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In this work, an analytical study has been performed on the Brushless Doubly Fed Machine's (BDFM) vibration due to the interaction of its fundamental magnetic fields, exerting bending forces in the back iron. The effects of rotor eccentricity on exacerbating the machine's vibration have been considered by assessing the stator back iron displacement function in the presence of rotor eccentricity. Finite element analysis is carried out for a 250 kW BDFM built in frame size D400 to validate the analytical methods. The stator back iron displacement is determined for an ideally-constructed machine as well as when the rotor has static and dynamic eccentricity. In addition, the prototype BDFM was tested at different operating conditions in order to examine its noise and vibration levels. A set of measurements was conducted to assess the main vibration component frequencies developed by the machine at different rotor speeds. It is shown that the main vibration components are created by bending set-up in the back iron, rotor eccentricity, and the components with time and space harmonic natures. The results obtained from finite element analysis and experimentally agree with the analytical assessment of BDFM vibration.
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1372 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 35, NO. 3, SEPTEMBER 2020
Vibration Analysis of Brushless Doubly Fed
Machines in the Presence of Rotor Eccentricity
Salman Abdi , Ehsan Abdi , Senior Member, IEEE, Hamid Toshani , and Richard McMahon
Abstract—In this work, an analytical study has been performed
on the Brushless Doubly Fed Machine’s (BDFM) vibration due
to the interaction of its fundamental magnetic fields, exerting
bending forces in the back iron. The effects of rotor eccentricity
on exacerbating the machine’s vibration have been considered by
assessing the stator back iron displacement function in the presence
of rotor eccentricity. Finite element analysis is carried out for a
250 kW BDFM built in frame size D400 to validate the analytical
methods. The stator back iron displacement is determined for an
ideally-constructed machine as well as when the rotor has static
and dynamic eccentricity. In addition, the prototype BDFM was
tested at different operating conditions in order to examine its
noise and vibration levels. A set of measurements was conducted
to assess the main vibration component frequencies developed by
the machine at different rotor speeds. It is shown that the main
vibration components are created by bending set-up in the back
iron, rotor eccentricity, and the components with time and space
harmonic natures. The results obtained from finite element analysis
and experimentally agree with the analytical theory of BDFM
vibration.
Index Terms—Brushless Doubly Fed Machine (BDFM),
Vibration analysis, Rotor eccentricity, Beam theory, Time
harmonics, Space harmonics, Rotor speed ripples, Finite element
analysis.
I. INTRODUCTION
THE Brushless Doubly Fed Machine (BDFM) is a variable
speed generator or motor, which in recent years has been
investigated as a possible replacement for the Doubly-Fed In-
duction Generator (DFIG) [1], currently used in majority of
large wind turbines. Similar to the DFIG concept, a BDFM
allows variable speed operation using a variable voltage, variable
frequency (VVVF) converter rated at only a fraction (30–50%)
of the generator rating [2], [3], but it also benefits from the fact
that it does not require any brush gear, eliminating this source
Manuscript received August 16, 2019; revised February 19, 2020; accepted
March 29, 2020. Date of publication April 15, 2020; date of current version
August 20, 2020. Paper no. TEC-00846-2019. (Corresponding author: Salman
Abdi.)
Salman Abdi is with the School of Engineering, University of East Anglia
(UEA), Norwich NR4 7TJ, U.K. (e-mail: s.abdi-jalebi@uea.ac.uk).
Ehsan Abdi is with the Wind Technologies Ltd, St. John’s Innovation Centre,
Cambridge CB4 0WS, U.K. (e-mail: ehsan.abdi@windtechnologies.com).
Hamid Toshani is with the Iran University of Science and Technology, Tehran
1684613114, Iran (e-mail: h_toshani@elec.iust.ac.ir).
Richard McMahon is with the Warwick Manufacturing Group (WMG),
University of Warwick, Coventry CV4 7AL, U.K. (e-mail: r.mcmahon.1@
warwick.ac.uk).
Color versions of one or more of the figures in this article are available online
at https://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEC.2020.2987100
Fig. 1. A schematic of the wind turbine drive train with a BDFM as generator.
of failure and reducing machine maintenance, which is the key
advantage of the machine [4]. The BDFM also shares with the
DFIG the ability to control the reactive power flow through the
machine.
