The proposed HS technique is composed of two procedures: harmonic series processing (HSP) and local spectra analysis (LSA). The HSP is to synthesize the fault related features in the spectrum, whereas LSA is to extract fault indicators for incipient IM air-gap eccentricity fault detection.

### 2.1 Harmonic Series Processing

In stator current signal based IM fault detection, the characteristic frequency components *f*
_{
e
} in Hz used to detect static and dynamic eccentricity defects [21] are given by

$$\begin{aligned} f_{e} & = f_{s} \left[ {\left( {kR \pm \alpha } \right)\frac{1 - s}{p} \pm \beta } \right] \\ & = \left( {kR \pm \alpha } \right)f_{r} \pm \beta f_{s} \\ \end{aligned}$$

(1)

where *f*
_{
s
} is the supply frequency in Hz; *R* is the number of rotor slots; *s* is the slip; *f*
_{
r
} is the rotor rotating speed; *k* = 1, 2, 3, …; *p* is the number of pole pairs; \(\beta = 1, \, 3, \, 5, \ldots\) is the order of the stator time harmonics; \(\alpha\) is the eccentricity order: \(\alpha = 0\) denotes static eccentricity and \(\alpha = 1\) denotes dynamic eccentricity.

The *K*th order fault harmonic series *f*
_{
e_K
} will be

$$f_{e\_K} = f_{o\_K} + \beta f_{s} ,\quad \beta = { 1},{ 3},{ 5}, \, \ldots$$

(2)

where the *K*th order origin frequency component \(f_{o\_K}\) is

$$f_{o\_K} = \left( {kR \pm \alpha } \right)f_{r}$$

(3)

To explore the relationship among different fault harmonic series, the fault harmonic series of interest are synthesized to reveal fault characteristic features. If the first *P* fault harmonics (\(\beta\) = 1, 3, 5, …, 2*P–*1) are considered for synthesis, the harmonic frequency band corresponding to the *K*th fault harmonic series will be extracted as

$$q_{K} = \left[ {f_{o\_K} , \, f_{o\_K} + 2Pf_{s} } \right]$$

(4)

and the bandwidth in Hz is \(2Pf_{s}\) +1.

To synthesize fault harmonic series, the frequency bands are converted to a discrete-point domain representation. The discrete-point closest to \(f_{o\_K}\) is considered as the first discrete-point in *q*
_{
K
}, defined as *D*
_{
k
}{1}. Suppose *u*
_{
f
} discrete data points represent unit frequency distance in the spectrum, the bandwidth of Eq. (4) in discrete-point domain will be formulated as

$$B_{w} = \left\langle {2Pf_{s} + 1} \right\rangle u_{f}$$

(5)

where \(\left\langle \cdot \right\rangle\) denotes the integer function. Then the harmonic frequency band of Eq. (4) in the discrete-point domain will be

$$D_{K} \{ i\} , \quad i = { 1},{ 2},{ 3}, \, \ldots ,B_{w}$$

(6)

The root-mean-square (RMS) operation is applied to synthesize harmonic frequency bands. Given the harmonic frequency bands of *M* interested fault harmonic series, the *i*th component in the synthesized spectrum *v* will be derived as

$$v\left\{ i \right\} = \sqrt {\frac{{\sum\nolimits_{j = 1}^{M} {D_{j} \left\{ i \right\}^{2} } }}{M}} \quad i = { 1},{ 2},{ 3}, \, \ldots ,B_{w}$$

(7)

### 2.2 Local Spectra Analysis

The *m*th harmonic (\(\beta\) = 2 *m–*1) of the *K*th order fault harmonic series will be derived from Eq. (2),

$$f_{K,m} = f_{o\_K} + \left( {2m - 1} \right)f_{s}$$

(8)

Through the synthesis operation in Eq. (7), the *m*th fault harmonic over all *M* harmonic frequency bands is synthesized to form a new fault frequency component. Then the *m*th synthesized fault frequency component in the discrete-point domain will be derived in the synthesized spectrum by the following representation:

$$v\left\{ {1 + \left( {2m - 1} \right)f_{s} u_{f} } \right\}$$

(9)

The local spectra encompassing the *m*th synthesized fault frequency component will be

$$\left[ {v\left\{ {1 + \left( {2m - 1} \right)f_{s} u_{f} - du_{f} } \right\}, \, v\left\{ {1 + \left( {2m - 1} \right)f_{s} u_{f} + du_{f} } \right\}} \right]$$

(10)

where *d* is the half bandwidth of the local spectra in the frequency domain. The mean value of this local spectrum is the fault indicator regarding the *m*th synthesized fault frequency component, denoted by *F*
_{
m
}.

To illustrate the operation of the proposed HSP and LSA processes, a simulated current spectrum with three air-gap eccentricity frequency harmonic series is illustrated in Figure 1. It is normalized by deducting mean and then being divided by its standard deviation. In Figure 1, the black arrow points 50 Hz supply frequency; magenta arrows with number 1, green arrows with number 2 and yellow arrows with number 3 denote three fault harmonic series respectively. The origin frequency components \(f_{o\_K}\) are 114.1 Hz, 142.4 Hz, and 170.7 Hz, respectively. The harmonics of defect features are computed using Eq. (2) with *f*
_{
s
} = 50 Hz and \(\beta\) = 1, 3, 5, 7, 9. Figure 1 shows that some fault characteristic frequency components are buried in the spectrum, and it is difficult to predict which characteristic frequency components will protrude in the spectrum. The extracted frequency bands represented in Eq. (6) of three fault harmonic series, and synthesized spectrum represented in Eq. (7) with fault related local spectra are demonstrated in Figure 2. The dashed vertical lines indicate the boundaries of local spectra encompassing synthesized fault frequency components. Figures 2(a), 2(b) and 2(c) represent three frequency bands computed by Eq. (4) with staring frequencies \(f_{o\_K}\) being 114.1 Hz, 142.4 Hz, and 170.7 Hz, respectively. Figure 2(d) shows results of the synchronization of spectra in Figures 2(a), 2(b) and 2(c) using root-mean-square calculation using Eq. (7). It is seen from Figure 2(d) that the fault characteristic features can be highlighted in the spectrum after the HSP operation, which could be used for IM fault detection. In this simulation, the supply frequency *f*
_{
s
} = 50 Hz; \(f_{o\_K}\) is the starting point of the spectrum in Figure 2(d). Based on Eqs. (2) and (3), the fault frequencies in Figure 2(d) should appear at 50 Hz, 150 Hz, 250 Hz, 350 Hz and 450 Hz. The frequency components outside the boundaries of local spectra are usually caused by the modulation of load variation and supply frequency, and the synchronization of different harmonic series local bands.

If both static and dynamic eccentricities occur simultaneously, the following characteristic frequency components could also be excited [22], given by

$$f^{\prime}_{e} = \left| {kf_{r} \pm f_{s} } \right|,\quad k = 1,{ 2},{ 3}, \, \ldots$$

(11)

The related harmonic series can be expressed as

$$f^{\prime}_{e1} = kf_{r} + f_{s} ,\quad k = { 1},{ 2},{ 3}, \, \ldots$$

(12)

$$f^{\prime}_{e2} = kf_{r} - f_{s} ,\quad k = { 2},{ 3}, \, \ldots$$

(13)

The local spectra of the first *Q* harmonics in Eq. (12) will be transformed to the discrete-point domain, and then synthesized into one spectrum in RMS form using Eq. (7). The mean value of the synthesized spectrum is considered as a fault indicator denoted by *F*
_{
d1}. The fault indicator derived from Eq. (13) will be denoted by *F*
_{
d2}.