This paper proposes a new diagnosis algorithm to detect broken rotor bars (BRBs) faults in induction motors. The proposed algorithm is composed of a frequency signal dimension order (FSDO) estimator and a fault decision module. The FSDO estimator finds a number of faultrelated frequencies in the stator current signature. In the fault decision module, the fault diagnostic index from the FSDO estimator is used depending on the load conditions of the induction motors. Experimental results obtained in a 75 kW threephase squirrelcage induction motor show that the proposed diagnosis algorithm is capable of detecting BRB faults with an accuracy that is superior to a zoom multiple signal classification (ZMUSIC) and a zoom estimation of signal parameters via rotational invariance techniques (ZESPRIT).
1. Introduction
Induction motors are widely used in electrical machines in the electrical and processing industries. There are various induction motor failures from stator, rotor, bearing, or other faults
[1
,
2]
. Induction motor failure results in severe damage to the motor itself, as well as to the motorrelated processes in the industrial plant.
Many studies based on the frequency domain analysis using the fast Fourier transform (FFT) have been performed in order to detect broken rotor bars (BRBs), stator windings, and mechanical faults
[3

6]
. Using a motor current signature analysis (MCSA), which is based on the FFT method, the spectrum of stator currents can be used to diagnose BRBs faults and stator winding faults with various faulty motor conditions
[3]
. The MCSA based on the FFT analysis of the sideband components provides the information to diagnose accurately and quantitatively the broken rotor bars in induction machines
[4]
. A FFTbased MCSA with advanced signalanddataprocessing algorithms is proposed for an online induction motor diagnosis system
[5]
. A detection method is proposed for stator winding faults in the lowfrequencyrange of the flux spectrum with low resolution that exploits an external stray flux sensor
[6]
. However, the frequency resolution of FFTbased methods is limited to the number of data points.
Based on the FFT, zoomFFT (ZFFT) techniques with window functions have been proposed to increase the frequency resolution
[7

9]
. ZFFT techniques reduce the computational time, save the memory size, and increase the accuracy in a specified frequency range
[7

9]
. However, the ZFFT techniques for the highfrequency resolution requires a long acquisition time
[8
,
9]
.
In the literature, timefrequency domain analyses based on wavelettransform techniques have been investigated to overcome the resolution problem encountered in fault diagnostic analysis
[10

12]
. Wavelettransform techniques, which are effective for stationary as well as nonstationary signal processing, provide a multiresolution signal analysis with high resolution in the time and frequency domains
[10

12]
. An automatic online diagnosis algorithm for brokenrotorbar detection has been proposed based on the wavelettransform application to the startup current transient
[10]
. By using the wavelet decomposition, fault diagnosis has been performed in both time and frequency domains
[11]
. Based on a continuous wavelet technique with frequency Bspline functions, a diagnosis method identifies as many BRB faultrelated frequency components as possible during large transients
[12]
. Wavelettransform techniques can be applied to detect faults during stationary and nonstationary conditions
[10

12]
. However, in stationary conditions, wavelettransform techniques have the disadvantage in the frequency resolution compared to the subspace methods.
Subspace methods such as a multiple signal classification (MUSIC) algorithm and an estimation of signal parameters via rotational invariance techniques (ESPRIT) have been applied to diagnose induction motor faults while overcoming the frequency resolution limitation of the FFT method in low signaltonoise ratios (SNRs)
[13

17]
. The MUSIC and ESPRIT compute an autocorrelation matrix of the signals, estimate a frequency signal dimension order (FSDO), use signal and noise subspaces, and convert the fault detection problem into a generalized eigenvalue problem
[13

17]
. The MUSIC was applied to estimate the harmonic components in order to improve induction motor fault detection
[13]
. The MUSIC and shorttime MUSIC were used for the analysis of the spacevector of voltages induced in stator windings, after supply disconnection, to detect BRBs faults in induction machines
[14]
. The MUSIC algorithm with zooming methods was proposed to detect fault sensitive frequencies based on a frequency analysis of the stator current
[15]
. Using the motor leakage flux spectral analysis, the MUSIC is applied to identify BRB faults in induction motors
[16]
. The ESPRIT with the zooming technique has been proposed to estimate both the frequencies and amplitudes of BRB fault harmonics using the stator current analysis
[17]
. However, the difference in the spectrum of healthy motors and those with BRB faults determined using the zoom MUSIC (ZMUSIC)
[15]
or zoom ESPRIT (ZESPRIT)
[17]
is not obvious when the load level is close to noload condition. To overcome this limitation, this paper proposes another fault diagnostic index using the result of the FSDO estimator, which is needed to perform the ZMUSIC
[15]
or ZESPRIT
[17]
.
