Performance Analysis of a Three-Phase Induction Motor under Mixed Eccentricity Condition

Abstract

A substantial portion of induction motor faults is eccentricity related. In practice, static as well as dynamic eccentricities happen to exist together. With this point in mind, an analytical approach to evaluate performance of a three-phase induction motor under mixed eccentric condition has been presented in this paper. Clear and step-by-step theoretical analysis, explaining completely the presence of certain harmonics in the line current spectrum in presence of eccentricity, is discussed. More importantly, it is shown for the first time that a link exists between the lowi and the high-frequency elements of these harmonics. It is also shown that these high-frequency components are not very strong in all types of machines. These results will be useful in generating rules and laws to formulate on-line tools for machine condition monitoring. Finite element results to substantiate the inductance values used in the simulation are also included. The analysis is validated by the line current spectrum of the eccentric machine obtained through simulation using modified winding function approach (MWFA) and experimentation.

Full text

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PERFORMANCE ANALYSIS OF A THREE PHASE INDUCTION MOTOR UNDER
MIXED ECCENTRICITY CONDITION
Subhasis NandiRajMohan BharadwajHamid A. ToliyatAlexander G. Parlos
Student Member, IEEEStudent Member, IEEESenior Member, IEEESenior Member, IEEE
Electric Machines and Motor Drives Laboratory
Department of Electrical Engineering
Texas A&M University
College Station, TX, 77843-3128
Fax: (409) 845-6259
E-mail:toliyat@ee.tamu.edu
Abstract: A substantial portion of induction motor
faults is eccentricity related. In practice, static as well
as dynamic eccentricity happen to exist together. With
this point in mind, an analytical approach to evaluate
performance of a three phase induction motor under
mixed eccentric condition has been presented in this
paper. Clear and step-by-step theoretical analysis,
explaining completely the presence of certain
harmonics in the line current spectrum in presence of
eccentricity, is discussed. The effects of the number of
rotor slots, machine pole numbers which influence the
current spectrum considerably; can also be fully
identified. These results will be useful in generating
rules and laws to formulate on-line neural network
based tools for machine condition monitoring. Finite
element results to substantiate the inductance values
used in the simulation are also included. The analysis is
validated by the line current spectrum of the eccentric
machine obtained through simulation using modified
winding function approach
experimentation.
(MWFA) and
I. Introduction
Eccentricity related faults constitute a major
portion of the faults related to induction motors.
Machine eccentricity is the condition of unequal
air-gap that exists between the stator and rotor [1-
2].
There are two types of air-gap eccentricity:
the static air-gap eccentricity and the dynamic air-
gap eccentricity. In the case of the static air-gap
eccentricity, the position of the minimal radial
air-gap length is fixed in space. Dynamic
eccentricity occurs when the center of the rotor is
not at the center of the rotation and the position
of minimum air-gap rotates with the rotor. [2]
gives a detailed description of both these types of
eccentricities .
In reality both
eccentricities tend to co-exist. An inherent level
of static eccentricity exists even in newly
manufactured machines due to manufacturing and
assembly method [3]. This causes a steady
unbalanced magnetic pull (UMP) in one
direction. With usage, this may lead to bent rotor
shaft, bearing wear and tear etc. This might result
in some degree of dynamic eccentricity. Unless
detected early, these effects may snowball into
stator to rotor hub causing a major breakdown of
the machine [4].
Air-gap eccentricity in induction machines is
usually detected by analyzing the stator line
current spectra. The equation that describes such
frequency component for eccentricity is given by
[2]
(
f kRn
p
static and dynamic
)
s
d
±
−
±




()1
ν
(1)
where nd= 0 in case of static eccentricity, and
nd= 12 3, , ... in case of dynamic eccentricity (nd
is known as eccentricity order), f is the
fundamental supply frequency, R is the number of
rotor slots, s is the slip, p is the number of pole
pairs, k is any integer, and ν is the order of the
stator time harmonics that are present in the
power supply driving
±±±
135,,
, etc.). The principal slot harmonics
are also given by
with nd= 0 ,ν = 1. In case one of these
the motor (ν =
the above equation

