Content uploaded by Ehsan Abdi
Author content
All content in this area was uploaded by Ehsan Abdi on Mar 24, 2021
Content may be subject to copyright.
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 4, DECEMBER 2015 1611
Optimization of Magnetic Circuit
for Brushless Doubly Fed Machines
Salman Abdi, Ehsan Abdi, Senior Member, IEEE, Ashknaz Oraee, and Richard McMahon
Abstract—This paper presents an optimized design method for
the magnetic circuit of brushless doubly fed machines (BDFMs).
The BDFM is an attractive electrical machine, particularly for
wind power applications, as a replacement for doubly fed slip-ring
generators. This study shows that the conventional design methods
for the BDFM stator and rotor back iron can be modified, leading
to a lighter and smaller machine. The proposed design concepts are
supported by analytical methods, and their practicality is verified
using two-dimensional finite-element modeling and analysis. Two
BDFMs with frame sizes D180 and D400 are considered in this
study.
Index Terms—Brushless doubly fed machine (BDFM), finite-
element analysis (FEA), magnetic circuit, magnetomotive force
(MMF), total harmonic distortion (THD).
I. INTRODUCTION
THE BRUSHLESS doubly fed machine (BDFM) shows
commercial promise as both a variable speed drive and
generator. As a generator, it is particularly attractive for wind
power generation as a replacement for doubly fed slip-ring gen-
erators which was first proposed by [1] and subsequent interest
has been mainly focused on this application [2], [3]. A wind
turbine incorporating a BDFM will have higher reliability and
lower maintenance costs by virtue of the absence of brush gear
[4]. Studies have shown that problems with brush gear are a sig-
nificant issue in wind turbine operation and reliability, and that
the problem will be more severe in machines deployed offshore
where there are stronger winds and accessibility is impaired. In
addition, the BDFM offers a key advantage as a variable speed
drive in that it requires only a fractionally rated converter.
The BDFM has its origins in the self-cascaded machine [5]
and comprises two electrically separate stator windings of dif-
ferent pole numbers, one connected directly to the grid, called
the power winding (PW), and the other supplied from a vari-
able voltage and frequency converter, called the control winding
(CW). A schematic of the BDFM and the way it is connected
to the grid is shown in Fig. 1. The pole numbers are selected
in a way to avoid direct transformer coupling between the sta-
tor windings. The rotor winding is then specially designed to
couple both stator windings. The machine is normally run in
Manuscript received February 16, 2015; revised April 16, 2015; accepted
July 19, 2015. Date of publication September 1, 2015; date of current version
November 20, 2015. Paper no. TEC-00144-2015.
S. Abdi, A. Oraee, and R. McMahon are with the Electrical Engineer-
ing Division, Cambridge University, Cambridge CB3 0FA, U.K. (e-mail:
s.abdi.jalebi@gmail.com; ao331@cam.ac.uk; ram1@eng.cam.ac.uk).
E. Abdi is with Wind Technologies Limited, St. Johns Innovation Park, Cam-
bridge CB4 0WS, U.K. (e-mail: ehsan.abdi@windtechnologies.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEC.2015.2468063
Fig. 1. Schematic of BDFM grid connection.
Fig. 2. 250-kW D400 BDFM (right front) on test bed.
a synchronous mode, with an appropriate controller, in which
the shaft speed is set by the frequencies supplied to the stator
windings [6]. In this mode, the BDFM operates in a similar way
to the doubly fed induction generator with the torque related to
the load angle and a grid side power factor which can be varied
by adjusting the CW voltage [7].
Till date, there have been several attempts to manufacture
large BDFMs, for example, in Brazil with a 75 kW machine [8],
China with the design of a 200 kW machine [9], and the 250
kW BDFM reported by Abdi et al., in [10] and [11], shown in
Fig. 2 on test bed. The latter was built in a frame size D400, as
a stepping-stone toward a megawatt scale BDFM wind turbine.
It is, therefore, desirable to optimize the weight and size of the
machine before a large-scale BDFM is constructed, making it
possible to retrofit the BDFM in existing wind turbines.
It was shown in [12] that the rotor flux density peak values
in different teeth are not equal and vary according to a sinu-
soidal function. A similar pattern exists for the rotor core back
0885-8969 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
1612 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 4, DECEMBER 2015
TAB LE I
SPECIFICATIONS OF THE PROTOTYPE BDFMS
Frame size 400 180
PW pole number 44
PW rated voltage 690 Vat50 Hz (delta) 240 Vat50 Hz (delta)
PW rated current 178 A (line) 9.5A (line)
CW pole number 88
CW rated voltage 620 Vat18 Hz (delta) 172 Vat25 Hz (delta)
CW rated current 73 A (line) 6.8A (line)
Speed range 500 r/min ±36% 750 r/min ±33%
Rated torque 3670 Nm 100 Nm
Rated power 250 kW at 680 r/min 7.8kW at 750 r/min
Efficiency (at full load) >95% >92%
Stack length 0.82 m0.19 m
sections. In this study, it is shown analytically that the rotor field
distribution does not depend on the operating speed and that the
peak flux density in different sections of the rotor including
teeth and back iron sections varies with angular position. There-
fore, some parts of the rotor back iron which do not effectively
contribute in the machine magnetic circuit can be removed to
reduce its weight. Performance results from the new rotor are
obtained in simulation and compared to the original design.
