Diagnosis of broken bars fault in induction machines using higher order spectral analysis

Diagnosis of broken-bars fault in induction machines using higher
order spectral analysis
L. Saidi a,b,n
, F. Fnaiech a
, H. Henao b
, G-A. Capolino b
, G. Cirrincione b
a
Universite´ de Tunis, Ecole Supe´rieure des Sciences et Techniques de Tunis, SICISI, 5 avenue Taha Hussein, BP 96, Montfleury, 1008 Tunis, Tunisie
b
University of Picardie ‘‘Jules Verne’’, Department of Electrical Engineering, 33 rue Saint Leu, 80039 Amiens cedex 1, France
a r t i c l e i n f o
Article history:
Received 30 November 2011
Received in revised form
18 July 2012
Accepted 15 August 2012
Available online 21 September 2012
This paper was recommended for
publication by Jeff Pieper
Keywords:
Broken rotor bars
Bi-spectrum analysis
No-load condition
Motor current signature analysis (MCSA)
Receiver operating characteristic (ROC)
a b s t r a c t
Detection and identification of induction machine faults through the stator current signal using higher
order spectra analysis is presented. This technique is known as motor current signature analysis (MCSA).
This paper proposes two higher order spectra techniques, namely the power spectrum and the slices of bi-
spectrum used for the analysis of induction machine stator current leading to the detection of electrical
failures within the rotor cage. The method has been tested by using both healthy and broken rotor bars
cases for an 18.5 kW-220 V/380 V-50 Hz-2 pair of poles induction motor under different load conditions.
Experimental signals have been analyzed highlighting that bi-spectrum results show their superiority in
the accurate detection of rotor broken bars. Even when the induction machine is rotating at a low level of
shaft load (no-load condition), the rotor fault detection is efficient. We will also demonstrate through the
analysis and experimental verification, that our proposed proposed-method has better detection
performance in terms of receiver operation characteristics (ROC) curves and precision-recall graph.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
For several decades both correlation and power spectrum (PS)
have been most significant tools for digital signal processing (DSP)
applications in electrical machines diagnosis area. The information
contained in the PS is sufficient for a full statistical description
of Gaussian signals. However, there are several situations for which
looking beyond the autocorrelation of a signal to extract the infor-
mation regarding deviation from Gaussianity and the presence of
phase relations is needed. Higher order spectra (HOS), also known as
polyspectra, are spectral representations of higher order statistics,
i.e., moments and cumulants of third order and beyond. HOS can
detect deviations from linearity, stationarity or Gaussianity in any
signal [1,2]. Most of the electrical machines signals are non-linear,
non-stationary and non-Gaussian in nature and therefore, it can be
more interesting to analyze them with HOS compared to the use of
second-order correlations and power spectra. In this paper, the
application of HOS on stator current signals for different rotor
broken bars fault severities and shaft load levels has been discussed.
So far, the large usage of induction machines (IM) is mainly
due to their robustness, to their power efficiency and to their
reduced manufacturing cost. Therefore, the necessity for the
increasing reliability of electrical machines is now an important
challenge and because of the progress made in engineering,
rotating machinery is becoming faster, as well as being required
to run for longer periods of time. By looking to these last factors,
the early stage detection, localization, and analysis of faults are
challenging tasks. The distribution of IM failures has been
reported in many reliability survey papers [3–8]. These main
faults include: (a) stator faults, which define stator windings open
or short- circuits and stator inter-turn faults; (b) rotor electrical
faults, including rotor winding open or short-circuits for wound
rotor machines or broken rotor bars (BRB) or cracked end ring for
squirrel cage machines; (c) rotor mechanical faults such as
bearing damages, static or dynamic eccentricities, bent shaft,
misalignment, and any load-related abnormal phenomenon [3].
Issues of preventive maintenance, on-line machine fault detec-
tion, and condition monitoring are of increasing importance [3–9].
During the last twenty years or so, there has been a substantial
amount of research dealing towards new condition monitoring
techniques for electrical machine and drives. New methods have
been developed for this purpose [5]. Monitoring machine line
current and observing the behavior in time-domain and
frequency-domain characteristics, from healthy to faulty condition,
has been always the first step of analysis. This technique is known as
machine current signature analysis (MCSA).
The MCSA technique has been widely used for fault detection
and diagnosis and it is considered as the most popular not only
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.08.003
n
Corresponding author at: Universite´ de Tunis, De´partement de Geńie Electri-
que, Ecole Supe´rieure des Sciences et Techniques de Tunis, SICISI, 5 avenue Taha
Hussein, BP 96, Montfleury, 1008 Tunis, Tunisie. Tel.: þ216 22169567;
fax: þ216 72486044.
E-mail address: lotfi.saidi@u-picardie.fr (L. Saidi).
ISA Transactions 52 (2013) 140–148

for electrical faults but also for mechanical faults. Moreover, it can
easily detect machine faults by using only current sensors which
are not really invasive. Other signals to be monitored could be
stator voltages, instantaneous stator power, shaft vibration, stray
flux, electromagnetic torque, speed, temperature, acoustic noise
and much more [3–9]. Features extracted from one or more of
these signals could be used as the fault diagnosis indexes.
