![BDFM Stator subjected to magnetic pull II. PROTOTYPE MACHINE CONSIDERED IN THIS STUDY The specifications of the 250 kW BDFM are shown in Table I. The D400 BDFM was constructed as a frame size D400 machine with the stack length of 820 mm. The stator windings were form wound from copper strips. The power winding was rated at 690 V, 178 A, at 50 Hz and the control winding was designed for 620 V at 18 Hz and rated at 73 A. Both stator windings were connected in delta. The rotor comprises six sets of nests each consisting of a number of concentric loops [19], the conductors being solid bars with one common end ring [20]. The magnetic properties for the iron were provided by the machine manufacturer. The D400 BDFM on test bed is shown in Fig. 2. TABLE I Specifications of the 250kW D400 BDFM](https://www.researchgate.net/profile/Ehsan-Abdi-2/publication/300412732/figure/fig3/AS:1004893175304202@1616596611921/BDFM-Stator-subjected-to-magnetic-pull-II-PROTOTYPE-MACHINE-CONSIDERED-IN-THIS-STUDY-The_Q320.jpg)
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Abstract-- This paper studies the unbalanced magnetic pull
(UMP) in the Brushless Doubly-Fed Machine (BDFM).
Analytical study is performed to derive the UMP, then, Finite
element (FE) analysis, which has been verified experimentally,
is used to verify the analytical method. The BDFM with
different types of rotor eccentricities including static and
dynamic eccentricities, are also modeled in FE method and
their resultant UMPs are obtained. The results are compared
with the case at which a perfectly constructed rotor is
considered. The study has been carried out on a prototype
D400 250 kW BDFM.
Index Terms-- Brushless doubly fed machine (BDFM), finite
element (FE) analysis, unbalanced magnetic pull (UMP), rotor
eccentricity, parallel winding.
I. INTRODUCTION
HE brushless doubly fed machine (BDFM) is of
interest as a variable speed generator or drive because
only a fraction of the output power needs to pass through
the power converter. The absence of brush-gear and slip-
rings makes the machine particularly attractive as a wind
turbine generator because brush-gear and slip-ring
problems in the widely used doubly fed induction generator
(DFIG) have been identified as a principal failure mode [1].
Studies indicate that the combination of a BDFM and a
two-stage gearbox in a wind turbine would have excellent
reliability and retain low cost [2].
The authors have successfully demonstrated a small-
scale BDFM in a working 20 kW wind turbine [3] and have
built a 250 kW prototype in a frame size D400 involving
construction and winding techniques appropriate to large
machines that has undergone witnessed tests over its full
load and speed range. Other groups also have reported large
BDFMs, for example in Brazil with a 75 kW machine [4]
The research leading to these results has received funding from the
European Union’s Seventh Framework Program managed by REA
Research Executive Agency (FP7/2007 2013) under Grant Agreement
N.315485.
S. Abdi, and R.A. McMahon are with the Electrical Engineering
Division, Cambridge University, Cambridge, CB3 0FA(e-mail:
s.abdi.jalebi@gmail.com, ram1@eng.cam.ac.uk).
E. Abdi is with Wind Technologies Limited, St. Johns Innovation Park,
Cambridge, CB4 0WS, U.K.(e-mail: ehsan.abdi@windtechnologies.com)
and China with the design of a 200 kW machine [5].
The BDFM is originally a single-frame self-cascaded
induction machine, in which two stator windings of
different pole numbers share the same iron circuit with a
rotor winding of related pole number [6]. The contemporary
BDFM has two stator windings connected to different
frequency supplies, producing different pole number
magneto-motive forces (MMF) with no direct coupling
between them, coupling being through the rotor only. The
separate stator windings facilitate double feeding, with one
winding connected to the grid called the power winding
(PW) and the other via a partially rated power electronic
converter called the control winding (CW), as shown in Fig.
1, without any winding utilisation penalty. The rotor
winding carries an MMF induced by the stator windings
and the rotor and stator windings are coupled by the flux
rotating in the common iron circuit.
Fig. 1. Schematic of BDFM grid connection.
As with all induction type machines, characterized by
relatively small air gaps, the strong magnetic fields across
the air gap exert considerable forces on the iron parts of the
machine. These cause time-varying deflections on the
machine surface which lead to vibration and acoustic noise.
One source of producing deflection is unbalanced magnetic
pull (UMP), which can be mathematically shown as the
interaction between two air gap flux waves with pole-pair
A Study of Rotor Eccentricities Effects on
Brushless Doubly Fed Machines Performance
S. Abdi, E. Abdi, Senior Member, IEEE, and R.A. McMahon
T
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numbers differing by one [7].