In experiments, the small slip induces small current flowing in the rotor even though under noload condition
[22
,
23]
. To detect BRBs faults under noload condition, the wavelet transform is proposed using transient current analysis to overcome the resolution problem of FFTbased methods
[22]
. A detection method based on particle swarm optimization can estimate BRBs faultrelated frequencies using stator current under noload condition
[23]
. However, performance analysis using detection and false alarm probabilities is required in order to provide a stable diagnosis for BRBs faults
[22
,
23]
.
In this paper, a robust fault diagnosis algorithm is proposed to detect BRB faults in induction motors. In this approach, a FSDO estimator, and a fault decision module are proposed. For the FSDO estimator, we use the minimum description length (MDL) criterion
[18]
to improve the performance of BRB fault detection for motors operating with no load or a finite load. On the basis of the FSDO estimator, we propose a fault decision module that uses the FSDO estimate as a fault diagnosis index. In this module, the predetermined threshold values depending on the load conditions are optimized using the detection and false alarm probabilities from the experimental results. Experimental results show that the proposed diagnosis algorithm achieves the optimal performance for BRBs faults in various load conditions.
2. Detection of Broken Rotor Bar
In this section, frequencies in the current spectrum are analyzed in order to diagnose BRB faults. Then, a data model for BRB faults is presented using the stator current.
 2.1 Frequency analysis of broken rotor bar
When an induction motor rotor bar is broken, the stator steadystate current has harmonic frequencies given by
[15
,
17]
where
s
is the perunit motor slip,
f_{s}
is the electrical supply frequency, and
k
=1,2,3,… is the harmonic frequency index. In (1), the frequency of interest for BRB fault detection,
f _{BRB}
, exists near the electrical supply frequency
f_{s}
, Therefore, BRB faults are verified with the presence of frequency components of interest,
f_{BRB}
, near the electrical supply frequency,
f_{s}
, in the current spectrum.
The characteristic frequency for BRB fault detection,
f_{BRB}
, is a function of the slip
s
, where the slip
s
depends on the load condition
[15
,
17]
. When the relative mechanical speed of the motor is almost the same as the motor synchronous speed, which occurs when the motor operates with no load, the slip
s
is very small and it is very hard to identify frequencies arising from BRB faults in (1) using the FFT method or the MUSIC algorithm
[15
,
17]
.
 2.2 Data model
Discrete timedomain measurement data for the stator current can be represented as follows:
where
s
[
n
] consists of sinusoids,
e
[
n
] represents white noise with a zero mean and a variance of
σ
^{2}
,
K
is the number of pure sinusoids, and
a_{i}
,
f_{i}
, and
ϕ_{i}
are the amplitude, the frequency, and the phase of the
i
th pure sinusoid, respectively. Using cos(2
πf_{i}n
+
ϕ_{i}
) =
from Euler’s formula
[19]
, (2) can also be expressed in complex exponential form as
where
,
for 1 ≤
i
≤
K
,
for 1 ≤
i
≤
K
,
for 1 ≤
i
≤
K
,
for 1 ≤
i
≤
K
, and
M
= 2
K
. Using
L
>
M
serial discretetime samples {
x
[
n
],
x
[
n
+1],⋯,
x
[
n
+
L
−1]} we can rewrite (3) in a compact matrix format as follows:
where
x
(
n
) = [
x
[
n
]
x
[
n
+ 1] ⋯
x
[
n
+
L
−1]]
^{T}
,
with
,
i
= 1, 2, ⋯,
M
,
and
e
(
n
) = [
e
[
n
]
e
[
n
+1] ⋯
e
[
n
+
L
–1]]
^{T}
. Here ,
F
is a
L
×
M
Vandermonde matrix of rank
M
and
f
(
) is the mode vector with frequency
, where
i
= 1, 2,⋯,
M
.