To date, several large BDFMs have been manufactured, for
instance in China with a 200 kW machine [5], in Brazil with
the design of a 75 kW machine [6], and in the UK with the
largest BDFM ever reported. The later was designed and built
in a frame size D400 and tested by the authors and some aspects
of the machine’s performance were reported in [7] and [8].
However, to achieve successful large scale BDFMs for wind
generation application with competitive design and performance
specifications over its counterparts, it is essential to improve
the machine design including the vibration and acoustic noise
characteristics.
The BDFM is operated with one of its windings, called the
power winding (PW), connected directly to the 3-phase grid.
The other winding, called the control winding (CW), may be
either open circuited or short-circuited, thus operating similar
to induction machines, but at two different speeds [9]. However,
the desirable mode of operation for a BDFM is the ‘synchronous
mode’ in which the control winding is connected to a variable
voltage variable frequency converter as shown in Fig. 1 [7]. The
machine’s operating point is set as with a standard synchronous
machine, but by adjusting the control winding frequency this
operation can be at any rotor speed, leading to variable speed
generator or drive operation [10].
The magnetic fields in an induction machine exert significant
forces on the machine’s stator and rotor iron parts. Such forces
can result in displacements and can be observed in the form
of vibration and noise in the machine. In addition, because the
BDFM has two main field components with different frequencies
and pole numbers resulted from two stator windings, the vibra-
tion pattern is more complex in the BDFM than in the induction
machine.
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ABDI et al.: VIBRATION ANALYSIS OF BDFM IN THE PRESENCE OF ROTOR ECCENTRICITY 1373
The presence of rotor eccentricity further increases the
iron displacements and hence vibration levels by essentially
modulating the magnetic fields produced by the stator wind-
ings and thus introducing further harmonic components [11].
Similar problem has been reported for other two-field electri-
cal machines, such as dual stator induction machine [12]. In
this work, an analytical study is conducted on the BDFM’s
modes of vibration caused by the interaction of its fundamental
magnetic fields exerting bending forces in the back iron. The
effects of rotor eccentricity, including both static and dynamic
eccentricities, on exacerbating the machine’s vibration are taken
into account. Finite element (FE) analysis was carried out for
a large-scale BDFM built in frame size D400 to validate the
analytical methods. The FE models were validated by the experi-
mental data obtained from flux measurements of stator teeth and
back iron. The stator back iron displacement is then determined
for an ideally-constructed machine as well as when the rotor
has static and dynamic eccentricity. The prototype BDFM was
also tested at different operating conditions in order to examine
its noise and vibration levels. A set of measurements was con-
ducted to assess the main vibration frequencies at different rotor
speeds.
II. PREVIOUS WORKS ON BDFM VIBRATION
There are only few studies carried out on the vibration analysis
of BDFMs. Logan et al. [13] derived equations for vibration
components magnitudes generated by the PW and CW mag-
netic fields as functions of air-gap flux density, pole numbers
and machine dimensions. In that study however, the effects of
field harmonics and eccentricity were neglected. Abdi et al.
in [14] proposed a new parallel winding design for the stator
PW and CW to mitigate the vibration level in the presence of
rotor eccentricity. Analytical and experimental analysis of the
BDFM vibration were performed in [15] noting that the vibration
amplitude may be decreased if special attention is given to the
choice of loops number in the rotor cage, number of power and
control windings poles and careful design of the stator core.
Dorrell et al. in [16] proposed a number of rotor designs with
suitable stator pole pair combinations in order to improve the
BDFM design by reducing the unbalanced magnetic pull (UMP).
In order to minimise the magnetising currents in induction
type machines, it is essential to construct the machine with
shortest practical air gap. However, these induction machines
experience strong magnetic fields across the air gap, which
can cause considerable forces exerted on the iron parts of the
machine. These can ultimately lead to time-varying displace-
ments on the machine’s surface and hence transmit noise to the
surrounding air.
The UMP is an undesirable characteristic in dual stator ma-
chines by which vibrations and acoustic noise can be produced.