Unlike the induction motor, which has a magnetic field
dominated by a single pole number component, the BDFM
has principle field components at two different pole
numbers and the interaction of these two components leads
to more complex deflection patterns than those occur in the
induction machine [8].
Fig. 2. 250 kW D400 BDFM (right front) on test bed.
The presence of rotor eccentricity in practical machines
further modulates the field patterns, exacerbating the
resulting vibration. An eccentric rotor motion occurs when
the rotor axis is not aligned with the axis of the stator bore.
Due to manufacturing tolerances, wear of bearings, and
other reasons, some degree of rotor eccentricity is always
present. Rotor whirling generates an electromagnetic force
also known as UMP that acts between the rotor and stator.
This force can be resolved into two components: the radial
force, acting in the direction of the shortest air gap; and the
tangential force, which is perpendicular to the radial force.
The amplitude and direction of the latter force depend on
the operating condition of the machine, whirling frequency
and rotor radius. Acting roughly in the direction of the
shortest air gap, UMP tends to further increase the
eccentricity magnitude and may cause serious damage to
the machine or even the whole drive. In addition, UMP acts
as a major source of vibration and acoustic noise.
A detailed study of UMP and its resultant displacement
in stator back iron for the BDFM is presented in this paper
taking the effects of rotor eccentricity into account.
Analytical methods are commonly used to study the
electromagnetic forces due to rotor eccentricity, for
example in [9-11], however such methods are not sufficient
when accurate assessment of rotor eccentricity, iron
saturation effects, and stator and rotor slotting is required.
In addition, the complex magnetic field pattern in the
BDFM air gap resulting from the two stator fields with
different pole numbers and frequencies makes an analytical
study difficult.
A 2-D time-stepping Finite Element (FE) modeling
which has been verified experimentally for a D400 250 kW
prototype BDFM is therefore utilized. FE analysis has been
widely used, for example in [12-16], to study UMP and its
resulting vibration in electrical machines. The drawback of
FE modeling is that in the presence of eccentricity, there is
no geometric symmetry, so the whole cross section of the
machine must be modeled. This results in an FE mesh with
a large number of elements as reported by others such as
[17], [18] making the analysis time consuming.
Fig. 3. BDFM Stator subjected to magnetic pull
II. PROTOTYPE MACHINE CONSIDERED IN THIS STUDY
The specifications of the 250 kW BDFM are shown in
Table I. The D400 BDFM was constructed as a frame size
D400 machine with the stack length of 820 mm. The stator
windings were form wound from copper strips. The power
winding was rated at 690 V, 178 A, at 50 Hz and the
control winding was designed for 620 V at 18 Hz and rated
at 73 A. Both stator windings were connected in delta. The
rotor comprises six sets of nests each consisting of a
number of concentric loops [19], the conductors being solid
bars with one common end ring [20]. The magnetic
properties for the iron were provided by the machine
manufacturer. The D400 BDFM on test bed is shown in
Fig. 2.
TABLE I
Specifications of the 250kW D400 BDFM
Frame size 400
PW pole number 4
PW rated voltage 690V at 50 Hz (delta)
PW rated current 178 A (line)
CW pole number 8
CW rated voltage 620 V at 18 Hz (delta)
CW rated current 73 A (line)
Speed range 500 rpm ± 36%
Rated torque 3670 Nm
Rated power 250 kW at 680 rpm
Efficiency (at full load) > 95%
Stack length 0.82 m
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III. MAGNETIC FORCES AND RESULTING DEFLECTION
The radial forces exerted by the air gap magnetic field
on the stator tooth tops are calculated. The effect of
tangential forces on the teeth which ultimately exert torque
to the machine’s shaft are not considered in this analysis
[9]. Fig. 3 shows the schematic of the BDFM stator. The air
gap magnetic field is essentially the superposition of two
field components, one with 2p1 poles, the mean absolute
flux density of rotating at rad/s and another with 2p2
poles, flux density rotating at . The total flux density
as a function of space angle and time is therefore:
Fig. 4. Air gap flux density obtained from (1) and FE modeling when the
rotor is centric and eccentric. Results are obtained from frozen fields at
different times, so the relative phase shift is arbitrary.
Fig. 5. Comparison of displacements in the stator back iron when the rotor
is centric obtained from FE modeling and analytical method
where and are the frequencies of the two stator
supplies, and, and are phase offsets. Any harmonic
field components created by the rotor structure, slotting,
saturation and rotor eccentricity are ignored in (1). In Fig.
4, a typical air gap field for a 4/ 8 pole BDFM at the point
where t and terms are zero is shown in black, i.e. the
smooth line. The field represents magnetic loading of 0.48
T, which is the design value for the 250 kW BDFM. The air
gap flux density obtained from 2-D linear FE modeling
which takes into account all the harmonic components
mentioned above is also shown in Fig. 4 for a centric rotor.
