## 2 Symmetrical Component Analysis of Electrical Winding Faults  
The imbalance in three-phase systems, under steady state condition, can be evaluated using the well-known method of symmetrical components analysis (SCA). In the case of the monitoring and diagnostics of different asymmetries in induction motors, the three-phase stator currents are transformed into a symmetrical component coordinate system, according to the following complex Fortescue's transformation:  
\[\left[ \begin{array}{c}I_{1}\\ I_{2}\\ I_{0}\end{array} \right]=\frac{1}{3}\left[ \begin{array}{ccc}1&a&a^{2}\\ 1&a^{2}&a\\ 1&1&1\end{array} \right]\cdot\left[ \begin{array}{c}I_{sA}\\ I_{sB}\\ I_{sC}\end{array} \right], \tag{1}\]  
where:  
\(I_{sA}\), \(I_{sB}\), \(I_{sC}\)--stator phase currents in steady state in phases A, B, C, respectively;  
\(I_{0}\), \(I_{1}\), \(I_{2}\)--zero-, positive- and negative-sequence components of the stator current in steady state, respectively;  
\(a=\exp(j2\pi/3)\). Equation (1) relates to sinusoidal phase current signals in steady state. In three-phase induction motors there is no zero-sequence current component \(I_{0}\), this is why the calculation of symmetrical components in the steady state is reduced to the calculation of the positive- and negative-sequence components \(I_{1}\), \(I_{2}\), from Equation (1). However, the supply of IMs from PWM inverters additionally introduces a series of harmonics that cause the distortion of voltages and currents, moreover these signals change over time. Thus the classical SCA cannot be used directly in this case. The concept of symmetrical components described in Equation (1) deals with the phasors and hence is a steady-state concept. However, in the technical literature it was extended to the concept of symmetrical components in the time domain, which is also referred to as instantaneous symmetrical components (ISCA) [40; 41]:  
\[\left[ \begin{array}{c}i_{1}(t)\\ i_{2}(t)\\ i_{0}(t)\end{array} \right]=\frac{1}{3}\left[ \begin{array}{ccc}1&a&a^{2}\\ 1&a^{2}&a\\ 1&1&1\end{array} \right]\cdot\left[ \begin{array}{c}i_{sA}(t)\\ i_{sB}(t)\\ i_{sC}(t)\end{array} \right], \tag{2}\]  
where:  
\(i_{sA}(t)\), \(i_{sB}(t)\), \(i_{sC}(t)\)--corresponding instantaneous current signals of the three phases,  
\(i_{0}\), \(i_{1}\), \(i_{2}\)--instantaneous zero-, positive-, and negative-sequence symmetrical components of the stator current. The instantaneous symmetrical components defined by Equation (2) are in general complex signals. According to another definition, which can also apply here, you can get real signals. The idea consists in the replacement of complex phasor \(a\) with a \(120^{\circ}\) phase-shift operator in the time domain [42]. Thus:  
\[\left[ \begin{array}{c}i_{1}\\ i_{2}\\ i_{0}\end{array} \right]=\frac{1}{3}\left[ \begin{array}{c}i_{sA}+S_{120}i_{sB}+ S_{240}i_{sC}\\ i_{sA}+S_{240}i_{sB}+S_{120}i_{sC}\\ i_{sA}+i_{sB}+i_{sC}\end{array} \right], \tag{3}\]  
where \(S_{x}\) stands for an \(x\)-degree phase-angle operator in the time domain. Another formulation can be derived based on a \(90^{\circ}\) phase-shift operator that is easier to implement [43]. The idea is to write:  
\[e^{\pm j120}=-\frac{1}{2}\pm\frac{\sqrt{3}}{2}e^{j90}, \tag{4}\]  
and the result is as follows:  
\[\left[ \begin{array}{c}i_{1}\\ i_{2}\\ i_{0}\end{array} \right]=\frac{1}{3}\left[ \begin{array}{c}i_{sA}-\frac{1}{2}( i_{sB}+i_{sC})+\frac{\sqrt{3}}{2}S_{90}(i_{sB}-i_{sC})\\ i_{sA}-\frac{1}{2}(i_{sB}+i_{sC})-\frac{\sqrt{3}}{2}S_{90}(i_{sB}-i_{sC})\\ i_{sA}+i_{sB}+i_{sC}\end{array} \right]. \tag{5}\]  
This concept can be defined both on a total signal (such as above) or on its single frequency component (for example its fundamental component). It can be shown that Equations (3) and (5) are equivalent with respect to the fundamental components. Taking into account that in the cage induction motor the zero-sequence current component does not exist, quantities \(i_{1}\), \(i_{2}\) can be presented as the sums of the real and imaginary parts:  
\[\left[ \begin{array}{c}i_{1}\\ i_{2}\end{array}