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harmonics is a multiple of three, it may not exist
theoretically in the line current of a balanced
three phase machine.
The above expression is to be used with
caution. Stated succinctly, for a healthy machine
to produce a spectrum of principal slot
harmonics, the pole pair number R
R is the number of rotor slots, n the harmonic
order number, and p the number of fundamental
pole pair) should be equal to the pole pair
number of the space harmonics produced by a
phase of the stator winding [5-6,14]. The same is
also true in case of eccentricity related
components, where the pole pair number can be
Rnpk
±±
,k = 012, , .
np
±
(where
II. Analysis of Mixed Eccentric Machine
Analysis on the same lines as [3] shows that
mixed eccentricity gives rise to two more current
components around the fundamental. They also
give rise to additional dynamic eccentricity
related components as described by (1) . The
analysis also shows that due to static or dynamic
eccentricity the amplitude of the fundamental
current is changed. This will in turn change the
amplitude of the principal slot harmonic. These
changes are not only dependent upon the
amplitude of the extra components eccentricity
produces but also their phase.
In the following analysis, the well known
transformation is applied. Stator frame of
reference is transformed into rotor frame of
reference by addingωrt (ωr= rotor speed) to the
stator angular position. Similarly, rotor frame of
reference is transformed into stator frame of
reference by subtracting ωrt from the rotor
angular position.
MMFs due to stator currents are of the form
Ap x
n
t
cos()
±ω
,(2)
where p
number of fundamental pole pairs;ω = line
frequency in rad/sec and x the angular position
from the stator frame of reference. The
permeance function with both static and
dynamic eccentricities can be approximately
expressed as
np
n=
; n = number of harmonic ; p =
PPPxPxt
r
≈++−
012
coscos()
ω
(3)
Consider the component of the air-gap flux
produced by the interaction of (2) only with the
static eccentricity component of (3). It is of the
form with respect to stator.
()
[]
()
[]
[]
AP
2
or
pxtpxt
nn
1
11coscos
−±++±
ωω
(4)
()()
[]
( )()
[]
[]
AP
2
pxttpxtt
nrnr
1
11 coscos
−+±+++±ωωωω
(5)
with respect to rotor.
The interaction of the MMF (due to the
current the flux as given by (5) produces in the
rotor) and the dynamic eccentricity component of
the permeance function of (3) gives rise to air-gap
flux components given by,
()()
[]
A P P
1
4
pxxptt
nnr
12
1
11cos
−++−±−
ωωφ +
()()
[]
A P P
1
4
pxxptt
nnr
12
1
11 cos
−−+−±−
ωωφ +
()()
[]
A P P
2
4
pxxptt
nnr
12
2
11 cos
++++±−
ωωφ +
()()
[]
A P P
2
4
pxxptt
nnr
12
2
11cos
+−++±−
ωωφ
(6)
It is clear from the above expression that at
least the first and the fourth term can induce
voltage in the stator as the pole pair numbers of
only these terms will match with those in (2) for
any p.
Adding and expressing these terms with
respect to stator (replacing x by x
(
p x
nr
4
[
4
t
r
−ω
]
) yields
)
[
A P P
1
t
1 2
1
cos
−−
ωωφ
?
+
( )
]
A P P
2
p x
n
t
r
1 2
2
cos
+± −
ωωφ
(7)
which clearly shows that extra components of
stator currents are introduced at ω
ωω−
ω+
r and
r. Proceeding on the same line, it can be

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shown that the air-gap flux produced by the
interaction of the MMF, due to the current (5)
produces in the rotor, and the static eccentricity
component of the permeance function of (4)
induces stator currents at ω . Thus the
fundamental component of the line current is
altered.
Similarly the component of the air-gap flux
produced by the interaction of (2) first with only
the dynamic eccentricity component of (3) and
later with static eccentricity component also
induce stator currents at ω
to interaction again with the dynamic eccentricity
component instead of the static eccentricity the
fundamental component of the current is altered.
Also, these ωω±
same process will sequentially yield components
such as
ω±
r. Similarly, due
r components through the
ωω± k
r
’
, k’=2,3,4…(8)
These current components at
ωωω
’=± k
r , k = 12 3, , ...
(9)
produce the additional harmonics as given by the
dynamic eccentricity related components in (1)
the following way:
The MMF produced by these components
can be described as
()
B p x
n
t
cos
’
±ω
(10)
i) Acting only on P0, the average value of the
permeance function to produce air-gap flux
components given by
()
BPp x
n
t
0cos
’
±ω
(11)
with respect to stator, or,
()
()
BPp x
n
tt
r
0cos
’
+±ωω
(12)
with respect to rotor. These components produce
reversed rotation (R-Pn) pole pair rotor MMF
harmonics[7]. These harmonics upon acting on
P0 produce air-gap flux components of the type
()
[]
B P
1
Rp x
n
ptt
nr
0
2
3
cos
’
−−−ωωφ
?
(13)
with respect to rotor, or,
( )()
[]
B P
1
Rpxtptt
nrnr
0
2
3
cos
’
−−−−ωωωφ
?
(14)
with respect to stator.
Simplifying the above expression after
substituting ω’ by ωω±
rand ωrby ()1− s
p
ω
gives
()
()(
)
B P
1 0
Rp x
n
Rk
s
p
t
2
3
1
1cos
−−±
−
±