3. BRB Fault Diagnosis Algorithm
In this section, a robust diagnosis scheme is proposed to detect BRB faults. The proposed diagnosis algorithm consists of a FSDO estimator, and a fault decision module. The FSDO estimator provides the number of frequency components in the measured data using the MDL criterion. A fault decision module uses the number of frequency components as the fault index to detect BRB faults.
 3.1 FSDO estimator
The autocorrelation matrix
R
_{x}
of the measurement data
x
(
n
) can be expressed as
where
E
[⋅] denotes the expected value,
denotes the signal correlation matrix, and
I
_{L}
denotes an
L
×
L
identity matrix.
To implement the FSDO estimator, we need an estimate,
, of
R
_{x}
in (5). By using
N
>
L
serial measured time samples {
x
[
n
],
x
[
n
+1], ⋯,
x
[
n
+
N
−1]} , a spatial smoothing (SS) method is applied to calculate
as follows [20]:
where
P_{s}
=
N
−
L
+ 1 . In (6), the eigenvalues of
is defined as {
λ
_{1}
≥
λ
_{2}
≥⋯≥
λ
_{L}
} and can be obtained using the eigenvalue decomposition. When
has the full rank of
M
, we obtain
Information theoretic criteria, such as the MDL or Akaike information criterion (AIC), have been used to obtain the number of frequencies
[18]
. The MDL criterion for the estimation of
M
is used in this paper, and is given as follows
[17
,
18]
:
The estimate of
M
can then be expressed as
where
k
= 0,1,⋯,
L
−1 .
This estimate of
M
can be used to make a decision for the BRB fault detection. For example, when the noload condition is satisfied, the slip
s
is 0.00022 and therefore the frequency estimation for harmonic frequencies in (1) cannot be performed precisely. However, when a faulty induction motor with a BRB fault operates in low load conditions, harmonic frequencies are introduced into the stator current due to the airgap flux. Therefore, even though the difference between
f_{BRB}
and
f_{s}
is smaller than the frequency resolution of the frequency estimation, harmonic frequencies are present in the stator current in (1) for BRB faulty motors. In the proposed method,
is defined as a fault diagnostic index to detect BRB faults. The procedure for BRB fault detection is described in the following subsection.
 3.2 Fault decision module
A functional block diagram of the proposed diagnosis scheme for BRB faults in induction motors is shown in
Fig. 1
. First, the measured data vector
x
(
n
) in (4) is used to calculate the autocorrelation matrix
in (6). The eigenvalues of the correlation matrix are then computed. Through the analysis of the eigenvalues by the MDL criterion, the FSDO estimator calculates and supplies the estimate of the number of frequencies
in (9) to the fault decision module.
Block diagram of the BRB fault diagnosis algorithm
In the fault decision module, we use the property that
for BRB faulty motors is larger than that in healthy motors, because the BRB fault introduces harmonic frequencies in (1) in different load conditions. The fault decision module uses the motor speed
R_{s}
and
from the FSDO estimator. During the initial stage, the motor speed
R_{s}
can be estimated using the rotorslot harmonics
[5]
. Let
M_{th}
be the threshold parameter predetermined from experimental results depending on the motor speed. At different load conditions,
in (9) is used for the BRB fault decision and is compared to
M_{th}
. This is because
increases even when the frequency estimation cannot be performed exactly due to the low load condition
[15
,
17]
. The proposed diagnosis algorithm detects BRB faulty motors with the optimal performance even at no load condition using
M_{th}
.
4. Experimental Results
In this section, we present our experimental setup and analyze the proposed BRB fault diagnosis algorithm with different motors and load conditions. For the fault decision module of the proposed method, the cumulative distribution functions (CDFs) for detection and false alarm probabilities obtained from experimental results. The performance of the proposed BRB fault diagnosis algorithm is compared with those of the ZMUSIC and ZESPRIT algorithms for different motors and under different load conditions.
 4.1 Experimental setup
The experiment tests were performed with 75kW squirrelcage induction motors.
Fig. 2
shows the induction motor test system, which is composed of a test motor, a load motor, an inverter, and a data acquisition system (DAS). The specifications of the test and load motors are described in
Table 1
. The inverter is connected to the load motor to control the load condition of the test motor. Three load conditions were considered: noload, 50%, and 80% of the full load. Under the noload, 50% load, and 80% load conditions, the slips are almost 0.00022, 0.0054, and 0.0091, respectively, and the speeds of the motor are 1799.6 rpm, 1791.3 rpm, and 1783.2 rpm, respectively. For motor conditions, three types of motors were used: a healthy motor, a motor with one BRB, and a motor with two BRBs. As shown in
Fig. 3
, BRBs faults were simulated by cutting the end of the rotor bar.
