## 2 Modeling Electrical Generator Faults Using the Current Signal
As described in reference [24; 25], the components due to stator failures (\(f_{sf}\)) are given by Equation (18), while according to reference [26], in the case of a healthy motor, the main components are the first and fifth harmonics. In the case of an unbalanced voltage, regardless of slip, this fault shows itself mainly in the first and third harmonics. According to reference [27], short circuits cause the components given by Equation (19). As described in reference [28], in a symmetrical stator, the CS contains the harmonics given by Equations (20) and (21), for which the harmonics determined by Equations (22) through (24) should be added, in case of asymmetry. As described in reference [29], another simple alternative for the early detection of stator faults depends on the magnitude of the negative sequence of the current, which allows us to obtain the negative impedance to be compared with the average winding impedance. \[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]7  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]8  
\[i_{sT}=i_{sT_{0}}+\sum A_{sT_{i}}cos(2\pi f_{v}t+\varphi_{T_{i}}) \tag{6}\]9  
\[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]0\[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]1  
\[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]2  
\[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]3  
\[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]4  
\[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]5  
As described in reference [30], rotor faults generate components below the supply frequency in the stator spectrum, according to Equation (25), where the first term does not contribute to increasing the supply current because it induces an MMF of zero sequence. However, the second term induces a set of three-phase currents at the supply frequency and contains a component displaced by twice the slip frequency, \(2 sf_{s}\). The fault causes a \(2 sp\omega_{mr}\) variation in rotor speed, causing a displacement of the lower component \((1-2s)f_{s}\) and the appearance of an upper component at \((1+2s)f_{s}\) modulated by the third harmonic of the stator flux. Other components that may appear are given by Equation (26). Because of the static, dynamic, or mixed eccentricity, the air gap is not uniform, and the forces applied to the shaft become unbalanced, causing friction between the stator and rotor. According to references [26; 31; 32], the eccentricity causes the appearance of harmonics whose sequence is given by (27). If the eccentricity is static, \(n_{d}\) is zero, while if it is dynamic, it is \(1,2,3,\dots\) However, according to reference [33], a difference between static and dynamic eccentricity does not always exist for all motor configurations, in addition to the fact that there are components that are not easily detectable. \[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]6  
\[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]7  
\[i_{a}(t)=i_{0}\sin\left(2\pi f_{s}t+\varphi_{0}\right)\]8  
As described in reference [34], mechanical faults can be classified as those that cause air gap eccentricity, load torque oscillations, or a combination of both. The first effect is due to unbalanced loads, shaft misalignment, and gearbox and bearing failures. The second effect is due to wear or failure of the bearings and to rotor imbalance caused by poor assembly, for example. The load torque oscillation component, Equation (28), affects the rotor position and stator current. The length of the air gap affects the permeance, flux density of the air gap, and MMF, which ultimately modulates the stator current signal. When there is air gap eccentricity, that failure can vary with time and the angle of the circumference \(\theta\), so, in the case of dynamic eccentricity, the length of the air gap is a function of \(\theta\) and \(t\), according to Equation (29). The use of \(\omega_{r}t\) in the last equation provides an expression for the static eccentricity.