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Abstract

The brushless doubly fed machine (BDFM) is an alternative to the doubly fed induction generator, widely used in wind turbines without use of brush gear and slip ring. Rotor design is important for achieving an optimal multi-MW BDFM. To date, nested-loop rotors have been exclusively used in various BDFMs, although they may not be ideal for larger machines. This paper studies, both analytically and experimentally, different rotors for the D160 and D400 frame size BDFMs. An optimization on the rotor equivalent circuit parameters is carried out and verified with different parameter extraction methods. A comparison of the performance of the suggested rotors is also presented.
IEEE TRANSACTION ON ENERGY CONVERSION
1
AbstractThe Brushless Doubly Fed Machine (BDFM) is an
alternative to the doubly fed induction generator, widely used in
wind turbines without use of brush gear and slip ring. Rotor
design is important for achieving an optimal multi MW BDFM.
To date, nested-loop rotors have been exclusively used in various
BDFMs, although they may not be ideal for larger machines. This
paper studies, both analytically and experimentally, different
rotors for the D160 and D400 frame size BDFMs. An
optimization on the rotor equivalent circuit parameters is carried
out and verified with different parameter extraction methods. A
comparison of the performance of the suggested rotors is also
presented.
Index TermsBrushless doubly fed machine (BDFM),
Electrical machines, Nested-loop rotor, Optimization.
I. INTRODUCTION
he BDFM is a promising replacement for the widely used
conventional DFIG since it offers improved reliability and
reduced capital and maintenance costs [1]. It retains the
low-cost advantage of the DFIG system as it only requires a
fractionally rated converter and does not use permanent
magnet materials. The BDFM requires no brushed contact to
the rotor, eliminating a common source of failures making it a
particularly attractive machine for offshore wind turbines.
The BDFM has two non-coupling stator windings with
different pole numbers, p1 and p2, (where p1 and p2 are the
power winding, PW, and control winding, CW, pole pairs
respectively) and a specially designed rotor, coupling both
stator fields. The BDFM rotor is thus an important component
of the machine as its winding carries the mmf induced by both
stator PW and CW, as shown in Figure 1. However, limited
literature is available on the rotor design of the BDFM.
Broadway and Burbidge considered rotor designs in [2] and
a p1+p2 bar cage rotor was identified as the simplest concept
resulting in principal fields with space harmonics. The nested-
Ashknaz Oraee and Richard McMahon are with Electrical Engineering
Division, Cambridge University, 9 JJ Thomson Avenue, Cambridge CB3
0FA, UK (e-mail: ao331@cam.ac.uk; ram1@cam.ac.uk).
Ehsan Abdi and Peter Tavner are with Wind Technologies Ltd, St John’s
Innovation Park, Cambridge CB4 0WS, UK (e-mail:
ehsan.abdi@windtechnologies.com; peter.tavner@durham.ac.uk).
loop rotor was proposed as a development of the p1+p2
fabricated bar cage comprising p1+p2 nests each with multiple
concentric loops. Nested-loop type rotors have been widely
used in recent BDFMs due to their higher torque production
and relatively simple structure [3]. Several series-wound and
nested-loop rotors have also been studied [4]. A comparison
between cage and nested-loop rotors was carried out in [4] and
various prototype rotors assessed experimentally. The design
and behavior of various 6 pole rotor windings for two different
sizes of 4/8 pole BDFMs was analyzed in [5] with predictions
from their equivalent circuits. Gorginpour et al. [6] proposed a
new rotor configuration for the BDFM with equal current in
all loops of a nest. The loops were connected in series
resulting in a reduced rotor leakage inductance but increased
rotor resistance. McMahon et al. [7] characterized BDFM
rotors with the winding factor method, a new analytical
parameter calculation method. It was assumed that there is no
coupling via harmonic fields and the magnetizing inductances
associated with harmonic fields were neglected. Rotor
parameters were then compared for both nested-loop and
series wound rotors. It was shown that the rotor resistance of
the series wound rotor is almost twice that for the nested-loop,
both having turns ratios close to optimum, resulting in lower
torque performance.
