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Diagnosis of Three-Phase Electrical Machines Using
Multidimensional Demodulation Techniques
Vincent V. Choqueuse, Mohamed Benbouzid, Yassine Amirat, Sylvie Turri
To cite this version:
Vincent V. Choqueuse, Mohamed Benbouzid, Yassine Amirat, Sylvie Turri. Diagnosis of Three-
Phase Electrical Machines Using Multidimensional Demodulation Techniques. IEEE Transactions on
Industrial Electronics, Institute of Electrical and Electronics Engineers, 2012, 59 (4), pp.2014-2023.
�10.1109/TIE.2011.2160138�. �hal-00654230�
2014 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 4, APRIL 2012
Diagnosis of Three-Phase Electrical Machines Using
Multidimensional Demodulation Techniques
Vincent Choqueuse, Member, IEEE, Mohamed El Hachemi Benbouzid, Senior Member, IEEE,
Yassine Amirat, and Sylvie Turri
Abstract—This paper deals with the diagnosis of three-phase
electrical machines and focuses on failures that lead to sta-
tor-current modulation. To detect a failure, we propose a new
method based on stator-current demodulation. By exploiting the
configuration of three-phase machines, we demonstrate that the
demodulation can be efficiently performed with low-complexity
multidimensional transforms such as the Concordia transform
(CT) or the principal component analysis (PCA). From a practical
point of view, we also prove that PCA-based demodulation is
more attractive than CT. After demodulation, we propose two
statistical criteria aiming at measuring the failure severity from
the demodulated signals. Simulations and experimental results
highlight the good performance of the proposed approach for
condition monitoring.
Index Terms—Condition monitoring, electrical machines, prin-
cipal component analysis (PCA), signal processing.
I. INTRODUCTION
T
HREE-PHASE electrical machines such as induction mo-
tors or generators are used in a wide variety of applica-
tions. To increase the productivity and to reduce maintenance
costs of these systems, condition monitoring and diagnosis
are often desired. A wide variety of condition monitoring
techniques have been introduced over the last decade. Among
them, motor current signature analysis (MCSA) [1] has several
advantages since it is usually cheaper and easier to i mplement
than other techniques. In steady-state configurations, MCSA
based on stationary spectral analysis techniques is commonly
used (fast Fourier transform (FFT) and multiple-signal clas-
sification [2]). However, in practice, the steady-state assump-
tion is often violated due to nonconstant-supply-frequency or
adjustable-speed drives. In these situations, several authors
have investigated the use of nonstationary techniques such as
time–frequency representations [1], [3]–[6], time-scale analysis
[7]–[10], and polynomial-phase transform [11] (see [12] for a
Manuscript received December 7, 2010; revised April 5, 2011; accepted
May 29, 2011. Date of publication June 20, 2011; date of current version
November 1, 2011.
V. Choqueuse, M. E. H. Benbouzid, and S. Turri are with the Brest Lab-
oratory of Mechanics and Systems (LBMS), EA 4325, University of Brest,
29238 Brest, France (e-mail: vincent.choqueuse@univ-brest.fr; mohamed.
benbouzid@univ-brest.fr; sylvie.turri@univ-brest.fr).
Y. Amirat is with the Brest Laboratory of Mechanics and Systems (LBMS),
EA 4325, University of Brest, 29238 Brest, France, and also with the Institut
Superieur de l’Electronique et du Numérique, 29228 Brest, France (e-mail:
yassine.amirat@univ-brest.fr).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2011.2160138
more complete review). The main drawback of these methods
relies on their computational complexity. Furthermore, these
representations usually suffer from poor resolution and/or ar-
tifact (cross terms, aliasing, etc.), which can lead to misleading
interpretations.
Other investigations on machine modeling have recently
demonstrated that many types of failure lead to stator-current
modulation with a modulation index which is directly pro-
portional to the failure severity [5], [13]–[17]. In particular,
it has been proved that load torque oscillations lead to stator-
current phase modulation (PM) [ 9], [13]–[15], whereas air-gap
eccentricity and rotor asymmetry lead to stator-current ampli-
tude modulation (AM) [5], [14]. Therefore, a straightforward
technique to monitor the behavior of an electrical machine is
based on stator-current demodulation.