It is also shown in this paper that the proposed analytical
calculation of the BDFM stator back iron by [13] leads to over-
estimation of the back iron depth. This is because it is considered
that the BDFM stator back iron needs to be deep enough to carry
two separate 2p1and 2p2pole magnetic fluxes. However, the
BDFM magnetic field pattern has no obvious polar symmetry
and the motion of the field is not a simple rotation. An alter-
native analytical method is proposed for the stator back iron
calculation and is validated by finite element analysis (FEA),
leading to an optimized value for the stator back iron depth.
II. PROTOTYPE MACHINES CONSIDERED IN THIS STUDY
The specifications of the prototype BDFMs used in this study
are shown in Table I. These BDFMs have four- and eight-pole
stator windings and were constructed in frame sizes of 400 and
180 with the stack length of 820 and 190 mm, respectively. The
stator windings in both machines were connected in delta. The
rotors comprise six sets of nests each consisting concentric loops
[14], the conductors being solid bars with one common end ring
[15]. The magnetic properties for the iron were provided by the
machine manufacturer.
III. ROTOR MAGNETIC CIRCUIT OPTIMIZATION
A. Variation of Peak Flux Density With Angular Position
A study was performed by Creedy [16] on the BDFM mag-
netic field characteristics, which found that the peak values of
the rotor magnetomotive force (MMF) are not equal in all parts
of the rotor and vary sinusoidally with angular positions. This
observation was used later by Liao et al., [17] to design a BDFM
with reluctance rotor. Williamson et al. in [18] found by per-
forming FEA that the movement of the field is not a matter of
simple rotation and does not have a distinguishing N–S pattern.
The characteristic of the BDFM field distribution, generated
by its stator and rotor windings, is represented by (1) using
the flux density distribution for different combinations of f1
and f2[19]. In this equation, the effects of iron saturation,
slotting effects, and finite distributions of windings conductors
are ignored
B(θ, t)= ˆ
B1cos(2πf1t−p1θ)+ ˆ
B2cos(2πf2t−p2θ+α)
(1)
where as follows.
1) ˆ
B1and ˆ
B2are the peak flux density values, in T, of the 2p1-
pole and 2p2-pole flux density waveforms, respectively.
2) f1and f2are the excitation frequencies, in hertz, of the
2p1-pole and 2p2-pole flux density waveforms, respec-
tively.
3) tis time in seconds (s).
4) θis the angular position with a range of [0◦: 360◦)
5) αis the phase angle offset and its value sets the relative
alignment of the 2p1- and 2p2-pole flux density distribu-
tions and it can only be modified by changing the stator
windings excitation phase angles because the relative po-
sitions of the windings are fixed.
The field distribution components in the rotor frame can be
obtained from (1) when
f1=f2.(2)
This is true when the 2p1-pole and 2p2-pole fields distribution
components rotate in the same direction relative to the rotor,
known as “Cumulative BDFM,” which is the preferred BDFM
type [5]. However, the values of f1and f2for the stator field
distribution components are related to the BDFM speed given by
N=60
f1±f2
p(3)
where Nis the shaft speed in r/min and pis either equal to
p1+p2or p1−p2for the cumulative and differential BDFM,
respectively. Therefore, the stator field distribution depends on
the operating speed and that of the rotor does not. Equation
(1) can be reduced to the sum of traveling and standing waves
when f1=f2
B(θ, t)= 2B1cos(2πf1t−p1+p2
2θ)×cos(p2−p1
2θ)
+(B2−B1) cos(2πf1t−p2θ).(4)
The peak value of (4) is not equal for different angular positions
because of the standing wave term, i.e.,
2B1cos p2−p1
2θ×cos 2πf1t−p1+p2
2θ(5)
whose amplitude at different θis not equal and varies si-
nusoidally. It can be therefore concluded that the rotor field
distribution does not depend on the operating speed and, in
addition, the peak flux density in different sections of the rotor
including different rotor teeth and back iron sections varies with
angular position as the absolute value of a sinusoidal function.