Figure 1: Brushless doubly fed machine configuration
Various BDFMs have been reported in the literature, some
specially designed for wind power applications [8,3].
Nevertheless, attempts have been made to manufacture larger
machines, beginning in Brazil with a 75 kW machine by
Stator Power
Winding (PW)
Fractionally
rated
converter
Grid
Rotor Winding BDFM
Stator Control
Winding (CW)
Rotor Design Optimization for the BDFM
Ashknaz Oraee, Student Member, IEEE, Richard McMahon, Ehsan Abdi, Senior Member, IEEE, and
Peter Tavner, Senior Member, IEEE
T
IEEE TRANSACTION ON ENERGY CONVERSION
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Carlson et al. [9] and more recently in China with the design
of a 200 kW machine by Liu and Xu [10].
This paper analytically examines various BDFM rotors,
mainly a cage rotor with p1+p2 bars and a nested-loop rotor by
modeling analysis on the rotor structure and winding
arrangements. The equivalent circuit parameters are extracted
using the winding factor method and the coupled circuit model
and verified using experimental results for existing BDFM
prototypes.
Optimization on the rotor equivalent circuit parameters are
performed with the aim of minimizing rotor resistance and
inductance whilst keeping the turns ratio close to its optimum
value. Design objectives and constraints are investigated
further for MW wind turbines generators. Moreover, rotor
equivalent circuit parameters are extracted for the optimized
design and machine performance is analyzed in this study.
Finally, the effects of loop span and pole pair combination
for different rotor configurations are investigated and a
comparison on the rotor parameters is also presented.
II. BDFM BACKGROUND AND CONFIGURATION
A. BDFM equivalent circuit
The operation of the BDFM can be described by a per-
phase equivalent circuit similar to the equivalent circuits of
two induction machines with interconnected rotors, shown in
Figure 2 [7]. R1 and R2 are the stator resistances, Lm1 and Lm2
are the stator magnetizing inductances and L1 and L2 are the
stator leakage inductances. Parameters are referred to the PW
using the modifier ‘’. The slips are defined as:
s1=
ω
1p1
ω
r
ω
1
(1)
s2=
ω
2p2
ω
r
ω
2
(2)
where ω1 and ω2 are the angular frequencies of the PW and
CW, and ωr is the shaft angular speed. The leakage
inductances cannot be measured directly therefore a simplified
equivalent circuit is proposed in [11].
Furthermore, the rotor can be characterized by the rotor
turns ratio nr, resistance Rr and leakage inductance Lr, the two
latter parameters are also shown in the referred per-phase
equivalent circuit of Figure 2. The rotor leakage inductance is
formed from conventional leakage elements and harmonic
inductance terms from the space harmonics generated by the
rotor. The harmonic nature of the BDFM rotor results in a
higher differential leakage component compared to other
machines.
Figure 2: Referred per phase equivalent circuit of the BDFM
B. Design objectives and constraints
The design objective is to produce a rotor with turns ratio
close to its optimum value. Moreover, for better machine
performance i.e. higher efficiency and controllability, the rotor
leakage inductance and resistance must be at a minimum.
The power rating of the BDFM is given in [4]. However,
experience has shown that using a peak flux density of
2𝐵!"#, as a single field machine, gives an underestimate of
the peak flux density in the airgap which can lead to excessive
saturation in the teeth. An alternative approach is to use a limit
of 𝐵!"# defined as:
(3)
Therefore, for the BDFM, power rating can be calculated
as:
P
c=2
π
d
2
!
"
#$
%
&
2
l
ω
rBsum Jc
p1+p2
p1(1+1
nr
)(1+nr
p2
p1
)
!
"
#
#
#
#
$
%
&
&
&
&
(4)
From the above relation, machine rating is determined by
the stack length l, airgap diameter d, magnetic loading B!"#,
electric loading J, pole pairs and the rotor turns ratio.
With the assumptions of unity power factor and small load
angle operation, the value of nr giving the maximum output
power may be obtained by solving !!!