Classical demodulation techniques include the square-law
demodulator, the Hilbert transform (HT) [18], the energy sep-
aration algorithm [19], and other approaches. Applications to
failure detection are available in [7], [14], and [20]–[27]. Inter-
estingly, for a balanced system, i t has been shown in [14], [21],
[22], [24], and [28] that the Concordia transform (CT), which
has been used for failure detection purposes in [14], [21]–
[24], and [28]–[38], can also be interpreted as a demodulating
tool. As compared to classical demodulation tools, CT exhibits
interesting properties such as lower complexity and lack of end
effect problems or other artifacts [28]. However, in practice, this
approach can lead to poor performance since a real machine
usually presents a small degree of imbalance [22].
Once the demodulation has been performed, demodulated
signals must be further analyzed to measure failure severity.
In the literature, many criteria and/or techniques have been
proposed to perform this task. In [14], [21], [22], [24], [25],
and [29], failure severity is measured through statistical criteria.
However, these criteria require knowledge of the fault f requen-
cies, which also depend on other parameters (speed or slip
information). To overcome this problem, more sophisticated
approaches have been proposed for failure detection. These
include neural networks [31], [36], [39], Bayesian classifiers
[23], fuzzy logic classifiers [33], [36], genetic algorithms [38],
and other classifiers [34]. However, these approaches are com-
putationally demanding, and their performances highly depend
on the representativeness of the training set.
In this paper, we address the condition monitoring problem
from a signal processing point of view. As failure severity is
proportional to the modulation index, we propose to use the
modulation-index estimate as a failure severity indicator. The
proposed approach is composed of two steps: a stator-current
0278-0046/$26.00 © 2011 IEEE
CHOQUEUSE et al.: DIAGNOSIS OF THREE-PHASE ELECTRICAL MACHINES USING DEMODULATION TECHNIQUES 2015
Fig. 1. Block diagram of the proposed technique. Symbols i
k
(t)(k =
1, 2, 3) correspond to the stator currents, |a(t)| and f (t) correspond to the
instantaneous amplitude and frequency, and
m corresponds to the modulation-
index estimate.
amplitude/frequency demodulation followed by a modulation-
index estimation. These steps are described in Fig. 1. To
perform demodulation, we propose to exploit the multidimen-
sional nature of three-phase systems through low-complexity
linear transforms. Then, we propose to estimate the modula-
tion indexes from the demodulated signals with two original
estimators.
This paper is organized as follows. Section II describes the
signal model of the s tator current under healthy and faulty
conditions. Section III investigates the use of CT and princi-
pal component analysis (PCA) to perform multidimensional
current demodulation, and Section IV describes the proposed
failure severity criteria. Finally, Section V reports on the per-
formance of the proposed approach with synthetic and experi-
mental signals.
II. S
IGNAL MODEL
In the presence of a fault, it has been shown in [5], [13],
[14], and [16] that the stator current is amplitude modulated
and/or phase m odulated. For AM and/or PM, the instantaneous
amplitude a(t) and phase φ(t) can be expressed respectively as
a(t)=α (1 + m
a
cos(2πf
a
t)) (1)
φ(t)=2πf
0
t + m
φ
sin(2πf
φ
t) (2)
where α is a scaling coefficient, f
0
is the supply current
frequency, and f
a
(f
φ
) is the AM (PM) modulating frequency.
The scalars m
a
and m
φ
correspond to the AM and PM indexes,
respectively. For a faulty system, the modulation indexes are
directly proportional to the failure severity. In particular, with-
out any fault, the instantaneous amplitude and frequency do not
vary with time, i.e., m
a
= m
φ
=0.
Let us consider a three-phase system. In the presence of a
fault, all three line currents i
1
(t), i
2
(t), and i
3
(t) are simulta-
neously modulated, and the currents can be expressed as
i
1
(t)=a(t)cos(φ(t)) (3a)
i
2
(t)=a(t)cos(φ(t) − 2π/3) (3b)
i
3
(t)=a(t)cos(φ(t)+2π/3) . (3c)
In the literature, most studies assume a perfect balance
configuration. However, healthy electrical systems are rarely
perfectly balanced. Furthermore, the balance assumption usu-
ally does not hold when a failure introduces some asymmetry.