Experimental setup
Characteristics of the induction motor used in experiments
Characteristics of the induction motor used in experiments
BRB faults: (a) one BRB and (b) two BRBs
The experimental data was gathered with a sampling rate
f_{d}
of 2 kHz. This can be achieved by downsampling 200 kHz sampled data from the DAS by an integer factor 100. This is because the BRB fault can be detected by the analysis of the frequency band near the electrical supply frequency
f_{s }
[15
,
17]
. For the FSDO estimator, the size of the correlation matrix,
L
= 2048 , and the size of the measured time samples for the SS method,
N
= 4096 , are used. The measurement data acquisition time is 2.048 seconds and the measurement data size is
N
= 4096 . In order to verify the proposed diagnosis algorithm, the results are analyzed for all possible experimental cases.
 4.2 Proposed BRB diagnosis algorithm
In the fault decision module, the fault diagnostic index
is exploited in order to detect BRB faults.
Fig. 4
shows CDFs of the FSDO estimate
for healthy and faulty motors at the different load conditions. For the noload condition, as shown in
Fig. 4(a)
, there is a significant difference between the CDF of a healthy motor and the CDFs of faulty ones. Using the threshold value with
M_{th}
= 163 obtained from the CDFbased experimental results shown in
Fig. 4(a)
, we can easily detect BRB faults in motors operating under the noload with the false alarm probability for
, i.e.,
= 0 , and the detection probability for
i.e.,
= 1 , which are defined as follows
[21]
:
CDFs of for the (a) noload, (b) 50% load, and (c) 80% load conditions. Threshold values are M_{th} = 163 , M_{th} = 128 , and M_{th} = 124 for noload, 50% load, and 80% load conditions, respectively.
and
where
H
_{0}
is the healthy motor hypothesis, and
H
_{1}
is the BRB faulty motor hypothesis. As shown in
Figs. 4(b)
and
4(c)
, the threshold values are determined to be
M_{th}
=128 for 50% load condition and to be
M_{th}
=124 for 80% load condition, respectively, using experimental results based on the CDFs of healthy and faulty motors. Using the threshold values, BRB faults under 50% and 80% load conditions can be easily detected using
with
= 0 and
=1. Therefore, for the fault decision module, the fault index
can be used for the noload and finite load conditions.
Fig. 5
shows the threshold values as a function of load rates,
D
, using experimental results. As predicted by the experimental results as shown in
Fig. 4
, the threshold value
M_{th}
decreases as the load rate
D
increases. This is because when the load rate D increases, the motor slip s increases and the BRBs faultrelated frequencies
f_{BRB}
in (1) are getting away from the electrical supply frequency, therefore the number of BRBs faultrelated frequencies from the autocorrelation matrix of measured data in (5) decreases. Moreover, lines with two different slopes can be calculated with a small set of samples. The absolute value of slope between noload and 50 % load conditions is larger than that between 50 % and 80 % load conditions.
Threshold values as a function of load rate D.
For comparison purposes, the ZMUSIC
[15]
and ZESPRIT
[17]
have been analyzed at noload condition. For the ZMUSIC, the frequency shift was 60 Hz, the acquisition time was 3 s, and the 100 Hz sampling frequency data, which can be obtained by downsampling 200 kHz sampled data from the DAS by an integer factor 2000. For the ZESPRIT, the acquisition time was 3 s, and the 200 Hz sampling frequency data, which can be obtained by downsampling 200 kHz sampled data from the DAS by an integer factor 1000. The fault index
is defined as the amplitude of lower side band harmonic (LSH) in bandwidth
BW1
of the pseudospectrum for the ZMUSIC, where the fault index is computed with respect to the noise level
[15]
. The fault index
for the ZEPSRIT is defined as the sum of amplitudes of LSH in bandwidth
BW
1 and upper sideband harmonic (USH) in bandwidth
BW2
[17]
.