!!!
=0. The turns ratio for
maximum output power is therefore given by:
nr
opt =p1
p2
( )
1/2
(5)
This is in contrast to the result obtained in [1]. The reason
for this change is using an alternative approach of 𝐵!"# limit
instead of peak flux density of 2𝐵!"#. The actual values of
n!!"#!are 0.71 and 0.63 for the 4/8 BDFM from equation (5)
and [1], respectively. An optimum rotor for the purpose of
maximum machine rating has minimum resistance and leakage
inductance with a turns ratio close to an optimum value.
However, equation (4) does not show the sensitivity of the
maximum machine rating to the variation of rotor turns ratio.
Table I shows how the rotor turns ratio varies within a range
I1R1j1L1
j1Lm1
V1 1
j1Lrj1L2
Rr/s1IrR2
s2
s1I2
s2
s12
VV 1
rV2
rj1Lm2
IEEE TRANSACTION ON ENERGY CONVERSION
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of 2% and 5% of the maximum output power for different pole
pair machines. It is evident from Table I that the maximum
machine rating is not sensitive to the rotor turns ratio. Hence,
the value of the turns ratio being close to its optimal can be
traded off against gaining other performance measures.
Table I: Variation of rotor turns ratio for maximum power
rating
BDFM (p1/p2)
(p1/p2)1/2
2% range
5% range
1/2
0.71
0.53-0.95
0.45-1.12
1/3
0.58
0.43-0.78
0.36-0.93
1/4
0.50
0.37-0.68
0.31-0.81
III. ROTOR MODELING
A. Rotor structure and winding arrangement
Cage rotor with p1+p2 bars is identified as the simplest
concept with an enclosing cage and concentric loops
connected to the end ring from one side. Figure 3a shows
winding configuration of a p1+p2 bar cage rotor with two loops
per nest. The nested-loop rotor was proposed as a
development of the p1+p2 fabricated bar cage comprising
multiple loops in p1+p2 nests. An example of a nested-loop
rotor winding arrangement with three loops per nest is shown
in Figure 3b.
Rotors have been designed and built for the 4/8 D160 and
D400 frame size BDFMs. Specifications of the nested-loop
and p1+p2 bar cage prototype rotors for the D160 and D400
BDFMs are given in Table II and Table III. The existing
prototype rotors are shown in Figure 4.
a. p1+p2 bar cage rotor
b. Nested-loop rotor
Figure 3: BDFM rotor arrangements
Table II: D160 BDFM specifications
PW pole pair
2
Rated speed, rev/min
700
CW pole pair
4
Stator slots
36
Stack length, mm
190
Rotor slots (nested-loop)
24
Rated power, kW
1.4
Rotor slot (bar cage)
24
Rated torque, Nm
40
Table III: D400 BDFM specifications
PW pole pair
2
Rated speed, rev/min
680
CW pole pair
4
Stator slots
72
Stack length, mm
820
Rotor slots (nested-loop)
60
Rated power, kW
250
Rotor nests
6
Rated torque, Nm
3670
a. p1+p2 bar cage rotors
b. Nested-loop rotor
Figure 4: BDFM rotor prototypes
B. Winding factor method
To assess the performance of the overall machine, rotor
parameters are required and they can be extracted using the
equivalent circuit model. The winding factor method predicts
rotor parameters by calculating rotor loop impedances referred
to the stator winding when CW is short-circuited. The
impedance of individual loops, n, in the nested loop rotor
taking into account rotor and stator coupling via space
harmonics is calculated as:
Nest
Rotor coreEnd Ring
End Ring
Nest
21
2
pp +
π
Middle loop Inner loop
Outer loopRotor coreEnd ring
IEEE TRANSACTION ON ENERGY CONVERSION
4
Zeff1
Zeff2
!
Zeffn
!
"
#
#
#
#
#
$
%
&
&
&
&
&
=
kw1,1
I1
'
(
)*
+
,
kw1,2
I2
'
(
)*
+
,
!
kw1,n
In
'
(
)*
+
,
!