In this study, balanced and unbalanced three-phase systems are
considered. Let us denote s(t)=[s
1
(t),s
2
(t),s
3
(t)]
T
the 3 ×
1 vector which contains the stator currents, where (·)
T
corre-
sponds to the matrix transposition. In this paper, we investigate
the two following systems.
1) A balanced three-phase s ystem, where the stator currents
are given by
s(t)=i(t)=[i
1
(t),i
2
(t),i
3
(t)]
T
. (4)
In particular, by using (3), one can easily verify that
s
1
(t)+s
2
(t)+s
3
(t)=0.
2) A three-phase system with unbalanced currents, where
the stator currents are given by
s(t)=Di(t)=[α
1
i
1
(t),α
2
i
2
(t),α
3
i
3
(t)]
T
(5)
where D is a nonscalar 3 × 3 diagonal matrix which
contains the “nonequal” diagonal entries α
1
, α
2
, and α
3
.
Without loss of generality,
1
we assume that the overall
energy of the system is conserved, i.e.,
3
k=1
α
2
k
=3.
In this study, the modulation indexes are employed as failure
severity indicators. From a signal processing viewpoint, the
condition monitoring problem is therefore translated into an
estimation problem. One should note t hat the estimation of the
modulation indexes can be simplified by using a demodulation
preprocessing step. In the following, the demodulation is
performed by using a linear transformation of the stator
currents s(t).
III. AM/FM D
EMODULATION USING
MULTIDIMENSIONAL TRANSFORM
In this section, we prove that the use of the three-phase
current can expedite the demodulation step. In particular, we
show that the CT and the PCA can be considered as low-
complexity techniques for current demodulation. Furthermore,
we demonstrate that the PCA has a larger domain of validity
than the CT.
A. CT
CT is a linear transform which converts the three-component
s(t) into a simplified system composed of two components.
By denoting y
(c)
(t)=[y
(c)
1
(t),y
(c)
2
(t)]
T
the two Concordia
components, CT can be expressed into a matrix form as
y
(c)
(t)=
y
(c)
1
(t)
y
(c)
2
(t)
=
2
3
Cs(t) (6)
where C is the 2 × 3 Concordia matrix which is equal to
C =
√
2
√
3
−
1
√
6
−
1
√
6
0
1
√
2
−
1
√
2
. (7)
One can verify that the Concordia matrix is an orthogonal
matrix since it satisfies CC
T
= I
2
, where I
2
is a 2 × 2 identity
matrix.
1
One should note that the overall energy can be absorbed into the coefficient
α in (1).
2016 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 4, APRIL 2012
By using (3), (4), and (6), it can be demonstrated that the
Concordia components of a balanced system are equal to
y
(c)
1
(t)=a(t)cos(φ(t)) a (8a)
y
(c)
2
(t)=a(t)sin(φ(t)) a. (8b)
These components y
(c)
1
(t) and y
(c)
2
(t) are called in-phase and
quadrature components in the signal processing community.
Let us define the complex signal z
(c)
(t) as
z
(c)
(t)=y
(c)
1
(t)+jy
(c)
2
(t). (9)
By using (8) and (9), one can verify that z
(c)
(t) is the ana-
lytical signal of s
1
(t), i.e., z
(c)
(t)=a(t)e
jφ(t)
. Therefore, the
instantaneous amplitude and frequency can be obtained from
the modulus and the derivative of the argument of z
(c)
(t),
respectively, i.e.,
a(t)=
z
(c)
(t)
(10a)
f(t)=
1
2π
d arg
z
(c)
(t)
dt
(10b)
where |·|and arg[·] correspond to the modulus and the argu-
ment, respectively. It is important to note that (10) only holds
for a balanced system, i.e., for s(t)=i(t).
B. PCA
PCA is a statistical tool that transforms a number of corre-
lated signals into a smaller number of principal components. In
[40], PCA is employed after a CT to detect a failure; however,
no mathematical analysis has been performed to give a physical
interpretation to the principal components. In [41], the PCA is
applied on the three-phase stator currents directly, but only t wo
principal components are extracted, without any mathematical
justification. In this section, we give a deep theoretical analy-
sis of PCA for balanced and (static) unbalanced three-phase
systems. We show why the PCA can be applied on the stator-
current signals directly. Furthermore, we demonstrate why the
PCA can only extract two principal components and why prin-
cipal components are strongly linked to in-phase and quadrature
components. Finally, as opposed to CT, we prove that the PCA
can be employed for signal demodulation whatever the balance
assumption.
