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DETECTION
FROTORLOTAND OTHERECCENTRICITYELATED
HARMONICSINA THREE PHASE INDUCTION MOTOR
WITH
IFFERENT ROTOR
CAGES
SubhasisNan
Student M ember,
IEEE
Hamid
A. Toliyat
Senior Member.
IEEE
Electric Machines and
Motor
Drives Laboratory
Department of Electrical Engineering
Texas
ABM
University
College Station,TX
7843-3 128
E-mail:[email protected]
Abslroct -
Detection of rotor slot and other eccentricity related harmonics
are
given in a compact form by
related harmonics in the line current of a thr phase
[3]
induction motor is important both fiom the viewpoint
of sensorless speed estimation
as
well
as
eccentricity
related fault detection. However, it is now clear that
not all
three
phase induction motors
are
capable
of
generating such harmonics
in
the line current. Recent where
n j
= 0
in case
of static eccentricity, and
research has shown that the presence
of
these n,, = I n
case
of dynamic eccentricity (nd is
harmonics
is
primarily dependent on the number of known as eccentricity
rotor slots and the of
pole pair of supply frequency,
R
is the number of rotor slots,s
the machine. While the number of fundamental
pole .
IS the
slip,
p
is
the
number of fundamental
pole
pairs of a three phase induction motor usually is
avoided due to increased current), the order of
the
stator time harmonics that
are
present
number of
rotor
slots can vary widely. Tbe present in the power supply driving
the
motor v =
paper investigates this phenomenon further and obtains f , f ,f , etc.). The principal slot harmonics
a hitherto unexplored theoretical
b,asis
for the
are also bv the
above eauation
Fax: 409) 845-6259
1)
P
the
between one to
four
fiigher pole p a ~ s
re
generally pain,
k
is any positive integer, and v is
the
w i t h n d = O , v = l , k = l .
W h e n o n e o f t h e s e
xperimentally verified results. Detailed coupled
magnetic circuit simulation results
are
presented
for
a
four
pole, three phase induction motor q 4 3 , and harmonics is
a
multiple
of three,
it may not exist
42
rotor
slots
under healthy, static, dynamic and mixed theoretically in
the
line Current
of
a
balanced
eccentricity conditions. The simulation is flexible
three
phase
three
wire machine.
enough to accommodate other po le , numbers also. How ever, the harmonics
as
described by (1)
These simulations are helpful
in
quantifying the
are
not present in
the
machine for all combination
predicted harmonics under different combinations
of
of p and
R.
This is due to the fact that
the
only
load, pole i
numbem, rotor Slots and eccentricity flux which can produce voltage in
a
three phase
conditions, thus making the problem oasier for drive
stator
winding is
one
that has
a
number
of
pole
oairs that the winding itself may Droduce
141.
esigners or diagnostic tools' develope&.
- .
However, in
a
s q u i i i cage a flux with any
number of pole pairs can induce a voltage.
To
be
. Introduction
machine
to produce a
the principle slot harmonics orPSH and the other
spectrum
of principal slot
the
poleeccentricity related harmonics is absolutely pair
number
R f n p
n
the
order
essential for most of
the sensorless
adjustable
number
hould b e equal to
the
pole pair number
speed induction motor drive schr:mes [ l] and
diagnosis of eccentricity related faults
[2].
The
Of the
'pace
harmonics
produced by
a
phase
Of
PSH,
and
the
static
and
dynmi,:
eccenhicity the
stator winding.
For
example, with
36
stator
The
presence rotor slot harmonics (also called precise
for a
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slots and full pitch three phase concen tric winding
p pi Po
(4)
and R = 44, p =
2 ;
one principal slot harmonic
can be seen.
The
same winding with R 4 3
or R These MMFs
acting
on Po,
PrOduw air-gaP flux
4 2
hould not ideally give any principal slot components given by,
harmonics. However, in presence of static
or
5 )
dynamic eccentricity
the
pole
pair number
changes from R f
np
to
R
f p f
.
This will
then introduce additional harmonics
as
given by
A P , c o s [ p , ( x + o , t ) ] f w t )
( 6 )
(1)
With only R
4 3
nd not With R
4 4 r
R
4 2
with respect
to
rotor.
