MONITORING ECCENTRICITY IN ELECTRICAL MACHINES
Parameter estimation, condition monitoring and diagnosis of electrical machines - Peter Vas OXFORD EDITION 2003
1 Detection of static and dynamic air-gap in induction machines
An induction machine can fail due to air-gap eccentricity can occur due
to shaft deflection, inaccurate positioning of the movement, and so on. If the
air-gap eccentricity is large, then the resulting unbalanced radial forces
(unbalanced magnetic pull) can cause rotor-to-stator rub, and this can result
in the damage of the stator core and stator windings. It is possible to detect
air-gap eccentricity in induction machines by using non-invasive techniques.
For this purpose it is possible to utilize the
monitored stator currents. Such a technique is now described.
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. Thus for
example, static eccentricity can be caused by the ovality of the core or by the
incorrect positioning of the stator or rotor at the commissioning stage.
Assuming that the rotor-shaft assembly is sufficiently stiff, the level of
static eccentricity does not change. Due to the air-gap asymmetry
the stator currents will contain well-defined components, and these can be
detected.
In the case of dynamic air-gap eccentricity, the centre of the rotor is
not at the centre of the rotation and the minimum air-gap rotates with the
rotor. It follows that dynamic eccentricity is time and space dependent (static
eccentricity is only space dependent). For example, dynamic eccentricity can be caused by a bent rotor shaft, wear of bearings,
misalignment of bearings, mechanical resonances at critical speed, and so on.
Due the dynamic eccentricity, side-band components appear around the slot
harmonics in the stator line current frequency spectra.
It can be shown by some changes in the notation that, in general, the
frequency components in the stator currents of an induction machine
which are due to air-gap eccentricity can be obtained as
(1)
where 
is the fundamental stator frequency, 
is any integer, 
is the number of rotor slots, and 
is the eccentricity order number, which for
static eccentricity is 
and for dynamic eccentricity is .
Furthermore 
is the slip, 
is the number of pole-pairs, and is the harmonic of the stator m.m.f time
harmonics (
). It
follows from that 
gives the principal slot harmonic frequency,
and due to the stator time harmonics there are 
frequency components, if follows from eqn (1)
that when the rotor slot number is higher, the resulting frequency components in
the stator currents due to eccentricity are increased.
Thus by detecting a stator line
current, and by using a frequency spectrum analyzer, it is possible to detect
the presence of air-gap asymmetries by utilizing eqn (1). For example, if 
Hz, 
(four-pole machine), 
, and 
, it follows
from eqn (1) that by considering , the
principal slot harmonic frequency is obtained as 
Hz and this is one of the stator current time
harmonics which is present due to static eccentricity. The other frequency
component due to static eccentricity can be obtained
as 
Hz. The two line current stator frequency
components due to dynamic eccentricity are obtained from eqn (1) by considering , and thus 
and
Hz are obtained.
The stator current can be monitored by using a simple clip-on current
transformer around one of the supply cables to the induction machine. It is possible
to utilize fully digital techniques for the monitoring
of eccentricity in induction motors. In such a case
the monitored line current signal is digitized by using an A/D converter and
the spectrum analysis can be performed by using FFT. To minimize leakage, the
sampling rate has to fulfill the Nyquist criteria (sampling to be performed at twice or more than twice the highest
frequency component). For detecting sidebands around principal frequency
components, ZOOM FFT must be used in order to obtain
the desired frequency resolution.
It should be noted that when a
signal is analyzed by conventional FFT techniques, a spectrum is produced,
which covers a range from 
samples are taken, and the resulting spectrum
normally consists of N=400 frequency
lines, which are evenly spaced. Thus the line spacing (resolution) is 
, or in
terms of the sampling frequency 
, it is 
. It follows
that when the analysis is performed in a given frequency range (0-
), in case
of normal FFT techniques, it is possible to increase the resolution only by
increasing the transform size. However, by using ZOOM FFT, the resolution can be increased without increasing the transform size, but
only a correspondingly smaller part of the original frequency range can be
analyzed at a time. In case of conventional FFT for a given transform size,
better resolution can be obtained either by decreasing , and thus
losing the high frequency information, or by increasing the transform size
(which requires more computational time). When using ZOOM FFT, the increased
resolution can be obtained without losing the high
frequency information, or increasing the transform size. If the resolution is
changed by a factor of N, from 
, then the
length of the signal (in the time domain) increases from 
, and the
high resolution spectrum gives a more detailed spectrum compared to the
baseband spectrum. ZOOM FFT can be implemented
(digitally) in two ways. Either the increased resolution is
obtained by shifting the frequency range of interest, and then by using digital
low-pass filtering, or by recording a longer time signal (increasing the
samples by a factor of N) and by transforming it by parts using a smaller
transform.
Finally in should
be noted that instead of monitoring a line current, it is also possible to
detect eccentricity in induction machines by analyzing the locus of the stator
current space phasor.
2 Non-invasive
monitoring of the air-gap of a hydroelectric generator
Air-gap eccentricity can occur on
any size or design of a hydroelectric generator. Eccentricity can cause a rotor
to stator rub without warning on a generator equipped with only conventional
instrumentation. A non-invasive on-line air-gap monitor is
now described, which is suitable for installation on almost any
generator, and which can simultaneously protect the generator against air-gap
failure resulting in rotor-stator contact while providing diagnostic data. The
system is non-sensitive to electromagnetic interference and has a 0.2 percent accuracy over a range of air-gap lengths 2 to
Optical sensors are used to
determine the air-gap length by measuring four time intervals (
) between
pairs of four pulses produced by two sensor beams each time they pass a pair of
retro-reflective strips on the opposing surface.
A LED light
source (which can be a laser diode), couples 50-200 continuous microwatts into
the optical fiber. The latter is
connected with fiber-optic terminals into a four-port splitter/coupler.
The splitter divides the light into two paths to the sensor head. In the sensor
head, two lenses collimate the light exiting from the fibres into two marrow
beams. The beams are projected at a known angle across
the air-gap. These beams will intersect each of the pairs of the
retro-reflective strips mounted on the opposing surfaces (while the rotor is
moving) causing light to be reflected back to the source lenses. The reflected
light-pulses are added together by the
splitter/coupler and the sum is sent to the photodetector diode. This light is
converted into a current by a photodiode and then into a voltage signal by an
electro-optical interface card.
By using , the
air-gap length can be described by the following linear relationship:
(2)