Let us define the 3 × 3 covariance matrix as
R
s
= E
s(t)s
T
(t)
(11)
where E[·] denotes the mathematical expectation. Using (3), one
can remark that i
3
(t)=−i
1
(t)−i
2
(t). Therefore, each compo-
nent of s(t) can be rewritten as a linear combination of t he two
components i
1
(t) and i
2
(t) whatever the balance assumption.
2
Using an eigenvalue decomposition, it follows that the 3 × 3
symmetric matrix R
x
contains one zero eigenvalue. Therefore,
R
x
can be decomposed under the following form:
R
s
= UΛU
T
(12)
2
For unbalanced systems, one can verify that s
1
(t)=α
1
i
1
(t), s
2
(t)=
α
2
i
2
(t),ands
3
(t)=−α
3
(i
1
(t)+i
2
(t)).
where U is a 3 × 2 orthogonal matrix (U
T
U = I
2
) containing
the two eigenvectors and Λ=diag(λ
1
,λ
2
) is a diagonal matrix
containing the two nonzero associated eigenvalues λ
1
and λ
2
.
The two principal components of s(t), denoted y
(p)
(t)=
[y
(p)
1
(t),y
(p)
2
(t)]
T
,aregivenby
y
(p)
(t)=
y
(p)
1
(t)
y
(p)
2
(t)
= β
s
Λ
−
1
2
U
T
s(t) (13)
where β
s
is a scaling term which is equal to
β
s
=
trace[R
s
]
3
(14)
with trace[·] being the sum of the diagonal elements. Using
(12) and (13), one can verify that the PCA components are
uncorrelated.
Under the assumptions that φ(t) is uniformly distributed in
[0 2π[
3
and that a(t) and φ(t) are independent, it is demon-
strated in the Appendix that the PCA components are equal to
y
(p)
1
(t)=a(t)cos(φ(t) − θ) (15a)
y
(p)
2
(t)=a(t)sin(φ(t) − θ) (15b)
where θ ∈ Z whatever the balance assumption.
Let us define the complex signal z
(p)
(t) as
z
(p)
(t)=y
(p)
1
(t)+jy
(p)
2
(t). (16)
By using (15) and (16), one can verify that z
(p)
(t) is a ro-
tated version of the analytical signal of s
1
(t), i.e., z
(p)
(t)=
a(t)e
jφ(t)−θ
. Therefore, the instantaneous amplitude and fre-
quency can be obtained from the modulus and the derivative of
the argument of z
(p)
(t), respectively, i.e.,
a(t)=
z
(p)
(t)
(17a)
f(t)=
1
2π
d arg
z
(p)
(t)
dt
. (17b)
As opposed to (10), it is interesting to note that (17) holds
whatever the balance assumption. Therefore, the PCA-based
demodulation is less r estrictive than the Concordia one.
IV. M
ODULATION-INDEX ESTIMATION
After demodulation, the analytical signal and the instanta-
neous amplitude and/or frequency must be properly analyzed
to assess failure severity. Many papers propose to monitor the
deviation of the analytical signal z(t) from a circle in the com-
plex plane [29]–[32], [34]–[37], [40]. This solution i s perfectly
valid if the failure leads to stator-current AM since the radius
|a(t)| varies with time. However, if the failure leads to PM,
this solution is no longer correct since the failure only affects
the rotational speed in the complex plane. In this section, we
propose to estimate the AM and PM indexes to assess the failure
severity. Using the demodulated signals, two original estimators
3
From a decision viewpoint, the uniform probability density function (pdf)
for the phase represents the most ignorance that can be exhibited by the fault
detector. This is called the least favorable pdf for φ(t) [42].
CHOQUEUSE et al.: DIAGNOSIS OF THREE-PHASE ELECTRICAL MACHINES USING DEMODULATION TECHNIQUES 2017
of the modulation indexes are provided. These estimators are
based on the method-of-moments (MoM) technique. Although
the MoM estimation technique has no optimal properties, it
produces an estimator that is easy to determine and simple to
implement [43].