These
components produce
for the
Same
fundamental pole pair. In fact, with
R n ) ,,,,le
pair rotor
MMF
harmonics
[71
R
4 2 this condition is similar to
the
case in
[I]
which acting upon
Po
produce air-gap flux
where R =58); the speed detection algorithms
components of
the
type
using principal slot harmonics
are
likely to fail.
When both static and dynamic eccentricitv are
A, Po2 S[ R - p,)X -p p , t TU t ) -
4
I (7)
A Po cos(p,x
f
t
with respect to stator,
o r ,
. .
present (mixed eccentricity), additional .
compon ents given by
[5-61
with respect to rotor, or,
A, P
COS[(R
p , ) ~
U, t )- p , t T o t ) -
4
,]
fW,I,
k
=
1 2 3 . .
1
(2)
8)
will be present in the stator current spectrum of
any
three
phase irrespective of
p
and
R,
where with respect to stator.
f,
is the rotational frequency of
the
machine.
The
above expression can be simplified by
w
as
owever, these additional components will give substituting
,
by 1
-4
rise to other additional current spectrums at
the
D
same frequency points
as
described by
(1)
for
dynamic eccentricity related components.
11. Mechanism of PSH and other The
relevant rotor space
MMF
harmonics also
Eccentricity R elated Harm onic Generation
generate air-gap flux components similar to
9).
in the Motor Line C urrent
Finally comparing
(8)
with 1) shows that
the
In the following analysis, the well
known
reference
is
transformed into
rotor frame of
reference by
addingw t U , =
rotor speed) to
the
stator angular position. Similarly, rotor
frame
of
reference is transformed into stator frame of
reference by subtracting
o , t
from
the
rotor
angular position.
a) Healthy machine:
MMFs
due
to stator currents
are
ofthe form
transformation is applied. Stator
frame
of PSH components given
by
only present when at least
one
element of the set
R pn )
lso belongs to the set pn
.
Now for a
balanced three phase winding n is usually given
by (other than
1
which implies fundamental)
10)
= 6 k f 1 , k = 1 , 2 , 3,....
Thus in order to observe
the PSH, R
is given by
R = Z p [ X m f q ) f r ] , mfq = 0 , 1 , 2 , 3 ,.... =O or 1.(11)
A c o s ( p , x f o t ) ,
3)
Clearly, in our case only R
= 44
satisfies 1
1).
where
Pn
= P
;
=
number
of
space
b) Mm hine wi th stat ic eccenhic i ty :
harmonic;w=line frequency in rad sec and x the
angular position from the stator frame
of
expressedas
reference. The permeance function, without
p ~
po qcoSx
considering any eccentricity, can
be
approximately expressed as
this case, the permeance function c m be
12)
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The
air-gap
flux
components produced by
(3)
described in (a) these components again generate
acting
on
12) is given by
.
[a]air-gap flux of type
AI:
A P , c o s ( p , x i w t ) + - ~ ~ s [ ( p , - I ) x f ~ t ]
2
A,-CO ( R - p . f d , ) x -
epy,
13)
+-cos[(p,
+ I ) X f O t ]
2
These
comp onents then produce
R pn) ole
19)
pair rotor h4MF harmonics as dascribed earlier. The following combinations of
P,,P,
d,
dz are
The
harmonics containing the 4 term now possible.
combines with the eccentric part of the permeance
function and those containing'the 4 term
I ) P , = P , , = P ~ , d l = l , d 2 = 0 , k l = l , 2 , 3 ...
combines with
the
average part. This results in
air-gap flux of the form,
2 ) P , = & , P , = q o r & , d , = 2 ,d 2
= I ,
k l =0 ,1 ,2 ,3 ...
Following
reasoning as n the earliercBse, Combination 1) and combination
3)
with
d2 = 0
the value
of
R
in
order to observe static only applicable for machines described by
(11).
eccentricity related
components
is given by: Combination
2 )
is
only
applicable for those
described by (15). For other types
of
machine
R
= 2 p [ 3 ( m f q ) f r f 1 (15) (for exam ple R = 4 2 , p = 2 in
our
case)
combination 3 ) with
d z
= 2 is possible. Since
here, m f g = 0,1,2,3 ,... and r == Oor 1
values of 4 ,
z
are usually small and d, = 4 ,
these comp onents of the air-gap flux will induce
very weak signal in the line current, thus making
detection difficult in presence of inheren t noise in
the line current spectrum. Combination 1) will
give rise to dynamic eccentricity like components
while combinations 2 ) and 3 ) will give rise to
both static and dynamic eccentricity like
Thus, only R=43 in our case will give rise to static
eccentricity components.
c)
Machine wi th dynamic eccenhic iw
:
With dynamic eccentricity
the
air -gap function
can be expressed as
P =
o + P 2 c o s ( x - w r ) ( '6) components.
