\]  
\[-\frac{1}{2}[A_{sM_{i}}\cos[2\pi(f_{s}+f_{v})t+\varphi_{M}]\]  
\[-A_{sT_{i}}\cos[2\pi(f_{s}+f_{v})t+\varphi_{T}]]\]  
\[i_{0}=\sqrt{i_{sM0}^{2}+i_{sT0}^{2}} \tag{8}\]  
\[\varphi_{0}=tg^{-1}\frac{i_{sT0}}{i_{sM0}} \tag{9}\]  
As described in reference [22], in a fault-free machine, the rotor and stator currents should be balanced. However, due to small differences in the winding geometry and the nonlinearity of the materials, an asymmetry arises that causes axial flow dispersion. Under these conditions, the distribution of the harmonics in the air gap undergo alterations that can be easily detected, so that we can detect broken rotor bars, one-phase failure, dynamic eccentricity, a negative sequence phase, and short circuits in the rotor and stator windings. According to the same author, in a three-phase machine, with a full pole-pass and fed by a balanced frequency \(\omega_{s}\), the spatial distribution of the harmonics of the MMF about the stator and as a function of the air gap flux is given by Equation (10). To apply these Equations to the rotor, \(\theta\) is given by (11) or (12). Substituting these Equations in the general term of (10) and expanding it to obtain the first terms, we obtain Equation (13), which provides the components of the frequency spectrum of the current induced in the rotor by the harmonics of the air gap. That is, the stator current spectrum (CS) includes the components of the supply current and those of the rotor. The presence of short circuits between turns produces an MMF with its own frequency spectrum that is superimposed on the main one to give rise to a new spectrum that is expressed by (14) and whose main term is (15). \[\Phi_{s}=\Phi_{1}\cos\left(\omega t-p\theta_{s}\right)+\Phi_{5}\cos\left(\omega t +5p\theta_{s}\right) \tag{10}\]  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]0  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]1  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]2  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]3  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]4  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]5  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]6  
As described in reference [14], the faulty and healthy squirrel-cage induction motor current is given by (16) and (17), respectively. According to reference [23], when there is a short circuit or static eccentricity in the stator, a negative sequence component appears, the amplitude of which depends on the percentage of shorted turns.