A. Estimation of m
a
Let us consider the AM signal model in (1). Under the
assumption that 2πf
a
t(mod 2π) is uniformly distributed in
[0 2π[, it is demonstrated in the Appendix that the variance of
the instantaneous amplitude is given by
σ
2
a
= E
(a(t) − µ
a
)
2
=
µ
2
a
m
2
a
2
(18)
where µ
a
= E[a(t)] is the statistical average of f(t). Therefore
m
a
=
σ
a
√
2
µ
a
. (19)
The corresponding MoM estimator, denoted m
a
, is obtained by
replacing the theoretical moments µ
a
and σ
a
by their natural
estimators.
B. Estimation of m
φ
Let us consider the PM signal model in (2). Taking the
derivative of φ(t) leads to the instantaneous frequency, which
is equal to
f(t)=
1
2π
dφ(t)
dt
= f
0
+ m
φ
f
φ
cos(2πf
φ
t). (20)
Under the assumption that 2πf
φ
t(mod 2π) is uniformly distrib-
uted in [0 2π[, it can be demonstrated that the variance of the in-
stantaneous frequency is (see the Appendix for a similar proof)
σ
2
f
= E
(f(t) − µ
f
)
2
=
m
2
φ
f
2
φ
2
(21)
where µ
f
= E[f(t)] is the statistical average of f(t). Therefore
m
φ
=
σ
f
√
2
f
φ
. (22)
The corresponding MoM estimator m
φ
is obtained by replacing
the theoretical moments by their natural estimators. One should
note that the criterion m
φ
depends on the modulating frequency
f
φ
. If this frequency is unknown, it can be replaced by its
estimate
f
φ
. This estimate can be obtained, for example, by
maximizing the periodogram of f (t) [44], [45].
V. P
ERFORMANCES
This section reports on the performances of the proposed ap-
proaches. Experiments were performed with a supply frequency
equal to f
0
=50Hz. Signals were sampled with a sampling
period of T
s
=10
−4
s, and the proposed technique was ap-
plied offline in Matlab. For discrete signals, a straightforward
adaptation of the proposed techniques is given by Algorithms 1
and 2, respectively. As compared to the continuous case, s(t) is
replaced by its discrete counterpart s[n]=s(nT
s
), where T
s
is
the sampling period and n =0, 1,...,N −1. Furthermore, the
instantaneous frequency is approximated by replacing the phase
derivative with a two-sample difference,
4
and the statistical
moments are replaced by their natural estimators. In particular,
f(n), R
s
, m
a
, m
f
, σ
2
a
, and σ
2
f
are r espectively given by
f(n)=
arg [z(n)] − arg [z(n − 1)]
2πT
s
(23)
R
s
=
1
N
N−1
n=0
s[n]s
T
[n] (24)
µ
a
=
1
N
N−1
n=0
a[n] (25)
µ
f
=
1
N
N−1
n=0
f[n] (26)
σ
2
a
=
1
N
N−1
n=0
(a[n] − µ
a
)
2
(27)
σ
2
f
=
1
N
N−1
n=0
(f[n] − µ
f
)
2
. (28)
The next sections present the performances of the proposed al-
gorithms with synthetic and experimental signals, respectively.
Algorithm 1 Concordia-based failure severity criteria
1) Extract N-data samples s[n].
2) Compute y
(c)
[n] with (6).
3) Compute the analytical signal z
(c)
[n] with (9).
4) Extract the AM demodulated signal a[n] with (10a).
5) Extract the FM demodulated signal f[n] with (23).
6) Compute
m
a
with (19), (25), and (27).
7) Compute
m
φ
with (22), (26), and (28).
Algorithm 2 PCA-based failure severity criteria
1) Extract N-data samples s[n].
2) Compute R
s
with (24).
3) Perform eigenvalue decomposition of R
s
as in (12).
4) Compute β
s
with (14).
5) Compute y
(p)
[n] with (13).
6) Compute the analytical signal z
(p)
[n] with (16).
7) Extract the AM demodulated signal a[n] with (17a).
8) Extract the FM demodulated signal f[n] with (23).
9) Compute
m
a
with (19), (25), and (27).
10) Compute
m
φ
with (22), (26), and (28).
A. Synthetic Signals
Synthetic signals s(n) were simulated by using the signal mod-
el in (3). Analysis of the algorithm performances with amplitude-
and phase-modulated signals is i nvestigated independently.
4
Before subtraction, a phase unwrapping operation must be applied to avoid
phase jumps between consecutive elements.