This fact is further investigated by performing time-stepping
finite element (FE) simulation in the next section.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
ABDI et al.: OPTIMIZATION OF MAGNETIC CIRCUIT FOR BRUSHLESS DOUBLY FED MACHINES 1613
TAB LE I I
PROTOTYPE BDFMSOPERATING CONDITIONS CONSIDERED IN THIS STUDY
BDFM Frame Size D400 D 180
Torque(Nm) 3600 103
Speed (r/min) 650 750
|V4−pole|(V) 690 240
f4−pole (Hz) 50 50
|V8−pole|(V) 606 175
f8−pole (Hz) 15 25
Fig. 3. Meshing of the rotor back iron; different colors show different distance
from the center.
B. Finite Element Analysis of Rotor Field Distribution
The D400 BDFM is modeled in its synchronous operating
mode, using time-stepping FE simulation taking the material
nonlinear characteristics into account. The operating conditions
are as described in Table II. The rotor is a nested loop design
with six nests, each comprising five loops. As the flux pattern
in a BDFM has 180◦symmetry, only half of the machine cross
section is analyzed [20].
The rotor back iron meshing for a single nest span, i.e., 60◦
is shown in Fig. 3. The elements with the same color have the
same distance from the rotor center. The peak flux density for
each color group is calculated and shown in Fig. 4. As can be
seen, the peak flux density in rotor back iron is highest and
lowest in the center of each nest and between two adjacent
nests, respectively. In addition, the regions closer to the rotor
shaft have much higher variation in the peak flux density. Thus,
larger weight reduction can be achieved closer to the shaft.
C. New Rotor Back Iron Design for the BDFM
A new back iron design has been proposed for the rotor based
on the flux distribution discussed above. The cross section of
the new design for the D400 BDFM rotor is shown in Fig. 5.
There are two region inside the dashed circle, Aand B, which
do not take part in the magnetic circuit, and therefore, their
design is mainly driven by mechanical restrictions and cooling
requirements, which is outside the scope of this paper.
The back iron region outside the dashed circle forms part of
the rotor magnetic circuit and, therefore, its design affects the
machine performance. As can be seen in Fig. 5, the regions
Fig. 4. Peak flux densities in the rotor back iron elements specified with
different numbers in Fig. 3.
Fig. 5. Cross section of the new designed rotor for D400 BDFM.
Fig. 6. Rows of elements between two chunks of the new designed rotor back
iron specified with different green colors.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
1614 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 4, DECEMBER 2015
Fig. 7. Peak flux density values in new designed rotor back iron elements. The row numbers are corresponding to the numbers specified in Fig. 6. (a) Peak flux
density values in rows 1, 2, and 3. (b) Peak flux density values in rows 4, 5, and 6. (c) Peak flux density values in rows 7, 8, and 9. (d) Peak flux density values in
rows 10 and 11.
further from nest center have thinner back iron depth since they
carry less magnetic flux. In the following sections, the magnetic
characteristics and performance of the machine will be assessed
for synchronous and induction modes of operations.
1) Synchronous Mode of Operation: The D400 BDFM is
operating in its synchronous mode of operation and at the
same operating conditions given in Table II. No noticeable
change was found in the values of stator currents and machine
torque when the new back iron design is used. In order to as-
sess the flux density in the new back iron design, the peak flux
density in the elements shown in green in Fig. 6 are plotted in
Fig. 7. As can be seen, the peak flux densities are below 1.8T
which is the design limit.
Fig. 8 shows the modulus of flux density and magnetic flux
lines when the BDFM is run in synchronous mode of operation
with its rated supplied voltages and rated frequencies as given
in Table II. As it can be observed, in the normal operating mode
and under rated conditions, no excessive saturation can be found
in machine iron regions confirming the practicality of the new
design method for rotor back iron.
2) Induction Mode of Operation: The D400 BDFM with
the new rotor design has been modeled in induction mode of
operation using the FEA in order to assess its magnetizing
characteristics. When only one stator winding is supplied and
the unsupplied winding is left open, the BDFM operates as
an induction machine and if the rotor speed is set to the syn-
chronous speed, only the field due to the excited winding will
exist and, hence, the BDFM magnetizing characteristics can be
assessed.
Fig. 9(a) and (b) shows the stator CW and PW magnetizing
characteristics obtained from the FE model. The rated PW and
CW voltages at which PW and CW rated magnetic fields are
achieved at the excited frequencies can be calculated from the
Fig. 8. Numerical computation of flux density magnitude color map with flux
line overlaid; for the D400 BDFM in synchronous mode of operation, obtained
from FE nonlinear analysis.
following equations [21]:
VCW
rms ldωCWNCW
ph kCW
wBCW
rms (6)
VPW
rms ldωPWNPW
ph kPW
wBPW
rms (7)
where lis the machine stack length, dis the mean air gap
diameter, ωis the supply angular frequency, Nph is the number
of turns per phase per pole pair, Kwis the winding factor, and
Brms is the rms value of a stator winding rated magnetic flux.