It was shown in [16] that UMP occurs when the stator windings
pole-pair numbers differ by one, so this vibration source can be
avoided by careful stator winding design. However, it was shown
in [11], [14] that another source of vibration known as ‘bending
set-up in the back iron’ occurs in all BDFMs irrespective of
pole-pair number combinations causing displacements of stator
back iron and leading to bending mode vibration in the machine.
It was shown in [13] that the dominant term in the calculation
of displacements in a BDFM is related to the difference of stator
pole pair numbers p1p2and angular frequencies ω1-ω2.This
term essentially represents the existence of two fundamental
field components in the machine generated by stator windings
with different frequencies and pole numbers. It was also shown
that the vibration component with angular frequency ω1-ω2
has the greatest contribution in the vibration of an ideally
constructed BDFM.
As with the induction machine, the eccentricity of the rotor
further modulates the field patterns, and therefore increases
vibration and noise levels in the machine [17]; however, these
effects are not straightforward to precisely determine through
analytical calculations. Nevertheless, it will be shown in this
paper that in BDFMs with static and dynamic eccentricities,
vibration components at angular frequencies of ωr,ω1-ω2and
ω1-ω2±ωrare developed. The effects of eccentricity in the
vibration spectrum are also discussed in section III.
There are other important factors that can contribute to the
vibration and noise in the BDFM, such as the presence of time
harmonics in stator and rotor winding currents, air gap magnetic
field space harmonics, and the dynamics of the mechanical
system (including natural frequencies of various components).
These effects are present in the experimental results shown in
section V, but their analytical investigation is outside the scope
of this paper.
III. MAGNETIC FORCES AND RESULTING DISPLACEMENTS
A. Perfectly Constructed BDFM:
As mentioned in Section II, vibration caused by bending set-
up in the back iron is occurred in BDFMs irrespective of pole
pair number combinations causing stator back iron displacement
and leading to bending mode vibration in the BDFM.
The magnetic field in a BDFM air gap comprises two fun-
damental field components, one with 2p1poles and the mean
absolute flux density of ¯
B1rotating at ω1rad/s, and another
with 2p2poles and ¯
B2flux density rotating at ω2. The net flux
density is essentially the superposition of the above two field
components and can be expressed as a function of time and
space angle [18]:
B(θ, t)=π
2¯
B1cos (p1θ+ω1t+φ1)
+¯
B2cos (p2θ+ω2t+φ2)(1)
where ω1and ω2are the frequencies of the two stator supplies,
and φ1and φ2are phase angles. In (1) any harmonic field
components generated by saturation, rotor structure, slotting,
and rotor eccentricity are ignored.
In [13] a theoretical analysis of vibration patterns in the
BDFM was proposed using beam theory as described for the
induction machine by, for example, Alger [19]. Generally, this
assumes that transverse deflections dominate shear deflections
and that the underlying longitudinal strains are small, so that
changes in curvature can be directly related to the bending mo-
ment on a given cross-section in a linear fashion [13]. Using this
method, the forces exerted by the magnetic field are calculated
and the resulting displacement of the iron is determined from
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1374 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 35, NO. 3, SEPTEMBER 2020
Fig. 2. A block diagram showing how stator back iron deflection is obtained
using air gap flux density waveform [13].
the procedure illustrated in Fig. 2. The magnitudes of the main
vibration components as function of the machine dimensions
and pole numbers can then be obtained. The algebraic expression
of vibration components gives an insight on how they may be
changed to reduce vibration during the design of BDFMs and
therefore having an analytical expression of the vibration pattern
is important for the BDFM design optimisation.
Based on the above approach, the resulting displacement in
the stator back iron from the exerted magnetic force produced
by the air gap magnetic field of (1) can be calculated as [14]:
v(θ, t)=Kv
×
1
(2p1)212¯
B2
1cos (2p1θ+2ω1t+2φ1)
+1
(2p2)212¯
B2
2cos (2p2θ+2ω2t+2φ2))
+2
(p1p2)212¯
B1¯
B2cos ((p1p2)θ+(ω1ω2)t
+(φ1φ2)) + 2
(p1+p2)212¯
B1¯
B2
cos ((p1+p2)θ+(ω1+ω2)t+(φ1+φ2))
(2)
Kvis dependent on material properties and machine geometry
and expressed as [13]:
Kv=3DaD3
cπ2
64Eymμ0y3
c
(3)
where Dais the air gap diameter, Dcis the median diameter of
the stator back iron, ycis the back iron depth and Eym is the
Young’s modulus of the material. Hence, Kvis constant for a
particular BDFM. The full derivation of (2) can be found in [13].