This paper focuses on BRB fault in IM which represent about
10% of total industrial induction motor faults [3], by using both
spectral and bi-spectral analysis of the stator current. MCSA has
been extensively used to detect BRB and end ring faults in IM
[9–11]. The side-band frequency components at (172ks)fs have
been used to detect such faults, (s is the rotor slip and fs is the
fundamental supply frequency, and one integer k¼1, 2, 3, y).
Although, the MCSA gives efficient results when the motor
operates under rated or high load conditions, some drawbacks have
been observed when the load shaft is light. Since the side-band
frequency components (172s)fs are very close to the grid frequency
fs, a natural spectral leakage can hide frequency components
characteristics of the fault [12]. In this case, the standard MCSA
method fails to detect BRB faults. This problem can be solved by
increasing the sampling frequency but load variations during sam-
pling may decrease the quality of the spectrum [9]. Hence a trade-off
must be performed between spectrum leakage and frequency
resolution which is difficult to be reached when using the Fast
Fourier transform (FFT). Consequently, MCSA method is strongly
dependent on the rotor slip condition and it has not been applied to
BRB faults under no-load condition.
Due to those limitations of FFT, in the literature other methods
for spectral analysis become potential options to the detection of
BRB such as the zoom-FFT (ZFFT) [9–11], in [11] a method based on
the multiple signal classification (MUSIC) has been proposed to
improve diagnosis of BRB fault in IM, by detecting a large number of
frequencies in a given bandwidth. This method is called zoom-
MUSIC, wavelet approaches [13], neural network [14]. In those
techniques, the computational time and the accuracy in a particular
frequency range are increased. However, as expected, the frequency
resolution is still affected by the time acquisition period and the load
effect (especially no-load condition) was not studied.
The analysis of faults at low slip is important in industrial
applications and would provide the following benefits [12]:
À Improving quality control of new motors,
À Allowing no-load analysis of all type of motors,
À Preventing confusion of faults with load-induced current
oscillation,
À Reducing the cost of fault analysis.
In this context, the MCSA method based on a Hilbert transform
(HT) was used to detect BRB faults at very low rotor slip conditions
[15]. The effectiveness of the method was confirmed by experimen-
tal data on an induction motor with one BRB fault. The performance
of the method was not evaluated for different numbers of BRB.
Besides this method requires high computation and large amount of
data is required for the high frequency resolution. In another study
[16], the HT is used to extract signal envelop and then the wavelet
transform is applied to estimate fault index around 2sfs frequency
component which reflects the energy variation of the phase current
related to the BRB fault. The main drawback of this approach is that
no clear methodology is proposed for the wavelet decomposition.
Various DSP techniques [4–9] have been extensively used for
feature extraction purposes and HOS has been one of them [17–24].
The bi-spectrum is the third order spectrum and it results in a
frequency–frequency–amplitude relationship which shows coupling
effects between signals at different frequencies [19]. This tool has
already demonstrated its efficiency in many detection applications
including machinery diagnosis for various mechanical faults [20].
Other applications include the usage of the bi-spectral analysis to
detect the fatigue cracks in beams [21], stator inter-turn faults [22],
and bearings [23]. It has been also used in the case of the coupling
assessment between modes in a power generation system [19]. The
application of HOS techniques in condition monitoring has been
already reported [24,25] and it is clear that multi-dimensional HOS
would contain more useful information than traditional two-
dimensional spectral analysis for diagnostics purposes.
The next section will give a brief development of the basic
bi-spectrum theory followed by a description of the experimental
set-up. The third section will develop the PS and the bi-spectrum
signal processing tools in order to characterize the stator current
frequency components in presence of BRB. The last section will be
dedicated to the analysis of experimental results, and ends with a
comparison of the two higher order spectra techniques (PS and
proposed BDS methods) by means of the ROC curves, before
giving some conclusions on the implementation of this advanced
DSP technique for electrical machines fault detection.
2. HOS analysis
2.1. Basic definitions
In this section, the theoretical concepts of the bi-spectrum
method will be recalled. Let us assume that: x(n), n¼0, 1,y,NÀ1
is a discrete current signal, zero-mean and locally stationary
random process with a length N. The autocorrelation function of
a stationary process x(n) is defined by:
RxxðmÞ ¼ EfxðnÞxðnþmÞg ð1Þ
where E{.} is the expected mean operator (or equivalently the
average over a statistical set) and m is a discrete time-delay. The
PS is formally defined as the Fourier transform (FT) of the auto-
correlation sequence:
PxxðfÞ ¼
Xþ 1
m ¼ À1
RxxðmÞ eÀj2pf m
ð2Þ
where f denotes the frequency.
An equivalent definition can be given by:
PxxðfÞ ¼ EfXðfÞXn
ðf Þg ¼ Ef9Xðf Þ2
9g ð3Þ
where Xn
denotes the complex conjugate of X and X(f) is the
discrete Fourier transform (DFT) of x(n), given by:
XðfÞ ¼
XNÀ1
n ¼ 0
xðnÞ eÀj2pfn
N ð4Þ
where N is the number of samples.