The VCW
rms and VPW
rms are calculated for D400 BDFM as 356 and
284 V and shown in Fig. 9(a) and (b) by solid horizontal lines,
respectively.
As can be seen from Fig. 9(a), the CW magnetizing charac-
teristic is similar for the original and the new back iron designs.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
ABDI et al.: OPTIMIZATION OF MAGNETIC CIRCUIT FOR BRUSHLESS DOUBLY FED MACHINES 1615
Fig. 9. D400 BDFM magnetizing characteristics with the new designed rotor. (a) The CW is supplied and PW is opened. (b) The PW is supplied and CW is
opened.
Fig. 10. Numerical computation of flux density magnitude color map with
flux line overlaid; for the D400 BDFM in PW induction mode of operation,
obtained from FE nonlinear analysis. The PW voltage and frequency are set to
400 V and 20 Hz, respectively.
From Fig. 9(b), the PW shows similar agreement below its rated
voltage (solid line), but when the voltage is set to a greater
value, heavy saturation occurs. As can be seen from Fig. 10
which shows the modulus of flux density and magnetic flux
lines at 400 V, 20 Hz, since the four-pole span is wider than the
nest span, the flux lines force their way through the narrow parts
of the new back iron which causes heavy saturation. However, in
real operating conditions, in the synchronous mode, none of the
stator PW and CW is required to provide substantially higher
flux density than its rated design value, hence, the saturation
condition mentioned above is not occurred.
IV. STATOR MAGNETIC CIRCUIT OPTIMIZATION
A. BDFM Specific Magnetic Loading
For a typical induction machines, designers use the specific
magnetic loading, based on the electrical steel chosen, to achieve
a balance between the effective use of the iron and undue satura-
tion [22]. However, evaluating the overall effect of the BDFM’s
two stator fields, with rms flux densities of B1and B2on its iron
circuit, is not straightforward as the magnetic flux circulating in
the BDFM is complex [7].
The conventional explanation for the operation of the BDFM
considers two independent flux systems related to the power and
CWs. However, in a BDFM the fluxes are coupled via the rotor
and the flux circulating in the machine is the resultant of MMFs
from the three windings acting on their side of the air gap: The
explicit MMF waves of p1and p2pole pairs of the stator power
and control windings, and the implicit MMF wave from the
rotor winding with components of p1and p2pole pairs plus
space harmonic content depending on the degree of distribution
of the rotor winding. The resulting air gap flux then comprises
components of p1and p2pole pairs plus space harmonics.
The specific magnetic loading is traditionally defined as the
mean absolute flux per pole in the air gap of a machine [23]
¯
B= lim
T→∞
1
TT
0
1
2π2π
0|B(θ)|dθ dt (8)
where B(θ)is the flux density in the air gap, assumed uniform
along the axis of the machine. Ignoring the harmonic fields, the
magnetic field in the air gap of the BDFM may be written using
(1) as
B(θ)=√2B1cos(ω1t−p1θ)+√2B2cos(ω2t−p2θ+α)
(9)
where αis an arbitrary phase offset. It was shown in [7] that the
magnetic loading of the BDFM regardless of the pole number
combinations can be defined as
¯
B=2√2
πB2
1+B2
2.(10)
B. Stator Back Iron Flux Densities
The maximum flux densities in the teeth and core back must
be chosen according to design criterion e.g., to avoid saturation
in the core, to minimize core losses, etc. The number of poles and
magnetic loading determine the depth of the stator back iron in
conventional single-field electrical machines. For an induction
machine, the back iron depth ycis calculated as [23]
yc=√2Bd
2p¯
Bc
=¯
Bπd
4p¯
Bc
(11)
where pis the number of pole pair, ¯
Bis the magnetic loading,
ˆ
Bcis the back iron maximum flux density, ycis the core depth,
and dis the air gap diameter. However, the BDFM has two stator
windings with different pole numbers and, hence, the magnetic
field pattern has no obvious polar symmetry and the motion of
the field is not a simple rotation. As a result, the above equation
cannot be utilized for the BDFM. In [13], the back iron flux
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
1616 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 4, DECEMBER 2015
Fig. 11. PW and CW currents for D180 BDFM in synchronous mode of
operation. The machine is run in full-load (T100 N ·m) and no-load (T0
N·m) conditions.