Equation (2) comprises four terms, the first two of which are sin-
gle field components, as would be expected in an induction ma-
chine. The latter two terms are dependent on the difference and
sum of the pole pairs, respectively. Since each term is inversely
proportional to the fourth power of the pole pairs, the term that
includes the difference of pole pairs, i.e. the third term with angu-
lar frequency of ω1ω2, can become relatively large in BDFMs
with a small difference between power and control winding pole
pairs. This will be investigated experimentally in Section V.
B. BDFM With Rotor Eccentricity:
The previous section showed that a perfectly constructed
BDFM experiences vibration modes in addition to those that
would appear in an equivalent induction machine because of the
interaction between two stator fields. In this section, the effect
of rotor eccentricity is studied as a source of noise and vibration.
Rotor and stator eccentricity can essentially introduce addi-
tional components of flux with different pole numbers from those
of the windings. Two types of eccentricity are considered in this
analysis: static eccentricity where the central axes of the stator
bore and the rotor shaft are offset, and dynamic eccentricity
where the centre axis of rotor lamination stack is offset from
the centre axis of rotor shaft. The air gap length as a result of
eccentricity is:
g(θ, t)=g0gscos (θ+φs)gdcos (θ+ωrt+φd)(4)
where g0is the average air gap length, gsand gdare the amplitude
of the static and dynamic eccentricity components respectively,
and ωris the rotor angular velocity. φsand φdare arbitrary angles
relative to the stator reference axis. The magnetic flux density in
the air gap as a result of the surface magnetizing current density
Jm(θ, t)may be derived from the Ampere’s law as:
B(θ, t)=μ0g1(θ, t)Da
2Jm(θ, t)(5)
where the MMF drops across the iron parts of the magnetic
path are neglected. The inverse of air gap function in (4) can be
expressed using Fourier series as:
g1(θ, t)=g1
0+g1
s1cos (θ+φs1)
+g1
d1cos (θ+ωrt+φd1)+...
=g1
0[1 + ds1cos (θ+φs1)
+dd1cos (θ+ωrt+φd1)+...](6)
where ds1and dd1are the static and dynamic eccentricity coef-
ficients, respectively. Given these coefficients are generally very
small, ds1and dd1may be approximated to gs/g0and gd/g0,
respectively [20].
The BDFM magnetizing current density contains p1pole pair
PW and p2pole pair CW components:
Jm(θ, t)= ˆ
Jm1cos (p1θ+ω1t+φ1)
+ˆ
Jm2cos (p2θ+ω2t+φ2)(7)
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ABDI et al.: VIBRATION ANALYSIS OF BDFM IN THE PRESENCE OF ROTOR ECCENTRICITY 1375
TAB L E I
AIR GAP MAGNETIC FLUX DENSITY COMPONENTS IN THE
PRESENCE OF ROTOR ECCENTRICITY
After substituting (6) and (7) into (5), the components of the
flux are shown in Table I, where
¯
Bi=Da
π
g1
0μ0
pi
ˆ
Jmi (8)
Consequently, the displacement function of the stator back
iron in the presence of rotor eccentricity can be derived from the
procedure described in Fig. 2. Essentially, the exerted magnetic
force produced by the air gap magnetic field components of (8)
can be obtained from a series of integrations in order to derive
the resultant back iron displacement. The displacement com-
ponents that arise are summarized in Table II, neglecting those
containing the product of two or more eccentricity coefficients
(e.g. ds1.ds2), which are relatively smaller than other terms.
The magnitudes of displacement components given in Table II
include the constant Kvgiven in (3). The first four components
given in rows 1–4 in Table II are also present in the analysis
of displacements for a perfectly constructed BDFM given in
(2). The remaining components (5–24 in Table II) are created
from eccentricities. As previously noted, the displacement terms
have magnitudes inversely proportional to the fourth power of
the pole-pairs and thus it is those with low number of pole pairs
that make significant contribution in the machine vibration.
