## 2 Modeling Electrical Generator Faults Using the Current Signal  
Due to its design, durability, and low cost, the use of the squirrel-cage induction machine predominates at the industrial, commercial, and domestic levels [10; 11]. Although there are several types of generators, according to reference [12], in the WI, the doubly fed induction generator (DFIG), and the squirrel-cage induction generator (SCIG) predominate. The voltage signals, current, magnetic field, magnetomotive force (MMF), torque, and power of an induction machine are characterized by its sinusoidal behavior [13]. Since the speed of the rotor depends on the coefficients of the associated differential equations, which vary with time, the behavior of the materials used in the construction of the motor is not constant over time but depends on the position of the rotor. Under these conditions, it is hard to analyze signals in a spatial system, which is why in-plane analysis is preferred. For this purpose, using the Clarke and Park transforms, a change of variables is made. With the first transformation, we go from a 3D system (abc) to a 2D plane (alpha-beta) that varies with the stator, while with the second one, we obtain a plane dq0 equivalent to a 2D plane that rotates at the same rotor speed but is offset by an angle \(\theta\). Since the three-phase induction machine generally does not use a neutral line, the main current does not have a homopolar component, and the three phases can be represented in the dq plane. In this plane, the stator remains fixed (direct axis d) in relation to a rotor plane (quadrature axis q) that rotates at speed \(\omega_{\chi}\)[13; 14; 15]. The transformation between the abc space system and the dq0 plane, when the latter is oriented at an angle \(\theta\) with reference to the axis that remains fixed, can be performed directly using Equations (1) and (2). When \(\theta\) is zero, these Equations become (3) and (4), respectively [15; 16; 17; 18]. \[\left[ \begin{array}{c}i_{qs}\\ i_{ds}\\ i_{0s}\end{array} \right]=\frac{2}{3}\left[ \begin{array}{ccc}\cos\theta&\cos( \theta-\frac{2\pi}{3})&\cos(\theta+\frac{2\pi}{3})\\ \sin\theta&\sin(\theta-\frac{2\pi}{3})&\sin(\theta+\frac{2\pi}{3})\\ 0.5&0.5&0.5\end{array} \right]\left[ \begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\end{array} \right] \tag{1}\]  
\[\left[ \begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\end{array} \right]=\left[ \begin{array}{ccc}\cos\theta&\sin\theta&1\\ \cos(\theta-\frac{2\pi}{3})&\sin(\theta-\frac{2\pi}{3})&1\\ \cos(\theta+\frac{2\pi}{3})&\sin(\theta+\frac{2\pi}{3})&1\\ \end{array} \right]\left[ \begin{array}{c}i_{qs}\\ i_{ds}\end{array} \right] \tag{2}\]  
\[\left[ \begin{array}{c}i_{qs}\\ i_{ds}\end{array} \right]=\frac{2}{3}\left[ \begin{array}{ccc}1&-\frac{1}{2}&- \frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \end{array} \right]\left[ \begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\end{array} \right] \tag{3}\]  
\[\left[ \begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\end{array} \right]=\left[ \begin{array}{ccc}1&0\\ -\frac{1}{2}&-\frac{\sqrt{3}}{2}\\ -\frac{1}{2}&\frac{\sqrt{3}}{2}\end{array} \right]\left[ \begin{array}{c}i_{ a}\\ i_{b}\\ i_{c}\end{array} \right] \tag{4}\]  
According to reference [16], the phase current of the DFIG can be expressed as a function of the flow and torque vectors (the torque angle is \(90^{\circ}\) over the flow), Equations (5) and (6), respectively, which can be represented in the dq plane, according to Equation (2). Since the variables of Equations (5) and (6) rotate at speed \(2\pi\)fs, they cannot be measured directly, so it is necessary to apply the inverse Park transform to obtain the phase currents according to Equations (7) to (9).