Trigonometric manipulations as described in h)
show that the air-gap flux contains components of
111.
Induction Motor A nalysis under
the form
Healthv and Eccentric Conditions usine
(17)
The W inding Function Approach
Modified Windine Function Am roach
..
Thus, in this case also R is giveln by (15) and Analysis of three phase induction machine
R=43 only will give rise to dynam ic eccentricity using w inding function approach
(WFA)
is well
com ponents. documented in literature
[S-91.
However, in
presence of air-gap eccentricity those equations
d)Machine wi th mixed ecce nt r i c i~ ~: are
not valid
as
he average value of the winding
The permeance function for
the
mixed function
no
longer
remains zero[~o].
Using
eccentricity ca se is given by the modified winding function approach
4
+ pz
t ,
I 8 ) MWFA),
the self inductances of the rotor
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7/23/2019 Detection of Rotor Slot and Other Eccentricity Related
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used to describe the
stator and rotor circuit 460V, 60 Hz ac source. The simulations were
equations. Us ing the modified w indin g function carried out assuming dynam om eter load of 8.6
N-
approach expressions for
these
inductances under m. at a slip o f a round
0.029.
The spectral
static and d ynamic eccentricity conditions can be estimates of the line currents have been
developed as described in [ l l ] . These inductance normalized w ith respect to their respective
values have been verified using finite element
[6].
fundamental components.
The
simulation results
In the presen ce of mixed eccentricity the air- presented s how ex cellent m atch with th e
gap can be modeled as
theoretically predicted harmonic com ponents.
ge ,Orm) go al cos -a, cos((-@,) (20) Fig.1 show s the power spectral density
(PSD) of the phase a current
of
healthy
where. ai,a7
are
the amo unt of static an d machines having 44.43and 42 bars. It can
be
, . - . , .
easily seen that the
PSH
s only present for the 44
bar machine.
The
other PSH is missing as it is
ynamic eccentricity respectively, go
the
... .
~~~~ ~ ~
average air-gap and @ a particular position related.
along the stator inner surface. The n, the inverse
air-gap function
,
g 6,,
,
is replaced with 4 , PSH
I
with
o
E
2
g
ao
a3= Ju12+
2a,a2
cose , , + U , (22)
.IW
-120
a2
sinsrm 12W 1250 13W 1350 1400 1450
) (23) 4
nl
+a2 coser,,,
o
6,,,
=
arctan
The inverse air-gap function can be
g
ao
approxima tely expressed as 2 nm
(24) -120
ge-1 4,@m)
A 2
co s ( h , ) 12W 1250
13W
1350
1400 1450
where,
A, =
1
(-F]25)
The
modified equations
for
only static and only
dynamic conditions can be obtained by setting
a, or
a,
equa l to ze ro respectively in (20-25).
Frequency(Hz)
Fig. PSD of phase
a
current ofhealthy machine.
From
top
to
boftom R=44, 3,
42. PSH
is Principle
Slot
Harmonic.
g o J i q i A 2
=
Fig.2 show the PSD for these motors with
38.46 static eccentricity. As predicted, only the
43 bar machine generates an exclusive signal in
IV. Simulation Results
presence of static eccentricity. Same is the case in
Detailed description of the modeling of three the presence of dy namic eccentricity of 20 (Fig.
phase induction motor using the coupled magnetic 3). It is to be noted that only the pole pairs given
approach is given in
[12].
Similar approach was by R -pn+ 1 will be able induc e voltage in the
followed here. The sim ulated ma chine has 36 stator. Hence only one line can Seen in the
stator slots with full pitch 3 phase concentric corresponding to dynamic eccentricity.
winding,
4
poles and a rated Power o f 3
HF.The The PSH
of the 44 bar machine does not change
stator windings are connected in star. Simulation much with either kind ofeccenh icity.
results were obtained using balanced 3 phase,
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7/23/2019 Detection of Rotor Slot and Other Eccentricity Related
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B 100
-120
12
, . .,.
.