In a 2/4 pole pair machine, as for the D400 BDFM, there are
six components with a single pole pair waveform (i.e. comps. 5,
6, 12, 15, 16 and 22 in Table II) in addition to the main displace-
ment component present in the ideal machine analysis (comp. 3
in Table II), that together give the following approximation for
the back iron displacement:
v(θ, t)=2KV¯
B1¯
B2cos (θ+(ω1ω2)t+(φ1φ2))
+ds1¯
B2
1+¯
B2
2cos (θ+φs1)
+ds1
16 ¯
B1¯
B2cos(2θ+(ω1ω2)t+(φ1φ2+φs1))
+dd1¯
B2
1+¯
B2
2cos (θ+ωrt+φd1)
+dd1
16 ¯
B1¯
B2cos (2θ+(ω1ω2+ωr)t
+(φ1φ2+φd1))] (9)
TAB L E I I
DISPLACEMENTS COMPONENTS IN THE PRESENCE OF ROTOR ECCENTRICITY
From (9) it can be realised that the most important vibration
frequency components expected in the BDFM vibration spec-
trum are:ωr,ω1ω2and ω1ω2±ωr. It should be noted that
the vibration component with frequency signature of ω1ω2
rad/s is created due to both bending set-up in the back iron (as
shown in (2)) and eccentricity (as shown in (9)). Hence, the
two mechanisms superimpose but not necessarily with the same
phase over the operating speed range.
IV. FINITE ELEMENT ANALYSIS OF BDFM MAGNETIC FIELDS
Table III gives details of the 250 kW BDFM used in this study.
Both stator PW and CW are connected in delta. The rotor is a
nested-loop design comprising six nests, each with five loops
[21]. All rotor loops are terminated with a common end-ring at
one end only [22].
Finite element (FE) analysis has been widely used, for exam-
ple in [23]–[26] to study UMP and its resulting vibration in elec-
trical machines. Finite element analysis of the 250 kW BDFM
was performed using the commercial software application EFFE
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1376 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 35, NO. 3, SEPTEMBER 2020
TABLE III
SPECIFICATIONS OF THE 250 KW D400 BDFM
[26]. The model was solved as a voltage-fed problem so that
simulation results can be compared directly to experimental
measurements. A 2-D analysis was performed by assuming that
the machine is infinitely long in the direction parallel to the
shaft to reduce the computational time. The end region leakage
effects were computed using the method presented in [27] and
incorporated into the analysis using lumped parameters. The
modelling was performed using the time-stepping method for
accurate analysis.
In the synchronous mode of operation, the PW is connected
directly to the grid and the CW is supplied with variable voltage
at variable frequency from a converter. The implementation of
BDFM synchronous operation in FE is particularly challenging
because the CW excitation voltage required to set a specific load
condition cannot be predetermined as the machine is not stable in
open loop. Therefore, a closed-loop controller was implemented
with details described in [28]. The PW and CW voltage sources
used in simulations only included the fundamental frequencies
and did not contain time harmonics.
In order to validate the numerical computation of magnetic
fields in FE models, the stator back iron and teeth flux densi-
ties obtained by FE analysis have been compared to the same
parameters measured using flux search coils. The search coils
were fitted into the BDFM as shown in Fig. 3 and the results are
shown in Fig. 4. The close agreement validates the use of FE
modelling in obtaining the air gap magnetic fields in the BDFM,
which may be subsequently used for the analysis of stator back
iron displacement.
The 250 kW BDFM is modelled in the synchronous operating
mode for the two cases of ideally constructed rotor as well as
when the rotor has different degrees of static and dynamic ec-
centricity. The flux density in the air gap is then obtained by post
processing of the FE models. The stator back iron displacement
can be subsequently determined using the procedure described in
Fig. 2. Different levels and types of rotor eccentricity are studied.
The back iron displacement for an ideally-constructed rotor as
well as when the rotor has static and dynamic eccentricity are
shown in Figs. 5 and 6.
