## 1 Introduction  
Three-phase squirrel cage induction motor plays an important role in industrial processing because of its low cost and high reliability. However, sudden failures of the induction motor may lead to the descent of rotation precision or even result in irreversible damage to the whole machine and have to stop to maintain. Therefore, eccentricity fault diagnosis is necessary to guarantee the output precision and prevent the aggravation of the fault. Airgap eccentricity is one of the most common types of fault occurring in the induction motor [1]. This type of fault leads to unequal airgaps between the stator and rotor, which has a bad influence on the output rotation accuracy of the motor. If this type of fault is not resolved, it would cause the rotor-stator rub and consequently damage of the whole motor. Figure 1 provides the rotor-stator rub fault caused by eccentricity. According to Figure 1, the scratches are obvious on the outer ring of rotor shown in the red circle. For that cause, the motor is inevitable to maintain and this would inflict economic damage. There are three forms of eccentricity including static eccentricity (SE), dynamic eccentricity (DE) and a combination of both that is called mixed eccentricity. In the SE, the rotor is displaced from the stator center, but the rotor rotates around its own center. In the DE, the rotor is also displaced from the stator center, but the rotor rotates around the center of the stator center. In most cases, the SE and DE simultaneously occur. In that case, the center of the stator, the center of the rotor and the rotation axis are displaced with respect to each other [2; 3]. In order to detect the eccentricity fault, several types of techniques are used, such as the monitoring of magnetic flux and vibration [4]. However, due to the high cost and difficult installation of sensors, the aforementioned methods are less recommended. In decade years, motor current signature analysis (MCSA) [5] is widely used to detect the fault in the electrical machine, because the stator current signal is relatively easy to obtain and do not need additional complex and expensive sensors. According to the theoretical models, the spectral components related to fault can be located [6]. The occurrence of airgap eccentricity can introduce the specific harmonic component in the stator current spectrum. Using MCSA, the harmonic frequency sideband associated with the eccentricity fault can be expressed as Equation (1) [1, 7, 8, 9]:  
\[f=\bigg{[}(kN_{r}\pm n_{d})\frac{(1-s)}{p}\pm v\bigg{]}f_{s} \tag{1}\]  
where \(k\) is an integer, \(N_{r}\) is the number of rotor slots, \(s\) is the slip, \(p\) is the number of pole-pairs, \(f_{s}\) is the fundamental frequency, \(n_{d}\) is an integer representing the eccentricity order, \(n_{d}=0\) represents the static eccentricity and \(n_{d}=1,2,3\cdots\) represents the DE, \(v\) is the stator time harmonics. Moreover, if the SE and DE simultaneously occur, specific harmonic can be observed around fundamental frequency given by [1,7,9]:  
\[f_{mix}=f_{s}\pm mf_{r} \tag{2}\]  
\[f_{r}=\frac{1-s}{p}f_{s} \tag{3}\]  
where \(m=1\), \(2\), \(3\cdots\). Using the MCSA, only the single-phase stator current needs to be measured in order to detect the eccentricity fault. Apart from the monitoring of single-phase current, instantaneous power is another signal that contains more information than the single-phase current. Due to the transfer from the fundamental frequency to around DC, the fault signal can be detected clearly and the signal-to-noise ratio (SNR) can be improved [1]. Because of these advantages, the spectrum of complex apparent power signature has been employed for diagnosis of the mixed airgap eccentricity in the induction motor [9]. The utilization of instantaneous active power, as well as reactive power signal accompanying with their power factor and phase angle, can help to detect the eccentricity fault and broken rotor bars, as well as discriminating them from mechanical load oscillation [10].