The third-order spectrum, called the bi-spectrum B(f1, f2) is
defined as the double DFT of the third order moment. It can be
defined as:
Bðf1,f 2Þ ¼
Xþ 1
k ¼ À1
Xþ1
l ¼ À1
Mx
3ðk,lÞWðk,lÞeÀj2p
N ðf 1k þ f2lÞ
ð5Þ
where W(k, l) is a two-dimensional window function used to
reduce the variance of the bi-spectrum and Mx
3ðk,lÞ is the third-
order moment of the process x(n), given by:
Mx
3ðk,lÞ ¼ Efxn
ðnÞxðnþkÞxðnþlÞg ð6Þ
where k and l are discrete time delays.
As equivalence, Eq. (5) can be expressed in terms of the FT of
x(n) as:
Bðf1,f 2Þ ¼ EfXðf1ÞXðf 2ÞXn
ðf1 þf2Þg: ð7Þ
L. Saidi et al. / ISA Transactions 52 (2013) 140–148 141

It can be noted that the bi-spectrum presents twelve symme-
try regions [1,2]. Hence, the analysis can take into consideration
only a single non-redundant region. Hereafter, B(f1, f2) will denote
the bi-spectrum in the triangular region T defined by:
T ¼ ðf1,f2Þ : 0rf2 rf1 rfe=2, f2 rÀ2f1 þf e, where fe is the
sampling frequency.
In order to minimize the computational cost, the direct method
[1,2] has been used to estimate the bi-spectrum of a stator current
signal as described in the following paragraph.
2.2. Bi-spectrum estimation
In general, the expected values coming from Eqs. (3)–(7) need
to be estimated from a finite quantity of available data. This may
be achieved by dividing a signal into M overlapping segments
with k as a subscript, k¼1, 2, 3,y,M. A windowing function has
been applied to each segment and the FTs of all segments are
averaged [1]. The aim is to reduce the variance of the estimate by
increasing the number of records M [1,2].
The estimated bi-spectrum ^Bðf1,f2Þ is given by:
^Bðf 1,f2Þ ¼
1
M
XM
k ¼ 1
Xkðf1ÞXkðf2ÞXk
n
ðf1 þf2Þ % E Xðf 1ÞXðf2ÞXn
ðf 1 þf 2Þ
È É
:
ð8Þ
The conventional PS concerns only the square of the magnitude
of the Fourier coefficients and it cannot give information about
phase coupling [1,2]. On the other hand, the bi-spectrum is complex
and, therefore, it has magnitude and phase. The bi-spectrum or bi-
coherence (normalized bi-spectrum) based on HOS emphasizes the
triple products of the Fourier coefficients and it is really able to
provide such information. However, the computational cost is
higher and the 3D-plot or the contours plots are very complex to
describe as compared with that of the one-dimension [23,24]. In
fact, it is impossible to compute the bi-spectrum from large data
sets. On the other hand, by using smaller data sets, the estimation
error will increase a lot [2]. The proposed method is directly
computed in one-dimensional bi-spectrum diagonal slice (BDS)
which condenses the 2D distribution into 1D curve. It requires less
computation and it can be used in real time [22].
The BDS of a process x(n) can be obtained by setting f1 ¼f2 ¼f
and it is defined as follows:
^Bðf 1,f2Þ f1 ¼ f 2 ¼ f ¼ ^Dðf Þ % EfX2
ðf ÞXn
ð2f Þg

ð9Þ
3. BRB faults diagnosis based on PS and BDS
In this section, laboratory experiments have been used to verify
the above HOS-based IM fault detection techniques. Two identical
IM have been used to compare the results with one being healthy
and the other one being faulty with rotor broken bars.
3.1. BRB fault
It has been evaluated that rotor failures account about 10% of
IM faults [3,4]. The objective of MCSA is to compute a frequency
spectrum of the stator current and to recognize the frequency
components which are distinctive of rotor faults. Eq. (10) gives
the frequency components which are characteristic of BRB [3,4]:
fBRB ¼
k
p

ð1ÀsÞ7s
!
fs ð10Þ
where fs is the supply frequency, p is the number of pole pairs, k is
an integer and s is the rotor slip. By considering the speed ripple
effects, it was reported that other frequency components, which
can be observed in the stator current spectrum, are given by [3,4]:
fBRB ¼ ð172ksÞfs ð11Þ
The side-band frequency components at (172s)fs has been used
to detect such rotor faults. While the lower side-band is fault-
related, the upper side-band is due to consequent speed oscillations.
It has been shown that the sum of magnitudes of these two side-
band frequency components is a good diagnostic index given by [4]:
IdB ¼ 20log10
Il
I

þ20log10
Ir
I
!
=2 ð12Þ
where Il, Ir and I are the amplitude of the lower, upper side-band
frequency components and of the fundamental frequency of the
stator current, respectively.