Fig. 3. Stator flux search coils fitted on to (a) stator tooth (b) stator back iron
Fig. 4. Measured and predicted flux density in (a) stator tooth (b) stator back
iron
From Figs. 5 and 6 the following can be concluded:
rIn both static and dynamic eccentricity cases, the back iron
displacement is substantially larger than when the rotor is
ideally constructed.
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ABDI et al.: VIBRATION ANALYSIS OF BDFM IN THE PRESENCE OF ROTOR ECCENTRICITY 1377
Fig. 5. Displacement in the stator back iron for ideally-constructed machine
and different levels of static eccentricity.
Fig. 6. Displacement in the stator back iron for ideally-constructed machine
and different levels of dynamic eccentricity.
rFor the same levels of eccentricity, the severity of back
iron displacement caused by rotor dynamic eccentricity is
considerably higher than that of static eccentricity.
rFor the case of ideally constructed rotor, the back iron
displacement (green lines in Figs. 5 and 6) have 4-pole
(p1p2pole-pair) space distribution. This corresponds to
the 3rd component of displacement function with angular
frequency of ω1ω2, as predicted in Table II. It confirms
the analysis of section III-A that the dominant component
of back iron displacement for the ideally-constructed ma-
chine has ω1ω2angular frequency and p1p2pole pair
number.
rThe displacement for all other cases with static and dy-
namic eccentricity, have 2-pole (p1–p
2+1pole-pair)
space distribution. This confirms the findings of section III-
B that in a BDFM with non-ideal rotor construction, the
dominant displacement components have angular veloci-
ties of ωr,ω1ω2and ω1ω2±ωr,which correspond
to the 11th, 12th, 21st and 22nd components predicted in
Table II, all with pole pair number equal to |p1–p
2±1|.
Fig. 7. 250 kW BDFM and the load machine on test rig.
Fig. 8. Location of accelerometers on the BDFM test bed for vibration
measurements.
V. E XPERIMENTAL ASSESSMENT OF VIBRATION IN THE BDFM
A. Vibration Test Set-Up
The 250 kW BDFM is shown in Fig. 7 on the experimental
rig. The machine’s control system includes grid-side inverter
(GSI) and machine-side inverter (MSI). The GSI was developed
with an embedded control system to stabilize the DC-link and
synchronize to the 690 V grid voltage. The MSI was developed
to control the PW real and reactive power using a Speedgoat
controller [7].
A set of measurements was conducted on the machine to
validate the vibration analysis carried out in the previous section.
The machine was instrumented with a number of accelerometers
positioned on the bedplate, frame and terminal box of the ma-
chine, as shown in Fig. 8, and was operated in the synchronous
mode of operation at a speed range of 320–620 rpm. Vibration
was measured using Bruel & Kjaer 2260 instruments from
the drive end of the machine with the specifications shown in
Table IV. The accelerometers’ signals were digitised using a
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1378 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 35, NO. 3, SEPTEMBER 2020
TAB L E I V
SPECIFICATIONS OF THE VIBRATION MEASUREMENT INSTRUMENTS
Fig. 9. Vibration velocity over the BDFM operating speed range.
data logger controlled by LabVIEW software. The fast Fourier
transform of the signals were also computed using a built-in
algorithm. The measured averaged root mean square (rms)vi-
bration velocity over a range of operating speeds is shown in
Fig. 9.
As can be seen from Fig. 9, for the rotor speed between
540-680 rpm, the rms values of vibration velocity remain within
the range of 2.87–4.01 mm/s. This is close to the standard limit,
based on the ISO 10816 standard for vibration assessment of
a 250 kW class M electrical machine. Nevertheless, for the
operating speed range of 320–520 rpm, the measured vibration
rms values are between 5.15 and 12.70 mm/s, which is above
the tolerable limit.
B. Measured Vibration Spectra
The spectra of vibration amplitudes in dB have been obtained
for the 250 kW BDFM at the operating conditions where the
machine’s vibration velocity is at its highest level. Fig. 10(a)
shows the vibration spectrum at the rotor speed of 440 rpm
and the PW and CW supplied frequencies of 50 Hz and 6 Hz,
respectively. The supply voltages were set to give magnetic
loadings of 0.32 and 0.38 T for the PW and CW, respectively,
in order to achieve the nominal field strength in the machine air
gap.