is modeled as the superposition of two fundamental back iron
fluxes which are related to B1and B2
Bc(θ, t)= √2B1d
2p1yc
cos(ω1t+p1θ)
+√2B2d
2p2yc
cos(ω2t+p2θ+δ)(12)
where δis the phase offset between the two back iron fluxes. As
can be seen from (12), the back iron peak flux density at different
angular locations is not the same because the progression of
the four-pole and eight-pole electrical angles is different. This
brings difficulty in defining a single value as a measure of the
back iron flux density. In [13], the back iron peak flux density
is represented by the average value of maximum back iron flux
densities found around the machine circumference over a period
which is the smallest integer multiple of the period of the two
supplies, given by
¯
Bc=1
TT
0
max(Bc(θ, t)) dx. (13)
The back iron depth of a BDFM is then considered as the sum of
back iron depths needed to accommodate the two fundamental
fluxes
yc=√2B1d
2p1¯
Bc
+√2B2d
2p2¯
Bc
=√2d
2¯
BcB1
p1
+B2
p2.(14)
The FEA of D400 BDFM performed in synchronous mode of
operation shows that the stator back iron depth obtained from
(14) is considerably larger than the required depth to avoid
saturation. In order to investigate potential weight reduction
in the stator back iron, the performance of prototype BDFMs is
analyzed in their synchronous and induction modes of operation
when their stator back iron depth is reduced. The stator PW and
CW currents in synchronous operating mode and in full-load
and no-load conditions are shown in Figs. 11 and 12 for D180
and D400 BDFMs, respectively. The PW and CW voltages and
frequencies are as described in Table II.
As it is clear in Figs. 11 and 12, there are no significant in-
creases in PW and CW currents in both full-load and no-load
conditions when the core back length is reduced up to 30%.At
this level of depth reduction, the maximum increase in a stator
current from its rated value is below 5%. In order to see the
effects of stator back iron depth reduction on machine magneti-
zation characteristics, D400 BDFM is modeled in the induction
Fig. 12. PW and CW currents for D400 BDFM in synchronous mode of
operation. The machine is run in full-load (T3600 N·m) and no-load (T0
N·m) conditions.
mode and results are shown in Fig. 13. The magnetizing cur-
rents remain effectively unchanged in the range of interest, i.e.,
below rated voltage, when 30% reduction is applied to stator
back iron depth. Therefore, from the above, 30% reduction in
the stator back iron depth can be applied without affecting the
performance of the prototype BDFMs, which were designed
based on (14).
C. Stator Time Harmonics Consideration
Stator back iron depth reduction increases not only the stator
current amplitudes, but also their harmonic contents. Total har-
monic distortion (THD) is used to compare the level of harmonic
content in the stator currents when different levels of back iron
reduction is applied compared to the original design [24]. The
THD for a stator current when j%reduction in stator back iron
depth is applied can be defined as
THD(Ij)= 1
Ifund
n
i=1
I2
j(fi)(15)
where fiis the ith harmonic frequency, Ij(fi)is the current
harmonic component at fi, and Ifund is the fundamental com-
ponent of the stator current. The PW and CW currents time
harmonic components for D400 BDFM, in synchronous mode
of operation and at full-load and no-load conditions, are shown
in Figs. 14 and 15, respectively. The THD is also calculated
for PW and CW currents for the above conditions, given in Ta-
ble III. The harmonic levels are investigated when 0%,30%,
40%, and 50% reduction in the back iron depth is applied. The
levels of harmonic components for the cases of less than 30%
depth reduction are very close to the original case and, hence,
are removed from further consideration. In 30% reduction case,
the THD is increased by up to 15% which is occurred in PW
current at no-load condition, compare to 0% depth reduction
case. The jump in THD levels for the cases of 40% and 50%
depth reduction is high enough to make these cases thoroughly
unacceptable.
D. Stator Back Iron Depth Calculation
The core back depth required in an electrical machine is re-
lated to the number of poles and its magnetic loading. However,
in a BDFM, the magnetic field pattern has no clear polar sym-
metry, i.e., it does not have a regular multipole distribution.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
ABDI et al.: OPTIMIZATION OF MAGNETIC CIRCUIT FOR BRUSHLESS DOUBLY FED MACHINES 1617
Fig. 13. D400 BDFM magnetizing characteristics when different stator back iron reductions are applied. (a) CW is supplied and PW is opened. (b) PW is
supplied and CW is opened.
Fig. 14. D400 BDFM stator (a) PW and (b) CW currents harmonic components when the machine operates at synchronous mode and full-load conditions.
Fig. 15. D400 BDFM stator (a) PW and (b) CW currents harmonic components when the machine operates at synchronous mode and no-load conditions.
Furthermore, the motion of the field is not a matter of simple
rotation. In this section, the fundamental air gap MMF waves
are employed to find the total MMF acting across the air gap. It
will then be shown that the response of the BDFM rotor wind-
ing structure to that MMF determines the field pattern in the
machine’s iron region. Hence, the fundamental flux wave of the
BDFM can be defined by the (p1+p2)-pole rotor design, link-
ing stator and rotor irons, rather than the 2p1- and 2p2-poles
stator windings.