3.2. Model of the BRB stator current
A simple model which characterizes the IM stator current,
with electrical rotor asymmetries includes the so-called side-
band frequencies and it can be expressed by [11]:
iaðtÞ ¼ if cosðotÀjÞþ
X
k
il,kcosððoÀof ,kÞtÀjl,kÞ
þ
X
k
ir,kcosððoþof ,kÞtÀjr,kÞ ð13Þ
with fs as the fundamental frequency of the power grid.
if is the fundamental value of the stator current amplitude
(index f), o is its angular frequency, j is its main phase shift
angle, of ,k ¼ 2kso, and il,k, ir,k and jl,k, jr,k (k¼1, 2, 3,y,) are the
magnitude and the phase of the left (index l) and right (index r)
sidebands components, respectively.
As of,k ¼ 2kso and o ¼ 2pf s, the expression (13) can be
rewritten as:
iaðtÞ ¼ if cosð2pfstÀjÞþ
X
k
il,kcosð2pfl,ktÀjl,kÞ
þ
X
k
ir,kcosð2pfr,ktÀjr,kÞ ð14Þ
where fl,k¼(1–2ks)fs and. fr,k¼(1þ2ks)fs
This simplified expression of the stator current signal has been
used extensively for IM fault condition monitoring. By checking
the magnitude of the side-band frequency components through
the spectrum computation, various faults such as rotor bar breakage
can be detected with a high level of accuracy [19].
From Eq. (14), the FT can be computed:
Iaðf Þ ¼
if
2
dðf 7fsÞe8jj þ
1
2
X
k
il,kdðf 7f l,kÞe8jjl,k
þ
1
2
X
k
ir,kdðf 7fr,kÞe8 jjr,k ð15Þ
where d(.) represents the Dirac delta function.
By ignoring contributions at negative frequencies which fall
outside the useful region of the bi-spectrum, Eq. (15) can be
written as:
Iaðf Þ ¼
if
2
dðf ÀfsÞejj
þ
1
2
X
k
il,kdðf Àfl,kÞejjl,k þ
1
2
X
k
ir,kdðf Àfr,kÞejjr,k :
ð16Þ
If the expression (16) is substituted in (7), the bi-spectrum
becomes:
Bðf1,f 2Þ ¼ Iaðf1ÞIaðf 2ÞIa
n
ðf1 þf2Þ
L. Saidi et al. / ISA Transactions 52 (2013) 140–148142

¼
1
8
if dðf 1ÀfsÞejj þ
P
k1
il,k1
dðf 1Àf l,k1
Þejjl,k1
þ
P
k1
ir,k1
dðf1Àf r,k1
Þe
jjr,k1
0
B
B
B
@
1
C
C
C
A
Â
if dðf2Àf sÞejj þ
P
k2
il,k2
dðf2Àf l,k2
Þejjl,k2
þ
P
k2
ir,k2
dðf2Àfr,k2
Þejjr,k2
0
B
B
@
1
C
C
A
Â
if dðf3Àf sÞeÀjj þ
P
k3
il,k3
dðf 3Àf l,k3
ÞeÀjjl,k3
þ
P
k3
ir,k3
dðf3Àfr,k3
ÞeÀjjr,k3
0
B
B
B
@
1
C
C
C
A
ð17Þ
The time domain signals considered are the stator current and
its FT which are corrected for the window length to give the
frequency domain operation. By using FT values in Amperes (A) in
Eq. (17), bi-spectrum values are normalized (divided by the bi-
spectrum maximum value), then the bi-spectra amplitudes
(Fig. 2, and Fig. 4) are given without unit and bounded by 1.
3.3. Numerical simulation
To evaluate the bi-spectrum performance, a numerical simula-
tion has been performed. This signal is similar to the measured
current for three broken bars at rated load. For generating this
signal coming from Eq. (14) for k¼1, a sampling frequency of
128 Hz, a data length by segment of 1024 and M¼6 (total length
6144) have been used. In addition, a Hanning window has been
applied to the data to reduce spectral leakages due to the
selection of the parameters in signal generation and computation.
In this way, this simulation will also help to determine the
analysis parameters for experimental stator currents:
iaðtÞ ¼ if cosð2pf stÀjÞþil,1cosð2pfl,1tÀjl,1Þþir,1cosð2pf r,1tÀjr,1Þ
ð18Þ
where if¼0 dB¼1A, il,1¼ À80.1 dB¼114.8 mA and ir,1¼ À78.8
dB¼98.9 mA, fl,1¼(1–2s)fs¼47.8 Hz and fr,1¼(1þ2s)fs¼52.2 Hz,
fl,1þfr,1¼2fs, and random phases j, jl,1, jr,1 with a uniform dis-
tribution between 0 and 2p.
Fig. 1 shows the different simulation results of the time
domain signal given by (18), its PS and its bi-spectrum. Hence,
the analysis can take into consideration only a signal non-
redundant bi-spectral region [1,2].
A deep insight in (17) shows the existence in the bi-spectrum of
peaks which do not depend on the studied fault. Indeed, it can be
revealed by the non-zero products among the three factors of (17).
In fact, if the three delta functions (d) of each product have the same
support, the result is non-zero and the corresponding peaks appear
in the bi-spectrum. From the simulation results shown in Fig. 1, it
can be seen the presence of two peaks at (fs, fs) and (fs, 0). Other peak
is positioned at the intersection between the lines f1¼fs, f2¼fs and
f3¼f1þf2¼78 Hz (50 Hzþ28 Hz). In order to analyze all these
pulses, and for reasons of the bi-spectrum symmetry, the straight
lines (vertical bi-spectrum slice) f1¼fs and f2¼fx are only considered.