−






ωφ
(15)
ii) Similar interaction between constant part of
the air-gap and static eccentricity and vice-versa
can give rise to components of the form
()
()
B P P
2
2
Rpx
Rs
p
t
n
0 1
4
1
1
1cos
−±−
−
±






−






ωφ
(16)
iii) In the same way the interaction between
constant part of air-gap and dynamic eccentricity
and vice-versa produces components such as
()
()
B P P
2
2
Rpx
Rks
p
t
n
02
5
1
1
1cos
()
’
−±−
±−
±






−






ωφ
(17)
iv) Finally the interaction between static and
dynamic eccentricity produces components
such
()
()
B PP
3 1 2
4
Rp k x
1
Rks
p
t
n
2
6
1
1 cos
()
−±−
±−
±






−






ωφ
(18)
where k1
0 2
= , and k2
Depending upon R pn
more of these components will produce high
frequency components in stator current as given
by (1).
012
= , , ,...
,
combinations one or
III. Analysis of Three Phase Induction
Motor under Eccentric Condition using
Modified Winding Function Approach
(MWFA)
Analysis of three phase induction machine
using winding function approach is well
documented in literature [9-10]. However, in

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presence of air-gap eccentricity those equations
are not valid any more as the average value of the
winding function no longer remains zero [11].
Using the modified winding function approach,
the self inductances of the rotor loops, stator
coils, as well as the mutual inductances between
the stator coils and rotor loops are calculated.
These inductances are then used to describe the
stator and rotor circuit equations. Using the
modified winding function theory expressions for
these inductances under static and dynamic
eccentricity conditions can be developed as
described in [12].
In the presence of mixed eccentricity the air-
gap can be modeled as
ggaa
e rmrm
( ,
φ θ
)coscos()
φφ θ−=−−
012
(19)
where, a a
12
,
are the amount of static and
respectively,g0
average air-gap andφ
along the stator inner surface. The inverse air-
gap function, germ
,
θ
[14].
dynamic eccentricity
the
a particular position
−1()
φ
is as described in
IV. Modeling of Three Phase Induction
Motor under Mixed Eccentricity Condition
Detailed description of the modeling of three
phase induction motor using the coupled
magnetic approach is given in [13]. However in
case of the mixed eccentric machine the
electromagnetic torque expression is given by,
TI
L
II
L
II
L
II
L
I
ms
t
ss
rm
ss
t
sr
rm
rr
t
rs
rm
sr
t
rr
rm
r
=+++








05 .
∂
∂θ
∂
∂θ
∂
∂θ
∂
∂ θ
(20)
V. Finite Element and Simulation Results
The simulated machine has 36 stator slots
with full pitch 3 phase concentric winding, 44
rotor slots, 4 poles and a rated power of 3 HP.
The simulation results were obtained using
balanced 3 phase, 460V, 60 Hz ac source.
Fig.1 show the plots of Lr a
Lr a
L
r br c
11
,
respectively for the same machine
with mixed eccentricity ( 15% static and 5%
dynamic) obtained through simulation and finite
element. As saturation is neglected in winding
1 for healthy and
1, L
function approach, the Carter’s coefficient has to
adjusted in order to match the machine
performance both under load and no-load. This is
the cause for difference in magnitude between
simulation and finite element results. Otherwise,
the similarities are very much apparent.
Fig.2 show the Power Spectral Density
(PSD) of the healthy and the faulty machine
around the fundamental
s = 0029.
. The load
dynamometer type.
components as given by (9) and the high
frequency components as given by (1) are all
present. With R = 44, and p = 2, air-gap flux
components as given by (15) and (18) are only
responsible for inducing these high frequency
components. Thus this method of simulation is
able to completely replicate the effects of
eccentricity in an actual machine. The static
eccentricity was 38.46%
eccentricity 5%. A more complete high frequency
spectrum with a higher
eccentricity is included in [14].
The magnitude of PSD obtained from the
first Lower and Upper side-bands (LSB and USB)
around fundamental and PSH, with different
values of eccentricities under loaded (as
described earlier) and unloaded conditions are
given in Fig.3. The values of static eccentricity
used are 38.46%, 25.64% and 12.82%. Similarly
dynamic eccentricity values are chosen to be 5,10
and 20%. It is interesting to note that only the
USB around the fundamental in the no-load case
is appreciably above the same in the loaded case.
and PSH with
beconsidered
The low
to
frequency
and dynamic
value dynamic
VI. Experimental Results
Experimental results were obtained on a
similar machine as used for the simulation. A
nominal 38.46% static
introduced by machining the bearing housing of
the machine end bells. The inherent machine
dynamic eccentricity (expected to somewhere
around 5%) was used in the experiments. The
results are plotted in Fig. 4. The value of slip is
same at s = 0029.
.
It can be seen that even with healthy end bells the
low frequency components as given by (9) can be
found around the
saturation can be a possible cause for these low
frequency sidebands. These components only
eccentricity was
fundamental. Machine