Assuming the MMF due to the stator windings PW and CW
as MMFgs, and the MMF due to the reaction of rotor winding
as MMFgr, the combined action of these MMFs sums to the
total air gap MMF wave of MMFgwhich develop the machine
magnetic flux and, therefore, the rotor torque. The stator MMF
acting on the air gap can be given as
MMFgs =kw1N1I1cos(ω1t−p1θ)
+kw2N2I2cos(ω2t−p2θ−γ).(16)
TABLE III
THD CALCULATED FOR PW AND CW CURRENTS IN FULL-LOAD AND
NO-LOAD CONDITIONS WHEN D400 BDFM OPERATES IN
SYNCHRONOUS MODE
THD(%)
Back Iron Depth Reduction IFL
PW IFL
CW INL
PW INL
CW
0% 10.68.618.917.2
30% 10.89.721.719.6
40% 11.81533.136.5
50% 17.419.869.538.9
(16) can be resolved as
MMFgs =4kw1N1I1cos γ.{cos((ω1+ω2)t/2
+(p1+p2)θ/2−γ)}.{cos((ω1−ω2)t/2
+(p1−p2)θ/2−γ)}(17)
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
1618 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 4, DECEMBER 2015
Fig. 16. Magnetic flux in the BDFM in synchronous mode of operation.
where
γ=(π/2)(kw1N1I1/kw2N2I2).(18)
The BDFM rotor winding must have q=p1±p2poles
to meet the BDFM rules. Most prototype BDFMs have q=
p1+p2poles on the rotor, because of difficulties in achieving
acceptable performance with q=p1−p2. MMFgr in this case
must take the form:
MMFgr =kwrNrIr{cos((ω1+ω2)t/2+(p1+p2)θ/2)}.
(19)
From (17), it can be seen that the MMF due to the stator
consists of two waves, one with p1+p2pole pairs rotating at
(ω1+ω2)/(p1+p2)rad/s and another of p1−p2pole pairs
rotating at (ω1−ω2)/(p1−p2)rad/s. If a q=p1+p2pole
rotor design is selected, the rotor reflects this action with an
MMF given by (19). With this design therefore, the MMF acting
across the air gap will be
MMFg=MMFgs −MMFgr
=4kw1N1I1cos γ.{cos((ω1+ω2)t/2
+(p1+p2)θ/2−γ)}.{cos((ω1−ω2)t/2
+(p1−p2)θ/2−γ)}−kwrNrIr
{cos((ω1+ω2)t/2+(p1+p2)θ/2)}.(20)
The result is that the rotor structure can only respond to one
of the two fundamental components of the stator MMF wave
and the rotor suppresses the other stator MMF wave, with con-
sequences to the PW and CW currents, resulting in end winding
and air gap leakage effects. This ensures that the (p1−p2)/2
wave effectively appears in the leakage path only. Therefore,
the fundamental flux wave of the BDFM, can be defined by
TAB LE I V
COMPARISON OF TWO METHODS TO CALCULATE STATOR CORE DEPTH,
PROPOSED BY [13] AND THIS PAPER
ycTerm ycValue Reduction in yc
D180 D400 D180 D400
Old Design √2d
2¯
BcB1
p1+B2
p221.4(mm)54.2(mm)−−
New Design ¯
Bπd
4p¯
Bc,p =p1+p214.5(mm)37.4(mm) 32% 31%
¯
B=2√2
πB2
1+B2
2
TAB LE V
REDUCTION LEVEL IN STATO R,ROTOR AND TOTAL MACHINE WEIGHTS,AND
MACHINE DIAMETER IN PROTOTYPE BDFMSOBTAINED BY THE NEW
DESIGN METHODS
BDFM Machine Stator Rotor Total
Diameter Weight Weight Weight
D180 4.7% 16.2% 20% 17.9%
D400 5% 17.5% 21% 18.8%
the q-pole rotor design, linking stator and rotor, and rotating at
(ω1+ω2)/(p1+p2).
Fig. 16 shows the magnetic flux in the iron circuit of D400
BDFM with four-/eight-pole stator winding configuration. As
can be seen in the figure, the magnetic field linking the stator and
rotor iron has a six-pole pattern as predicted above, although,
all poles’ flux strengths are not identical.
This finding can be used to obtain an optimum design value
for stator core back length if the BDFM is treated as an in-
duction machine with p1+p2poles and the magnetic loading
of 2√2
πB2
1+B2
2. This is summarized in Table IV. The stator
back iron depth in the new design method is about 32% and
31% smaller than in original design method in D180 and D400
BDFMs, respectively. This level of reduction has been shown
in previous sections to an acceptable limit before the machine
is prone to saturation effects. The analytical method proposed
in this section is based on the fundamental air gap MMF waves
including the MMF due to the stator PW and CW, and the MMF
due to the reaction of rotor winding, hence, it is generally valid
for all BDFMs.