Air gap non-uniformity was present in the prototype 250 kW
BDFM, partly due to the rotor shape not being purely circular
and partly due to the presence of rotor eccentricity. Experimental
measurements showed that the rotor axis was off centre by
approximately 15% of the air gap length. In addition, rotor
whirling was observed when experimental tests were carried
Fig. 10. Vibration amplitude spectrum at the machine’s drive end at 0 kW
(no-load), 100 kVAR, (a) 440 rpm, (b) 320 rpm.
out at rated operating conditions, showing the presence of both
static and dynamic eccentricities at normal operation.
Fig. 10(a) shows three distinct peaks at 56, 48.7 and 63.3
Hz frequencies. The 56 Hz vibration component is consistent
with ω1ω2component predicted in section III-A as the main
vibration component caused by the bending set-up in the back
iron of the ideally constructed BDFM. It is also consistent with
ω1ω2component predicted in section III-B as a result of
rotor eccentricity. The other two frequencies, i.e. 48.7 Hz and
63.3 Hz, however, are consistent with ω1ω2±ωrvibration
components created solely by the presence of rotor eccentricity.
It is important to note that there is always a degree of rotor
eccentricity present in a real machine due to manufacturing
imperfection, which may cause significant vibration and noise
issues in the BDFM as discussed in Section II.
Fig. 10(b) shows the vibration spectrum at the rotor speed of
320 rpm and the PW and CW supplied frequencies of 50 Hz and
18 Hz, respectively. The main peak vibration frequencies are 68,
63 and 73.3 Hz, which correspond to ω1ω2and ω1ω2±
ωr, similar to what was found at 440 rpm.
The time and space harmonic contents in the BDFM torque,
PW and CW currents, and active powers and their contribution
in the machine vibration were studied in [29], [30]. Fig. 10 also
demonstrates these harmonic-driven sources of vibration, which
are denoted with different colours and numbers. Table V shows
different vibration mechanisms and their associated numbers
and colours. These additional vibration components generated
from harmonic effects can exacerbate the vibration and noise
levels and therefore, special design considerations such as rotor
design optimisation and use of magnetic wedges in rotor slot
openings need to be employed to reduce the harmonic effects.
The vibration spectrum for a range of synchronous speeds was
also obtained where the output active and reactive power were
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ABDI et al.: VIBRATION ANALYSIS OF BDFM IN THE PRESENCE OF ROTOR ECCENTRICITY 1379
TAB L E V
THE ASSOCIATED COLOR TO EAC H SOURCE OF VIBRATION
FREQUENCY IN FIG.10
Fig. 11. Dominant frequencies in the vibration spectrum. For each speed the
diamond, triangle and rectangle are the frequencies of largest, second largest
and third largest components, respectively. The solid, dashed and dotted lines
correspond to ω1ω2,ω1ω2+ωr,andω1ω2ωr, respectively.
kept constant at 230 kW and 98 kVAR, respectively. For each
speed, the three dominant frequencies with largest magnitudes
were determined and are shown in Fig. 11. The solid, dashed and
dotted lines in Fig. 11 correspond to ω1ω2,ω1ω2+ωr,
and ω1ω2ωr,respectively.
As can be seen, the largest displacement component at most
speeds has the frequency of ω1ω2, which corresponds to the
source of vibration that includes the effects of both back iron
bending set-up and rotor eccentricity. There are also reasonable
correlations between the frequencies of second and third largest
displacement components and the predicted rotor eccentricity
vibration sources with ω1ω2+ωr,and ω1ω2ωr,re-
spectively. However, the variation in the frequency of second
and third largest components seen in Fig. 11 suggests that other
vibration mechanisms are taking part in the BDFM’s vibration
and noise patterns. These frequencies correspond mostly to the
vibration mechanisms noted in Table V.