Fig. 6
shows the CDFs of fault indices
and
for ZMUSIC
[15]
and ZESPRIT
[17]
, respectively, for noload condition where the slip
s
is 0.00022 and the motor speed is 1799.6 rpm. For fault indices, the bandwidth
BW
1 is between 57 and 60 Hz, and the bandwidth
BW
2 is between 60 and 63 Hz. For the ZMUSIC as shown in
Fig. 6(a)
, the CDF of
for the healthy motor is between those for the faulty motor with one BRB and two BRBs.
CDFs at noload condition. (a) Fault index in BW1 = [57, 60 Hz] for the ZMUSIC [15], (b) Fault index in BW1 = [57, 60 Hz] and BW2 = [60, 63 Hz] for the ZEPSRIT [17].
For the ZESPRIT as shown in
Fig. 6(b)
, the CDF of
for the healthy motor is close to those for faulty motors with one BRB and two BRBs. Compared to
Fig. 4(a)
, the ZMUSIC and ZESPRIT cannot detect the BRB fault frequencies at the noload condition due to the faultrelated frequency
f_{BRB}
is almost the same with the electrical supply frequency
f_{s}
.
Furthermore, under the 50% and 80% load conditions as shown in
Figs. 4(b)
and
4(c)
, the proposed BRB fault diagnosis algorithm has better performance for the detection probability
and false alarm probability
compared to those of ZMUSIC and ZESPRIT
[17]
. Therefore, compared to the ZMUSIC and ZESPRIT
[17]
, the proposed BRB fault diagnosis algorithm accurately detects BRB faulty motors under no load and finite load conditions.
5. Conclusion
In this paper, an accurate and efficient diagnosis algorithm is proposed for BRB fault detection in threephase squirrelcage induction motors under noload and finite load conditions. Fault diagnostic index from the FSDO estimator is proposed considering the stator current signature based on all fault sensitive frequencies. Experimental results show that the proposed diagnosis algorithm provides the optimal performance under noload and finite load conditions of induction motor.
The main advantages of the proposed diagnosis algorithm are that it can correctly identify BRB faults using a few samples and can be easily implemented using software. The main drawback, which is also found in other highresolution spectral analysis methods such as the MUSIC, ZMUSIC and ZESPRIT, is that the FSDO estimator is limited to the steadystate operating condition of the induction motor.
Acknowledgements
This work was supported in part by the “Development of Motor Diagnosis Technology for the Electric Vehicle” project of Korea Electrotechnology Research Institute (KERI) and in part by the Human Resources Development program (No. 20114010203030) of the Korea Institute of Energy Technology Evaluation and Planning(KETEP) grant funded by the Korea government Ministry of Trade, Industry and Energy.
BIO
DonHa Hwang received B.S., M.S., and Ph.D. degrees in Electrical Engineering from Yeungnam University in 1991, 1993, and 2003, respectively. He is currently a principal researcher at Korea Electrotechnology Research Institute (KERI), Changwon, Korea. His main research interest are design, analysis, monitoring, and diagnosis of electric machines.
YoungWoo Youn received B.S. and M.S. degrees in Communication Engineering from Information and Communication University, Daejeon, Korea, in 2005 and 2007, respectively. He is currently a researcher at power apparatus research center at Korea Electrotechnology Research Institute (KERI), Changwon, Korea. His research interests are in condition monitoring and signal processing.
JongHo Sun received B.S., M.S., and Ph.D. degrees in Electrical Engineering from Pusan National University in 1986, 1988, and 2001, respectively. Currently, he is a principal researcher at Korea Electrotechnology Research Institute (KERI), Changwon, Korea. His interests are diagnosis techniques for electric power equipments.
YongHwa Kim received a B.S. in electrical engineering in 2001 from Seoul National University, Seoul, Korea, and a Ph. D. in electrical and computer engineering from Seoul National University, Seoul, Korea, 2007. From 2007 to 2011, he was a senior researcher with the Korea Electrotechnology Research Institute (KERI), Geonggido, Korea. From 2011 to 2013, he was an assistant professor at the Division of Maritime Electronic and Communication Engineering, Mokpo National Maritime University, Korea. Since March 2013, he has joined the faculty with the Department of Electronic Engineering at Myongji University, Korea. His general research interests include communication systems, motor drives and diagnosis, and digital signal processing. Currently, he is particularly interested in communications and digital signal processing for Smart Grid.
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