The magnetic field in the air gap exerts an inward force on
the stator tooth top and an equal and opposite outwards
force on the rotor tooth tops. At any point the force is
related to the magnetic field strength by [21]
!
This force will cause the stator back iron to deflect,
which is estimated using 1-dimensional beam theory as
proposed by Alger [22]. The force must be balanced by
elastic deformation of the back iron and the frame. The
force in the air gap may be replaced by an equivalent force
at the central axis of the back iron (the dashed line in Fig.
3) given as a function of the mechanical angle by
"
where is the air gap diameter and is the diameter at
the center of the back iron. The force can be considered as
the superposition of two components, the average force
which causes a small deflection constant in and can be
neglected [8], and the space varying component given by
#
Next, the deflection resulted from this applied force can
be calculated. The force is resisted mainly by the stator
back iron, as the machine’s frame makes negligible
contribution to resisting the magnetic forces due to its much
lower bending stiffness [8]. The shear stress S in the beam
is related to the force by
$
The shear stress is the differential of the bending
moment, M and therefore
%
Finally the deflection, due to the loading is given
by the double integral of the bending moment
&
where is the Young’s Modulus of the material, yc is the
core back depth and l is the stator stack length. In the
present study, the air gap flux density, , is obtained
from FE modeling. Then from (2) to (7), the stator back
iron displacement is obtained. Fig. 5 shows the deflection
obtained from the method described above and from the
experimentally verified analytical method given in [8]. As
can be seen, there is close agreement between the two
results, validating the approach used in this paper to
calculate deflection.
IV. EFFECTS OF ROTOR ECCENTRICITIES
The radial forces acting on the surface of the rotor are
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Centric rotor by FE
Centric rotor by Equation (1)
33% rotor eccentricity by FE
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very large but cancel each other when the rotor axis is
aligned with the stator axis. Similarly, tangential forces are
balanced such that only an axially rotating moment is
produced. If the rotor is eccentric, then UMP occurs. The
phenomenon can be described as an imbalance of the radial
and tangential forces acting on the rotor (or stator) surface
such that a net radial force is developed. This can result in
vibration and noise, and increase the possibility of the stator
and rotor contact.
Fig. 6: Static eccentricity when the rotor center (B) is not aligned with the
stator center (A). The position of B is constant with time and the level of
eccentricity is ’a’.
The UMP due to eccentricity takes two forms: static and
dynamic. Static UMP is caused by the rotor axis being
positioned parallel to, rather than being on, the stator axis
as a result of manufacturing tolerances. In this case, the
rotor is rotating around its own axis, see Fig. 6. Dynamic
UMP occurs when the rotor is precessing about the stator
bore center but not its own center, see Fig. 7. This could be
produced by manufacturing tolerances or rotor whirl near a
critical speed. Different levels of static and dynamic
eccentricities when the rotor is off-center by 33%, 21%, and
7% of the nominal air gap length are considered in this
study. The level of eccentricity referred to in the following
text is always expressed as percentage of the nominal air
gap length.
'
""*
The air-gap magnetic flux density distribution frozen in
time for a perfectly constructed BDFM with uniform air
gap and specifications given in Table I, and the case when
the rotor is statically eccentric by 33%, are shown in Fig. 4.
It is obvious from the figure that some regions of the non-
uniform air gap experience high level of flux density, up to
3 T. As shown in Fig. 8, this high flux density level results
in a deflection which is significantly i.e. 18 times larger
than the case of an ideally constructed machine. Therefore,
the mitigation of UMP is important in the BDFM if
standard manufacturing tolerances are allowed.
Parallel connection of stator coils is widely used to
reduce UMP in electrical machines with non-uniform air
gaps and its effects have been discussed in the literature, for
example in [23]. Effectively, the variation of reluctance due
to uneven air gap length causes circulating currents in the
parallel paths, which improve the air gap flux distribution,
hence reducing the deflection and UMP.
Connecting the stator coils in parallel in a BDFM is not
as straightforward as in other electrical machines. The main
reason is that parallel connection of stator coils may lead
direct coupling of stator windings if not considered
carefully. The large circulating currents produced by direct
coupling will cause significant losses and degrade machine
performance.
Fig. 7: Dynamic eccentricity when the rotor center (B) is not aligned with
the stator center (A). The position of B varies with time, but the level of
eccentricity ’a’ remains constant.
V. RESULTS AND DISCUSSION
The 250 kW BDFM when operating in the synchronous
mode is modeled in FE using a time stepping method. The
FE models are verified by experimental results reported in
[23]. The PW and CW are supplied at 50 Hz and 15 Hz
respectively at their rated voltages. Fig. 9 shows the
magnetic flux in the cross section of the machine. Centric
rotor as well as different static and dynamic eccentric rotors
described in section IV are studied. For each model, the
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33% Eccentric Rotor
Centric Rotor
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space distribution of air gap flux density is extracted. Using
the air gap flux density and (2) to (7), the stator back iron
deflection is derived.