It should be noted that due to the excessive field harmonics
existing in the machine main air gap field, mainly caused by
slotting effects, stator winding configurations, and rotor nested
loop structure; larger magnetic loading than the value obtained
by (10) is expected for a BDFM. Consequently, in the machine
design stage and when the back iron depth reduction is applied,
nonlinear FEA should be employed to assess the possibility of
saturation in the machines magnetic circuit, specially in stator
teeth and back iron.
V. CONCLUSION
In this study, new design methods have been investigated
for the BDFM in order to optimize its magnetic circuit, hence,
reducing its size and weight. BDFMs are attractive machines
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
ABDI et al.: OPTIMIZATION OF MAGNETIC CIRCUIT FOR BRUSHLESS DOUBLY FED MACHINES 1619
for wind power generation as a replacement for doubly fed
slip-ring generators. It is, therefore, desirable to optimize the
weight and size of the machine before a large-scale BDFM is
constructed making it possible to retrofit the BDFM in existing
wind turbines.
It has been shown that the peak flux density in the rotor
magnetic circuit including rotor back iron varies with angular
position, independent of the rotor speed. It has, therefore, been
shown that some parts of the rotor back iron do not contribute
in the machine magnetic circuit and, hence, can be removed
to reduce its weight. This finding has been used to design a
new rotor back iron in which p1+p2chunks were removed,
reducing the rotor weight by about 20%.
This study also revealed that the conventional analytical meth-
ods for the BDFM stator back iron depth calculation lead to over-
estimations. It has been shown that the fundamental flux wave
of the BDFM can be defined by the (p1+p2)-pole rotor design,
linking stator and rotor, and rotating at (ω1+ω2)/(p1+p2).
Therefore, an appropriate value for the back iron depth can
be derived when a BDFM is considered as an induction ma-
chine with (p1+p2)pole pairs and the magnetic loading of
2√2
πB2
1+B2
2. This value is 30% smaller than the proposed
value by conventional methods, proved by FEA of two pro-
totype BDFMs. The summary of results from the new design
methods for both rotor and stator are shown in Table V, showing
significant reduction in the total weight and size.
REFERENCES
[1] A. Wallace, R. Spee, and G. Alexander, “Adjustable speed drive and
variable speed generation systems with reduced power converter re-
quirements,” in Proc. ISIE IEEE Int. Symp. Ind. Electron., Jun. 1993,
pp. 503–508.
[2] C. S. Brune, R. Spee, and A. K. Wallace, “Experimental evaluation of a
variable-speed, doubly-fed wind-power generation system,” IEEE Trans.
Ind. Appl., vol. 30, no. 3, pp. 648–655, May/Jun. 1994.
[3] R. A. McMahon, X. Wang, E. Abdi-Jalebi, P. J. Tavner, P. C. Roberts, and
M. Jagiela, “The BDFM as a generator in wind turbines,” in Proc. 12th
Int. Power Electron. Motion Control Conf., Sep. 2006, pp. 1859–1865.
[4] H. Arabian-Hoseynabadi, H. Oraee, and P. J. Tavner, “Wind tur-
bine productivity considering electrical subassembly reliability,” Renew.
Energy, vol. 35, pp. 190–197, 2010.
[5] A. R. W. Broadway and L. Burbridge, “Self-cascaded machine: A low-
speed motor or high frequency brushless alternator,” Proc. IEE, vol. 117,
pp. 1277–1290, 1970.
[6] S. Shao, E. Abdi, and R. McMahon, “Stable operation of the brushless
doubly-fed machine (BDFM),” in Proc. 7th Int. Conf. Power Electron.
Drive Syst., Nov. 2007, pp. 897–902.
[7] R. A. McMahon, P. C. Roberts, X. Wang, and P. J. Tavner, “Performance
of BDFM as generator and motor,” Proc. IEE Elect. Power Appl., vol. 153,
no. 2, pp. 289–299, Mar. 2006.
[8] R. Carlson, H. Voltolini, F. Runcos, P. Kuo-Peng, and N. Baristela, “Per-
formance analysis with power factor compensation of a 75 kw brushless
doubly fed induction generator prototype,” in Proc. IEEE Int. Conf. Elect.
Mach. Drives, 2010, pp. 1502–1507.
[9] H. Liu and L. Xu, “Design and performance analysis of a doubly excited
brushless machine for wind power generator application,” in Proc. IEEE
Int. Symp. Power Electron. Distrib. Gener. Syst., 2010, pp. 597–601.
[10] E. Abdi, R. McMahon, P. Malliband, S. Shao, M. Mathekga, P. Tavner,
S. Abdi, A. Oraee, T. Long, and M. Tatlow, “Performance analysis and
testing of a 250 kw medium-speed brushless doubly fed induction gener-
ator,” IETRenew.PowerGener., vol. 7, no. 6, pp. 631–638, 2013.