In (17) the following expressions appear:
À In the first factor Ia(f1 ¼fs): d(0)¼1 and d(72ksfs)¼0;
À In the second factor Ia(f2 ¼fx): d(fx Àfs) and d(fx Àfs 72ksfs);
À In the third factor In
a(f1 þf2 ¼fsþfx): d(fx) and d(fx 72ksfs).
The first factor is the unity and in order to have non-zero
results, the other two factors have to contain only pulses with the
same support. This is possible if only if 2ks¼1, assuming that the
pulse width of Kronecker is equal to zero, the consideration is that
this is an approximation and the presence of rounded numerical
computation errors makes this formula acceptable at the first
approximation. So we can write 2ksE1. This approximation reflects
the fact that the position of three peaks remains unchanged for
different values of slip s, that is to say that for each value of s it is a
new different current harmonic that gives these three impulses.
Thus, giving in the second factor the terms d(fxÀfs), d(fx) and
d(fxÀ2fs) and in the third factor d(fx), d(fxþfs) and d(fxÀfs). Hence, in
the bi-spectrum the peaks d(fxÀfs) and d(fx) can appear and they
correspond to f1¼fs and f2¼fs located at (fs, fs) and to f1¼fs and f2¼0
located at (fs, 0) with amplitudes i2
f =8
P
k
il,k and i2
f =8
P
k
ir,k, respec-
tively. In general il,k4ir,k, this explains that the peak on ðfs, f sÞ is the
highest. At the sampling frequency of 128 Hz, two symmetrical
pulses are detected at (fs, 28 Hz) and (28 Hz, fs), that is at the
intersection between the line f3¼f1þf2¼fsþ28 (here called decima-
tion line) and the lines f1¼fs and f2¼fs. By changing the sampling
frequency, the decimation line translates and the peak at (fs, 28 Hz)
disappears and can be relocated at others frequencies without
disturbing the region of interest (Table 1). It can be concluded that
these peaks are only an artifact of the numerical technique.
Fig. 1 show only the peaks which do not depend on the studied
fault. Nevertheless, other peaks giving information about BRB fault
can be observed in the one-dimensional bi-spectrum diagonal slice
Fig. 1. A simulated signal given by (18), and its, power spectrum (a) and bi-spectrum (b).

with f1¼f2¼f and given by:
Dðf Þ ¼
i3
f
8
dðf ÀfsÞejj þ
X
k
i3
l,k
8
dðfÀfl,kÞejjl,k þ
X
k
i3
r,k
8
dðfÀfr,kÞejjr,k
ð19Þ
3.4. Experimental set-up
The tested three-phase squirrel-cage IM (Fig. 2) is used in
measuring stator current signals of two three-phase 18.5 kW IM,
The characteristics of the two three phase induction motors (healthy
and faulty rotors) used in our experiment are listed in Table 2. One IM
is undamaged (0BRB) and it is defined as the reference condition
whereas the other one is with a synthetic rotor fault with one broken
rotor bar (1BRB) or three broken rotor bars (3BRB) at different load
levels. The synthetic rotor fault was obtained by drilling a small hole
of 3-mm diameter in all the rotor bar depth. The load has been
simulated by a magnetic brake which is coupled between the two
machines shafts used to create the desired shaft load of operation.
For experiments, one current sensor of 20 kHz frequency band-
width is used. The analog signals are passed through a low-pass
anti-aliasing filter with a cut-off frequency of 2 kHz. The current
signals are collected at a sampling frequency of 16,384 Hz for all the
experiments by means of a 12-bit A/D converter. The measurements
were carried out for different load conditions starting from 20% up
to 100% of the rated torque. In order to minimize the FFT leakage
effect, the Hanning window has been applied. The further signal
processing is performed by using the MATLAB environment in
order to generate power spectra and bi-spectra and the LabView
software for the data acquisition.
3.5. Data analysis
The PS and the bi-spectrum of one phase current have been
computed by using a 1024-point DFT length, a Hanning smoothing
window and no overlap. Moreover, computations have been
performed by using the formulas (3) and (8). As a first approach
a
b
c
Induction machine 18.5 kw
(Healthy machine)
Magnetic
brake
Control unit
Selector
Coupling
Labview in PC
Digital stator
current signal
Induction machine 18.5 kw
(Faulty machine)
Analogstatorcurrentsignal
ia
ib
ic
Variable voltage
source
iaia
Acquisition board
12 bit
Current
sensors
adapting and anti-
aliasing filters
3-phase 50 Hz
power source
Fig. 2. Configuration of the experimental set-up for BRB detection.
Table 1
Effect of decimation on the position of f3¼f1þf2.
Sampling frequency [Hz] f3¼f1 þf2 [Hz]
128 78; (50þ28)
256 206; (112þ94)
512 462; (250þ212)
1024 973; (498þ475)
Table 2
Induction motor characteristics used in the experiment.