Table II
The rms value of displacement obtained from FE modeling
for static and dynamic eccentricities
Eccentricity Displacement (μm)
7% 21% 33%
Static 43 125 214
Increase 3.5 times 10 .5 times 18 times
Dynamic 29 67 137
Increase 2.5 times 5 times 11.5 times
Different levels and types of rotor eccentricity are
studied in this paper. The back iron displacement for static
(Fig. 6) and dynamic (Fig. 7) eccentricities are shown in
Figs. 10 and 11, respectively. The rms values of
displacement for the above cases are also calculated and
shown in Table II. As it is clear significant increase (up to
18 times in static eccentricity) can be observed compare to
when the rotor is ideally constructed. In addition, the
amount of increase in static eccentricity is higher in all
different levels compare to when the dynamic eccentricity
is existed.
%
&
VI. CONCLUSION
This paper has provided the analysis of the UMP caused
by the fundamental fields in the BDFM which allows its
magnitude to be calculated and compared with those in
conventional machines. The theoretical predictions have
been validated numerically using FE modeling and
analysis. The presence of two components of the magnetic
field in the air gap with different pole numbers give rise to
the UMP magnitude that do not occur in single field
machines like the induction motors. This high amount of
UMP can be a key source of vibration and noises in the
BDFM.
The rotor eccentricity, caused by manufacturing
tolerances, wear of bearings, and other reasons, further
modulates the field patterns and exacerbating the resulting
vibration. Rotor whirling generates an electromagnetic
force and hence UMP that acts between the rotor and stator,
resulting in much more vibration and noise, and increase
the possibility of the stator and rotor contact. Different
types of rotor eccentricities including static and dynamic
eccentricity have been studied in this paper and their effects
are considered in terms of the stator back iron
displacement. The displacement is showing to be scaled up
by several times in different levels of rotor eccentricity,
compare to where the centric rotor is modeled.
Consequently, actions must be taken in order to mitigate
the UMP and its resulting vibration in a BDFM to
acceptable levels. More accurate design and manufacturing
process for the BDFM rotor and its bearings to reduce the
air gap non-uniformity is required. Furthermore, reinforcing
the stator back iron stiffness by increasing the back iron
thickness, and paralleling the stator winding coils can be
employed in order to have a BDFM with less vibration and
noise levels.
VII. REFERENCES
[1] H. Arabian-Hoseynabadi, H. Oraee, and P. J. Tavner, “Wind turbine
productivity considering electrical subassembly reliability,”
Rnewable Energy, no. 35, pp. 190–197, 2010.
[2] P. Tavner, A. Higgins, H. Arabian, H. Long, and Y. Feng, “Using an
FMEA method to compare prospective wind turbine design
reliabilities.” Poland: Proc. European Wind Energy Conf. (EWEC
2010), 2010, pp. 1-10.
[3] T. Logan, J. Warrington, S. Shao, and R. A. McMahon, “Practical
deployment of the brushless doubly-fed machine in a medium scale
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Electronics and Drive Systems, November 2009.
[4] R. Carlson, H. Voltolini, F. Runcos, P. Kuo-Peng, and N. Baristela,
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machine,” IET Electric Power Applications, vol. 7, pp. 1 – 10, 2014.
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VIII. BIOGRAPHIES
Salman Abdi received his BSc degree from Ferdowsi University,
Mashhad, Iran, in 2009 and M.S.c degree from Sharif University of
Technology in 2011, both in electrical engineering. He is currently
working toward the PhD degree at Cambridge University in electrical
machines design and modelling. His main research interests include
electrical machines and drives for renewable power generation.
Ehsan Abdi (SM’ 2012) received his BSc degree from Sharif University
of Technology in 2002 and his MPhil and PhD degrees from Cambridge
University in 2003 and 2006 respectively, all in Electrical Engineering.
Currently, he is the Managing Director of Wind Technologies Ltd where
he has been involved with commercial exploitation of the brushless doubly
fed induction generator technology for wind power applications. His main
research interests include electrical machines and drives, renewable power
generation, and electrical measurements and instrumentation.
Richard McMahon received the degrees of BA (in Electrical Sciences)
and PhD from Cambridge University in electrical engineering, in 1976 and
1980 respectively. Following postdoctoral work on semiconductor device
processing he was appointed University Lecturer in Electrical Engineering
at Cambridge University Engineering department in 1989 and became a
Senior Lecturer in 2000. His research interests include electrical drives,
power electronics and semiconductor materials.
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