[11] S. Abdi, E. Abdi, A. Oraee, and R. McMahon, “Equivalent circuit pa-
rameters for large brushless doubly fed machines,” IEEE Trans. Energy
Convers., vol. 29, no. 6, pp. 706–715, Sep. 2014.
[12] D. Mathekga, “Field distribution characteristics, non-linear characteris-
tics, and thermal model of the brushless doubly fed (induction) machine,”
Ph.D. dissertation, Eng. Dept., Univ. Cambridge, Cambridge, U.K., 2014.
[13] X. Wang, R. A. McMahon, and P. J. Tavner, “Design of the brushless
doubly-fed (induction) machine,” in Proc. IEEE Int. Conf. Elect. Mach.
Drive, May 2007, pp. 1508–1513.
[14] R. McMahon, P. Tavner, E. Abdi, P. Malliband, and D. Barker, “Char-
acterising brushless doubly fed machine rotors,” IET Elect. Power Appl.,
vol. 7, pp. 535–543, 2013.
[15] R. A. McMahon, E. Abdi, P. Malliband, S. Shao, M. E. Mathekga, and P.
J. Tavner, “Design and testing of a 250 kw brushless DFIG,” in Proc. 6th
IET Int. Conf. Power Electron., Mach. Drives, Mar. 2012, pp. 1–6.
[16] F. Creedy, “Some developments in multi-speed cascade induction motors,”
Inst. Elect. Eng. J., vol. 59, pp. 511–537, 1920.
[17] Y. Liao, L. Xu, and L. Zheng, “Design of a doubly fed reluctance mo-
tor for adjustable speed drives,” IEEE Trans. Ind. Appl., vol. 32, no. 5,
pp. 1195–1203, Sep. 1996.
[18] S. Williamson and A. C. Ferreira, “Generalised theory of the brushless
doubly-fed machine—Part 2: Model verification and performance,” Proc.
IEE Elect. Power Appl., vol. 144, pp. 123–129, 1997.
[19] S. Williamson, A. C. Ferreira, and A. K. Wallace, “Generalised theory
of the brushless doubly-fed machine—Part 1: Analysis,” Proc. IEE Elect.
Power Appl., vol. 144, pp. 111–122, 1997.
[20] A. M. Oliveira, P. Kuo-Peng, N. Sadowski, F. Runcos, R. Carlson, and
P. Dular, “Finite-element analysis of a double-winding induction motor
with a special rotor bars topology,” IEEE Trans. Magn., vol. 40, no. 2,
pp. 770–773, Mar. 2004.
[21] E. Abdi, “Modelling and instrumentation of brushless doubly-fed (induc-
tion) machines,” Ph.D. dissertation, Eng. Dept., Univ. Cambridge, Cam-
bridge, U.K., 2006.
[22] M. G. Say, The Performance and Design of Alternating Current Machines,
2nd ed. London, U.K: Sir Isaac Pitman & Sons Ltd., 1948.
[23] J. Pyrhonen, T. Jokinen, and V. Hrabovcova, Design of Rotating Electrical
Machines. Hoboken, NJ, USA: Wiley, 2007.
[24] F. Xiong and X. Wang, “Design of a low-harmonic-content wound rotor
for the brushless doubly fed generator,” IEEE Trans. Energy Convers.,
vol. 29, no. 1, pp. 158–168, Mar. 2014.
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 engi-
neering. He is currently working toward the Ph.D.
degree in electrical machines design and modelling
from Cambridge University, Cambridge, U.K.
His main research interests include electrical ma-
chines and drives for renewable power generation.
Ehsan Abdi (SM’12) received the B.Sc. degree from
the Sharif University of Technology, 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 brush-
less doubly fed induction generator technology for
wind power applications. His main research inter-
ests include electrical machines and drives, renew-
able power generation, and electrical measurements and instrumentation.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
1620 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 4, DECEMBER 2015
Ashknaz Oraee received the B.Eng. degree in elec-
trical engineering from Kings College London, Lon-
don, U.K., 2011. She is currently working toward the
Ph.D. degree in electrical machine design and opti-
mization at Cambridge University, London.
Her research interests include electrical machines
and drives for renewable power generation.
Richard McMahon received the B.A. degree in elec-
trical sciences and the Ph.D. degree in electrical engi-
neering from Cambridge University, London, U.K.,
in 1976 and 1980, respectively.
Following postdoctoral work on semiconductor
device processing, he was appointed as the University
Lecturer in the Department of Electrical Engineering,
Cambridge University, in 1989, and became a Senior
Lecturer in 2000. His research interests include elec-
trical drives, power electronics, and semiconductor
materials.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:30:41 UTC from IEEE Xplore. Restrictions apply.