Description Induction motor
Manufacturer LEROY-SOMER
Rated power 18.5 kW
Input voltage 380 V
Full load current 35.90 A
cos j 0.87
Supply frequency 50 Hz
Number of poles 4
Number of rotor bars 40
Number of stator slots 48
Rated slip 0.024
Rated speed 1456 rpm
Connection D-connected

(Fig. 3), the PS of the IM stator current has been represented in the
frequency bandwidth [0 Hz, 1000 Hz] for the healthy case, for 1BRB
and for 3BRB. As expected many frequencies have been detected
because of the large ratio of signal-to-noise (SNR) being around
150 dB. Then, the same cases have been investigated by using the bi-
spectrum in the frequency bandwidths [0 Hz, 60 Hz] (Fig. 4). The total
number of samples in each measurement was N¼131,072 samples
(N¼Tacq.fe) at a sampling frequency of 16,384 Hz (Dt¼61 ms). The
frequency spacing between successive samples of the DFT called the
frequency resolution has been Df¼0.125 Hz (Df¼fe/N [Hz]) corre-
sponding with an acquisition time Tacq¼8 s(Tacq¼1/Df). All the
frequency component magnitudes have been normalized and
expressed in dB scale with respect to the fundamental frequency of
the power grid set at 0 dB (1 pu).
There are some limitations on the amount of information which
can be extracted from spectral analysis. A spectral peak at a
particular frequency may come from several possible faults.
A rotating component may produce a number of spectral peaks at
harmonics of the rotational frequency and they can have a constant
phase which cannot be detected by the PS. These PS limitations have
led to use other diagnostics techniques such as the bi-spectrum to
extract the possible non-linear characteristics of the signal. This
concept will be discussed in the following section.
4. Experimental results
In this section, both PS and BDS analysis will be used to
distinguish the fault severity in an IM with BRB at any level of
shaft load. In fact, it is very important to detect BRB at low load
level since it is well known that the influence of any rotor fault is
reduced when the shaft load is low due to the fact that the rotor
currents are reduced. Therefore, at low load levels, the influence
of rotor currents on the stator side is reduced and the detection
becomes more difficult.
4.1. PS analysis
The data coming from the stator current measurement have been
sampled at the frequency 16,384 Hz which leads to a data set with
N¼131,072 samples for each variable. The spectra of motor current
signals at various loads (0%, 25%, 50%, 75%, and 100% of rated load)
and at different numbers of broken bars (0BRB, 1BRB, and 3BRB), are
shown in Fig. 5, indicate that the amplitude of the (172s)fs
components are noticeable and increase with fault severity. Indeed,
in the case of a healthy machine, apart the fundamental frequency of
the power grid, it is expected that all the other frequency compo-
nents will have small magnitudes compared to faulted cases.
However, one BRB produces frequency components at (172ks)fs
with large magnitudes (Fig. 5). These magnitudes are increasing
-150
-100
-50
0
-150
-100
-50
0
Amplitude[dB]
100 200 300 400 500 600 700 800 900 1000
-150
-100
-50
0
Frequency [Hz]
Fig. 3. Stator current PS of a healthy IM (a), with 1BRB (b) and with 3BRB (c), in the frequency bandwidth [0 Hz, 1000 Hz] at rated load (s¼2.4%).
Fig. 4. Stator current bi-spectrum of a healthy IM (a), with 1BRB (b) and with
3BRB (c) in the frequency bandwidth [0 Hz, 60 Hz] at rated load (s¼2.4%).

with the fault severity and the higher the load is the larger is the
fault visibility. It can be observed that at no load when the slip is too
small, the fault-related frequency components have overlaps with
the fundamental frequency coming from the grid. Otherwise, as the
motor becomes more heavily loaded, the rotational frequency slows
and the slip frequency increases. The greater the slip, the greater is
the frequency separation observed between (172ks)fs and the
fundamental frequency. The lighter the load, the larger the ratio
between line frequency amplitude and that of the (172ks)fs side-
band; especially, as load moves below 50% of full load. From 50% to
100% load this effect is less significant.
4.2. Bi-spectrum analysis
In order to provide a higher resolution in the sampled signal,
the data is divided into K segments (K¼4) according to the step of
the bi-spectrum estimation. The overlapping was zero for two
consecutive segments with two different fault conditions such as
for healthy and for one BRB mode.
The three-dimensional graph of the bi-spectrum is difficult to
be described and to be analyzed (Fig. 4). However, the BDS has
been computed for both healthy and faulty modes (Fig. 6). It has
been shown that the BDS method gives similar results compared
to PS computed for the same cases at different shaft load levels.
However, the detection is even smoother than in the case of PS
and it is much more visible. This is due to the fact that the
bi-spectrum can effectively filter out the Gaussian noise [9].
Comparing the BDS results plotted in Fig. 6, reveals that there
are clear differences in the (172ks)fs sideband component values
between the healthy and faulty at different load conditions.
Unlike the PS-based method, the proposed BDS-method deter-
mines the BRB fault at absolute no-load condition.