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changed marginally when the static eccentricity
was introduced. While the upper sideband
changed from -53.89 to -52.14 dB from healthy to
eccentric case, the other sideband remained the
same. This shows error in fault diagnosis can
occur if decision is taken only on the basis of low
frequency components.
However, the high frequency components
clearly identified the presence of eccentricity.
The USB increased from -95.15 to -90.24 dB,
whereas the LSB increased from -89.06 to -84.03
dB when the eccentric end bells were used.
These results show that a pragmatic approach to
detect eccentricities requires examination of both
high frequency and low frequency components.
0100200300
-0.1
-0.05
0
0.05
0.1
Lr1a(mH)
Sim
0100200300
-0.1
-0.05
0
0.05
0.1
FE
0100200300
-0.1
-0.05
0
0.05
0.1
Lr1a(mH)
Sim
0100200 300
-0.1
-0.05
0
0.05
0.1 FE
0 100 200300
-0.1
-0.05
0
0.05
0.1
Lr1b(mH)
Sim
0100200300
-0.1
-0.05
0
0.05
0.1
FE
0 100200300
Angle (degrees)
-0.1
-0.05
0
0.05
0.1
Lr1c(mH)
Sim
0100200300
-0.1
-0.05
0
0.05
0.1
FE
Fig 1. Plots of mutual inductance between rotor loop
1 and stator phases obtained from simulation and
finite element analysis. From top to bottom row:
healthy machine (‘a’ phase); mixed eccentric machine
(phases ‘a’, ‘b’, ‘c’ respectively) with 15% SE and 5%
DE.
0 50 100150
-100
-50
0
PSD(dB)
Fundamental
1300 1350 1400 1450
-120
-100
-80
-60
-40
PSH
0 50100 150
-100
-50
0
PSD(dB)
Fundamental
f+fr
f+2fr
f-fr
f-2fr
1300 1350 1400 1450
-120
-100
-80
-60
-40
PSH
L
U
Frequency(Hz)
Fig.2 Simulated spectrum of the of phase ‘a’ current
of machine under load around the fundamental and
PSH. Upper row: Healthy, lower row: with mixed
eccentricity (38.46% SE, 5% DE).
10 2030
-60
-50
-40
-30
-20
PSD(dB)
DE=20%
102030
-60
-50
-40
-30
DE=10%
102030
-70
-60
-50
-40
DE=5%
102030
-60
-50
-40
-30
PSD(dB)
DE=20%
102030
-60
-50
-40
DE=10%
1020 30
-70
-65
-60
-55
-50
DE=5%
10 2030
-90
-85
-80
-75
-70
PSD(dB)
DE=20%
102030
-95
-90
-85
-80
-75
DE=10%
102030
-100
-95
-90
-85
-80
DE=5%
102030
-90
-85
-80
-75
-70
PSD(dB)
DE=20%
102030
-95
-90
-85
-80
-75
DE=10%
102030
-100
-95
-90
-85
DE=5%
% Static Eccentricity
Fig 3. Simulated variation of the first side band
components at no load and under load at different
levels of mixed eccentricity. From top to bottom row:
LSB around fundamental, USB around fundamental,
LSB around PSH, USB around PSH. Solid line implies
unloaded condition, and broken line loaded condition.