## 2 Methodology
### The Dragon Transform and Fault Severity Quantification  
The technique used for signal analysis and statistical examination in this paper is the so-called Dragon Transform [16]. This paper uses the harmonics due to rotor bar faults in inductions motors--they are present in the machine's stator current and describe unique trajectories in the time-frequency plane - for diagnosis purposes. Unfortunately, in inverted-fed machines these harmonics appear too close to the path described by the first harmonic fixed by the power supply. This makes it necessary to use a signal processing technique precise enough to follow the entire trajectory of these harmonics despite the proximity of the high-energy first harmonic. The Dragon Transform technique has this ability because it traces those trajectories as very thin lines in the time-frequency plane. This transform, which is explained in [16], is an atom-based correlation technique whose time and frequency resolutions allow the fault trajectories to be perfectly delineated and distinguished from the first harmonic, however close they may be. The atoms defined in [16] are known as Dragon atoms because their energy in the time-frequency plane follows the path of the harmonic component to be detected, i.e., the atoms' shape adapts perfectly to the trajectory of the harmonic to be followed. This kind of atoms is based on any modulated window, as the Gabor function (in the next equation, the atom is based on a gaussian window in brackets) [18]:  
\[\phi(t)=\left(C_{\sigma}e^{-\frac{(t-t_{c})^{2}}{2\sigma^{2}}}\right)e^{2\pi f _{c}(t-t_{c})}, \tag{1}\]  
where \(\sigma\) is a deviation parameter which characterizes time dispersion; \(t_{c}\) and \(f_{c}\) are the time and frequency at the center of the atom; and \(C_{\sigma}=1/\left(\sqrt[4]{\pi}\sqrt{\sigma}\right)\) is a normalization constant. The atom adapts to the trajectory of the signal to be detected if the exponential complex angle is defined so that its derivative is equal to the frequency of the component to be detected. In this way, these atoms can precisely follow the evolution of the harmonic over time. Therefore, each of the atoms' energy dispersion occurs in the direction of the component's evolution. Once the family of atoms is defined covering the entire time-frequency plane, these atoms are correlated with the signal. This correlation defines the Dragon Transform as follows [16]:  
\[\langle h,\phi\rangle=\int_{-\infty}^{+\infty}h(t)\phi^{*}(t)dt, \tag{2}\]  
where \(h(t)\) is the signal to be analyzed. The result of the Dragon Transform application is illustrated in Figure 1 with an actual signal, which is the startup current of an inverter-fed induction motor with a broken rotor bar. Inverters permit programming of the starting characteristics. In this case, the voltage supply's first harmonic follows a linear ramp up to 50 Hz in ten seconds. Figure 1a shows the stator current during a controlled starting and the operation of the motor in a stationary state after the starting. Figure 1b shows the motor slip \(s\), which is calculated as follows:  
\[s=\frac{n_{1}-n}{n_{1}}, \tag{3}\]where \(n\) is the motor speed and \(n_{1}\) is the synchronous speed, calculated as \(60f_{FC}/p\) (\(f_{FC}\) is the first harmonic frequency and \(p\) is machine's pole pairs). In an inverter-fed motor, the slip \(s\) is always very low, even during the startup transient, as it is shown in Figure 1b. Figure 1c shows the expected trajectory of the first component (controlled by the inverter) and the predicted paths of the fault-related harmonics (\(f_{BBH}\)) whose values are given by:  
\[f_{BBH}=(1\pm 2s)f_{FC}, \tag{4}\]  
This equation defines a pair of harmonics around the first component separated by \(2 sf_{FC}\), and known as \(LSH\) (lower side harmonic, \((1-2s)f_{FC}\)) and \(USH\) (upper side harmonic, \((1+2s)f_{FC}\)). The result of the correlation of the signal in Figure 1a with a suitable family of Dragon atoms is shown in Figure 1d, where color is used to represent each frequency component's energy.