The different magnitudes of frequency components related to
BRB have been computed in the three previous cases (healthy,
1BRB, 3BRB) by using the formula (12) for the average value IdB
(Fig. 7). It can be seen that even when the machine is operating at
no load the healthy rotor can be clearly distinguished from the
faulty case (this difference is indicated by dashed circles in Fig. 7).
The BDS technique discussed is characterized by a higher
sensitivity than the PS, especially when the machine is operating
at no-load.
In the next section, we present the detection performance
evaluation results by comparing the PS-based method and our
proposed BDS-based method using ROC curves, which make the
results more convincing.
4.3. Detection performance evaluation: Need for ROC analysis
When comparing two or more diagnostic tests, ROC curves are
often the only valid method of comparison [26]. An ROC curves is
a detection performance evaluation methodology, and demon-
strates how effectively a certain detector can separate two groups
in a quantitative manner [26–28]. An ROC curve shows the trade-
off between the probability of detection or true positives rate
(tpr), also called sensitivity and recall versus the probability of
false alarm or false positives rate (fpr). ROC curves are well
described by Fawcett in [26]. The tpr and fpr are mathematically
expressed in (20) and (21), respectively.
tpr ¼
true positives
true positivesþfalse negatives
ð20Þ
fpr ¼
false positives
false positivesþtrue negatives
ð21Þ
40 45 50 55 60
-150
-100
-50
0
Amplitude[dB]
40 45 50 55 60
-150
-100
-50
0
Amplitude[dB]
40 45 50 55 60
-150
-100
-50
0
Amplitude[dB]
40 45 50 55 60
-150
-100
-50
0
40 45 50 55 60
-150
-100
-50
0
40 45 50 55 60
-150
-100
-50
0
40 45 50 55 60
-150
-100
-50
0
40 45 50 55 60
-150
-100
-50
0
40 45 50 55 60
-150
-100
-50
0
0BRB 0%
1BRB 0% 3BRB 0%
3BRB 50%1BRB 50%0BRB 50%
0BRB 25%
1BRB 25%
3BRB 25%
40 45 50 55 60
-150
-100
-50
0
Amplitude[dB]
40 45 50 55 60
-150
-100
-50
0
Frequency [Hz]
Amplitude[dB]
40 45 50 55 60
-150
-100
-50
0
40 45 50 55 60
-150
-100
-50
0
Frequency f [Hz]
40 45 50 55 60
-150
-100
-50
0
40 45 50 55 60
-150
-100
-50
0
Frequency [Hz]
0BRB 100% 3BRB 100%1BRB 100%
0BRB 75% 1BRB 75%
3BRB 75%
48.37 Hz
-128 dB
51.63 Hz
-126.3 dB
49 Hz
-124 dB
49.4 Hz
-88 dB
48.87 Hz
-119.3 dB
48.87 Hz
-104.4 dB
48.87 Hz
-81.28 dB
49.25 Hz
-124.1 dB
48.88 Hz
-140.7 dB
49.5 Hz
-88.3 dB
48.25 Hz
-121.6 dB
48.13 Hz
-117.1 dB
48.2 Hz
-78.87 dB
47.62 Hz
-126 dB
47.5 Hz
-102.8 dB
47.8 Hz
-80.1 dB
51 Hz
-122 dB
50.6 Hz
-86.4 dB
51.13 Hz
-117 dB
51.12 Hz
-102.6 dB
51.13 Hz
-79.7 dB
50.75 Hz
-121.2 dB
50.88 Hz
-135.7 dB
50.5 Hz
-91.61 dB
51.75 Hz
-123.6 dB
52 Hz
-130.5 dB
51.8 Hz
-76.16 dB
52.38 Hz
-124.2 dB
52.5 Hz
-102.5 dB
52.2 Hz
-78.8 dB
Fig. 5. Zoomed stator current PS for three modes: healthy, 1BRB and 3 BRB under different shaft load levels (0%, 25%, 50%, 75% and 100% of rated load).

Additional term associated with ROC curves is,
specificity ¼
true negatives
false positivesþtrue negatives
¼ 1Àfpr ð22Þ
sensitivity ¼ recall ð23Þ
For each case (Healthy, 1BRB and 3BRB) and for each load level
condition (0%, 25%, 50%, 75% and 100% of rated load), a series of 10
independent Monte-Carlo experiments are conducted. For each
experiment, the probability of false alarm and the probability of
detection are obtained by counting detection results at each side-
band frequency components (172s)fs amplitude out of 150 inde-
pendent Monte-Carlo experiments by both the PS-based standard
method and our proposed BDS-based method.
Each point on the ROC curve corresponds to a speciﬁc pair of
sensitivity and (1-speciﬁcity) and the complete curve gives an over-
view of the overall performance of a test. When comparing ROC
curves of different tests, good curves lie closer to the top left corner
and the worst case is a diagonal line (shown as a dashed line in Fig. 8).