VI. CONCLUSION
In this paper, various vibration mechanisms present in the
BDFM’s main mode of operation, the synchronous mode, were
studied. It was shown that one of the main sources of vibration is
created by bending set-up in the back iron, which occurs in any
BDFM irrespective of pole pair combinations, causing a stator
back iron displacement. It was also shown analytically that the
displacement component with an angular frequency and pole-
pair number of ω1ω2and p1p2(e.g. 4-pole displacement
component for a 4/8 pole prototype BDFM) can become very
large in BDFMs with a small difference between power and
control winding pole pairs and hence is the dominant vibration
component in an ideally-constructed BDFM.
In addition, it was shown that the rotor eccentricity introduces
additional components of back iron displacement and results
in BDFM vibration with the dominant frequencies of ω1ω2
and ω1ω2±ωrand pole-pair number |p1–p
2±1| (e.g.
2-pole displacement component for a 4/8 pole BDFM). The
analytical theory was validated through deriving the back iron
displacement in a 250 kW BDFM from the air gap magnetic
fields obtained from the finite element analysis.
A set of vibration tests were carried out on the prototype
BDFM in order to assess the machine’s vibration pattern. The
experimental results confirmed the dominant vibration frequen-
cies of ω1ω2and ω1ω2±ωras predicted analytically for
an eccentric BDFM. A degree of air gap non-uniformity was
present in the prototype machine due to the uneven air gap length
caused by manufacturing tolerance as well as rotor eccentricity.
Therefore, it is essential to implement appropriate design
considerations in order to mitigate the vibration and noise levels
before a large-scale BDFM is constructed for wind turbine
drive trains. These may include increasing the air gap length,
introduction of damping in the rotor winding, introduction of
damping in the stator winding with parallel paths, and isolation
of the stator frame from the stator core.
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Salman Abdi received the B.Sc. degree from Fer-
dowsi University, Mashhad, Iran, in 2009, and the
M.Sc. degree from the Sharif University of Technol-
ogy, Tehran, Iran, in 2011, both in electrical engineer-
ing. He then completed the Ph.D. degree in electrical
machines design and modeling from Cambridge Uni-
versity, Cambridge, U.K, in 2015. He is currently an
Assistant Professor in Electrical Engineering at the
University of East Anglia (UEA), Norwich, UK. His
main research interests include electrical machines
and drives for renewable power generation and auto-
motive applications.
Ehsan Abdi (Senior Member, IEEE) received the
B.Sc. degree from the Sharif University of Technol-
ogy, Tehran, Iran, in 2002, and the M.Phil. and Ph.D.
degrees, from Cambridge University, Cambridge,
U.K., in 2003 and 2006, respectively, all in electrical
engineering. He is currently the Managing Director
of Wind Technologies Ltd., Cambridge, where he has
been involved with commercial exploitation of the
brushless doubly fed induction generator technology
for wind power applications. He became a Senior
Member of the IEEE in 2012. His main research
interests include electrical machines and drives, renewable power generation,
and electrical measurements and instrumentation.
Hamid Toshani received the B.S. degree in electrical
engineering from the Ferdowsi University of Mash-
had, Mashhad, Iran, in 2009 and the M.Sc. degree in
control engineering from Iran University of Science
and Technology, Tehran, Iran, in 2012. He is currently
a Ph.D. Student majoring in Control Engineering in
the Department of Electrical Engineering, Iran Uni-
versity of Science and Technology, Tehran, Iran. His
research interests include robust control, optimiza-
tion techniques and intelligent approaches in control
systems.
Richard McMahon received the B.A. degree in
electrical sciences and the Ph.D. degree from the
University of Cambridge, Cambridge, U.K., in 1976
and 1980, respectively. Following post-doctoral work
on semiconductor device processing, he became a
University Lecturer in electrical engineering with the
Engineering Department, University of Cambridge,
in 1989, where he was a Senior Lecturer in 2000. In
2016, he joined the Warwick Manufacturing Group
(WMG), University of Warwick, Coventry, U.K., as
a Professor of power electronics. His current re-
search includes electrical machines, power electronics and the electrification of
transport.
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... So, the rotor should produce two magnetic fields with different pole pairs at the same time. Existing rotor types include reluctance rotor, nested-loop cage rotor and wound rotor [13][14][15][16][17][18][19][20][21]. These structures can modulate the magnetic motive force (MMF) of the rotor by adjusting the rotor reluctance or regulating the rotor currents. ...
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