120
1ZtW 1250 13W 1350 I4W 1
FrequenWHz)
Fig2 PSD
of
phase 'a' current of static eccentric
machine. From top
to
bottom R=4 4, 43, 42. SEC is
Static Eccentricily Compo,wnt.
(38.46% static and
20%
dynamic). Though the
f v,
omponeqts
are
present for
a11
the
machines, the high frequency components for the
42 bar machine is almost submerged by the noise
floor as was predicted by the theoretical analysis.
The actual line c k n t spect rum of such a
machine is likely to be even worse and hence will
not bo suitable for speed estimation or fault
diagnostic purposes by using the higher order
harmonics.
From the theoretical and simulation analysis
it
is clear that machines
of
the class given by
1
1)
(in our case the 44 bar machine) is better for
sensorless speed estimation purpose as he
PSH
is
always present.
The
machines of the class given by (15) (the
43 bar machine in this case) are better from fault
diagnosis view point as they give different
signatures under static, dynamic and mixed
eccentricity conditions.
a
0
50
150
1m
I . ' 4
I
-120
.
12m 1250 13W 1350 14W 1450
Frequency(H2)
Fig.3 PSD of phase 'a' current
of
&mmic eccentric
machine. From top
to
bottom R=4 4,,43,4 2. DEC is
Dynamic Eccentricity Component.
Frequency H2)
Fig.4 PSD of phase a currenf of mixed eccenlric
machine aroundfu ndamental. From top
to
bottom
R=44, 43, 42.
V.Conclusions
Figs. 4 and show the line current spectnun
The effects of
pole
pair and rotor slot numbers
of the machines around the fundamental and the on the presence of
different harmonics under
PSH
region respectively with mixed eccentricity healthy and eccentric conditions are presented.
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L-rw
120
12m
1250 ISW
1350
i4cm 1450
1201
I ,
12w 1250 1 x 4 1350 14cm
14.50
Frequency(Hz)
Fig.5 PSD
of
phase
a
urrent of mixed eccentric
machine around PSH, SE C. From top
to
bottom
R=44, 3, 42.
2.
3.
4.
5.
6.
7.
8.
presented. Simple and concise theory leading
to equations that describes the necessary
relationship between the pole pair and rotor slot
numbers required for
the
presence of PSH and
eccentricity related harmonics, has been 9.
developed. Detailed simulation results
corroborating the developed theory are included.
These results clearly set up the
norm
in selecting
motors for sensorless speed estimation and
eccentricity related fault diag nosis purposes.
Acknowledgment
This material is based in part upon work
supported by the Texas Advanced Research
Program under Grant
No.
95-PO83 and by the
Department of Energy under Grant
No.
DE-
FG07-981D13641.
pp. 128-135,
New Orleans,
Louisiana, Oct. 5-8,
1997.
J. R Cameron, W. T. Thomson, and A. B. Dow,
Vibration and current monitoring for detecting
air
gap eccentricity in large induction motors,
IEE Proceedings,
pp. 155-163, Vol ,133,
Pt.
B,
N0.3, May 1986.
P. Vas, Parameter Estimation, Condition
Monitoring, and Diagnosis of Electrical Machines,
Clarendron Press, Oxford, 1993.
G ron,
Equivalent circuits of electric machinery,
John Wiley Sons. nc. ,New
York,
1951.
D. G Dorrell, W.T. Thomson and S Roach,
Analysis of airgap
flux
current, vibration signals
as
a function of the combination of static and
dynamic airgap eccentricity in 3-phase induction
mofors,
IEEE Trans. Ind Appln.,
vol. 33, No.1,
pp. 24-34, 1997.
S.
Nandi
,
RajMohan Bharadwaj, H.A. Toliyat,
A.G. Parlos, Performance analysis of a three
phase induction motor under incipient mixed
eccentricity condition, to appear in IEEE
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P. L. Alger, The nature of induction machines,
Gordon and Breach, New
York,
1965.
X. Lou, Y.
Liao, H
A.
Toliyat,
A.
El-Antably,
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,
M.
S.
Arefeen,
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arlos,
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10. H.A. Toliyat, N.
A.
AI-Nuaim, A novel method
for modeling dynamic air-gap eccentricity in
synchronous machines based
on
modified winding
function theory, presented in IEEE-PES Summer
Meeting, July, 1997.
11. S.
Nandi, H.A. Toliyat and A.G. Parlos,
Performance analysis of a single phase Induction
motor under eccentric condition,
IEEE-IAS
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