Fig. 8 shows the ROC curves of the PS-based method and our BDS
proposed method. The two curves are obviously not identical; detec-
tion algorithm based on BDS-method is better than PS-method, in the
40 45 50 55 60
-150
-100
-50
0
|D(f,f)|[dB] 0BRB 0%
40 45 50 55 60
-150
-100
-50
0
1BRB 0%
40 45 50 55 60
-150
-100
-50
0
3BRB 0%
40 45 50 55 60
-150
-100
-50
0
|D(f,f)|[dB]
0BRB 25%
40 45 50 55 60
-150
-100
-50
0
1BRB 25%
40 45 50 55 60
-150
-100
-50
0
3BRB 25%
40 45 50 55 60
-150
-100
-50
0
|D(f,f)|[dB]
0BRB 50%
40 45 50 55 60
-150
-100
-50
0
1BRB 50%
40 45 50 55 60
-150
-100
-50
0
3BRB 50%
40 45 50 55 60
-150
-100
-50
0
|D(f,f)|[dB]
0BRB 75%
40 45 50 55 60
-150
-100
-50
0
1BRB 75%
40 45 50 55 60
-150
-100
-50
0
3BRB 75%
40 45 50 55 60
-150
-100
-50
0
Frequency f [Hz]
|D(f,f)|[dB]
0BRB 100%
40 45 50 55 60
-150
-100
-50
0
Frequency f [Hz]
1BRB 100%
40 45 50 55 60
-150
-100
-50
0
Frequency f [Hz]
3BRB 100%
49.25 Hz
-144 dB
50.75 Hz
-143.3 dB
49.12 Hz
-139.3
49.37 Hz
-125.38
48.87 Hz
-121 dB
48.625 Hz
-128.87dB
49Hz
-93.71dB
48.88 Hz
-105.7 dB
48.88 Hz
-110.6 dB
49.5 Hz
-49.84
48.25 Hz
-134.1 dB
48 Hz
-121.6 dB
48.25 Hz
-82.78 dB
47.75 Hz
-132 dB
47.75 Hz
-109.20 dB
47.75 Hz
-92dB
50.87 Hz
-128.09 dB
50.62 Hz
-124.33 dB
51.13 Hz
-122.65 dB
51.125 Hz
-122dB
51 Hz
-98.61 dB
51.12 Hz
-123.8dB
50.88 Hz
-120.1 dB
50.63 Hz
-94.58 dB
51.75 Hz
-138.8 dB
52 Hz
-123.5 dB
51.75 Hz
-79.86 dB
52.25 Hz
-128dB
52.27 Hz
-112.05 dB
52.25 Hz
-87.25 dB
Fig. 6. Zoomed stator current BDS for three modes: healthy, 1BRB and 3 BRB under different shaft load levels (0%, 25%, 50%, 75% and 100% of rated load).
0 25 50 75 100
-150
-140
-130
-120
-110
-100
-90
-80
-70
-60
-50
Load level [%]
FaultindexI[dB]
0BRB
1BRB
3BRB
0BRB
1BRB
3BRB
Fig. 7. Fault index using the (172s)fs components for different shaft load levels
for both PS (solid line) and BDS (dashed line).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
False positive rate
Truepositiverate
Worst case
PS
BDS
Fig. 8. ROC curves comparison: The PS-based method and the proposed BDS-
based method.

high sensitivity rate. The ROC curves of the PS-based method show
degraded performance (lower tpr for the same fpr), on the other hand,
the ROC curves of the BDS-based method show slight degradation.
Another quantitative measure of the ROC curve performance is
the precision-recall (PR) graph [28]. The PR measures the fraction
of positive examples that are correctly labeled. In PR space, one
plots recall on the x-axis and precision on the y-axis. Recall is the
same as tpr, whereas precision measures that fraction of examples
classified as positive that are truly positive.
Fig. 9 shows the PS curves, which indicate that the BDS detector
is a relatively good detector in terms of PR graph, and illustrates
how the precision of both methods change as the relative recall (tpr)
increases. The precision of the PS-based method drops sharply
beyond the tpr¼0.14, but the precision of the BDS-based method
is almost not affected by the increasing tpr.
5. Conclusion
In this paper, a short overview of the bi-spectral analysis has
been presented and a method using the HOS to detect BRB faults
has been presented. Data from real tests have been used to
express the ability of the bi-spectrum and more exactly the DSB
analysis and its associate phase for condition monitoring of three-
phase IM with rotor faults at no-load. The form of bi-spectral slice
is similar to the graph of PS and the main difference is that this
slice has inside information about the phase and the non-linear
system characteristic. Therefore, it can distinguish in between
many operating conditions even with low influence in the PS.
The robustness of the proposed method is explored by running a
series of independent Monte-Carlo experiments. The evaluation of
results using ROC curves makes the results more convincing.
By using the bi-spectral analysis for IM condition monitoring, the
sensitivity of fault detection can be easily improved without changing
the data acquisition. Alternatively, the Wigner bi-spectrum and the
wavelet bi-spectrum can be used for condition monitoring in non-
stationary cases met during transients and in many real applications.
Acknowledgments
The authors would like to thank an anonymous referee for the
insightful comments that have significantly improved the article.
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0 0.2 0.4 0.6 0.8 1
0.75
0.8
0.85
0.9
0.95
1
Recall
Precision
PS
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Fig. 9. Precision-recall graph comparison: The PS-based method and the proposed
BDS-based method.

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