"
#
#
#
#
#
#
#
#
#
#
$
%
&
&
&
&
&
&
&
&
&
&
(6)
where Ix is the current flowing through each individual rotor
loop and kw is the winding factor for a single loop in a nest of
a particular loop. Furthermore, the equivalent impedance
presented by one nest for the ith harmonic pole pair is given
by:
Zeq =kw1i
2
Zeffi
i=1
n
"
#
$
$
%
&
'
'
1
(7)
A transformation from p1+p2 phase system of the nested
loop rotor to a three-phase system of the stator is then
required. Note that these parameters are at rotor frequency, i.e.
speed dependent.
The turns ratio for the nested loop rotor is calculated using
mmf balance with one stator winding open circuit, as given in
[6]:
nr=
kww1ikww2i
Zi
kww2i
2
Zi
i=1
n
(8)
To extract rotor parameters for the p1+p2 bar cage rotor, the
phase relationship between coils must be investigated. The
phase change between the peak flux in the coils is:
2
π
p1+p2
2
(9)
For the 4/8 pole BDFM, the phase change between the
loops is !!
! or 120 degrees. The same phase shift will apply
between the emfs induced in the bars of the p1+p2 cage rotor.
The induced emfs in the bars by the p1 pole pair field must
have the same phase sequence as those induced by the p2 pole
pair field for a functional BDFM. Therefore, the currents
flowing in the set of loops or in the bars must have the same
phase delay. For the p1+p2 bar cage rotor, shown in Figure 4a,
to produce the same mmf pattern each slot must contain the
same amp-conductors. Since the two rotors produce identical
mmf patterns, the rotor turns ratio must also be identical.
Hence, in the winding factor method this is equivalent to
multiplying the effective impedance of the outer loop by the
scaling factor of equation (10).
scale =(cos(0) cos( 2
π
p1+p2
2
))2+(sin(0) sin( 2
π
p1+p2
2
))2
(10)
For example, in the 4/8 BDFM this factor evaluates to 3.
The effective loop impedance is then calculated as:
Zeffbarcage =Zeff1:n1zeffn×scale
[ ]
(11)
Therefore, the equivalent impedance presented by one nest
for a bar cage rotor changes to:
Zeq =kw1i
2
Zeffbarcage,i
i=1
n
"
#
$
$
%
&
'
'
1
(12)
For the p1+p2 bar cage rotor, equation (8) must be modified
to:
nr=
kww1ikww2i
Zbarcage,i
kww2i
2
Zbarcage,i
i=1
n
(13)
C. Coupled circuit model
An alternative method of obtaining the equivalent circuit
parameters is the coupled circuit model that relates the stator
and rotor voltages and currents by using a transformation into
complex sequence components. The coupled circuit approach
is based on the model reduction procedure starting from a
coupled circuit model which leads to a single set of dq
parameters for the rotor, independent of the operating speed.
The coupled circuit model of the BDFM for various rotors is
studied in [12].
The rotor inductance matrix, Mrr, is calculated as:
Mrr =Mrr
coupled coils +Mrr
leak
(14)
where M!!!"#$ is the conventional leakage component and
M!!!"#$%&'!!"#$% is the magnetizing and mutual inductance
IEEE TRANSACTION ON ENERGY CONVERSION
5
component. The conventional leakage component of the rotor
inductance arises from leakage effects due to magnetic flux
not linking stator and rotor conductors. The latter consists of
slot, overhang and zig-zag, for each rotor loop and can be
calculated using methods described in [13]. M!!!"#$ for each
nest of the nested loop rotor is given by:
Mrr
leak =
Lloop 0!0
0Lloop " #
# " " 0
0!0Lloop
!
"
#
#
#
#
#
$
%
&
&
&
&
&
(15)
where Lloop is the diagonal matrix of nest leakage
inductances.
The rotor resistance is found by calculating the resistance of
the bar using its cross-sectional area and length. The rotor
resistance matrix is therefore calculated by finding the
resistance of each rotor loop with zero off-diagonal terms
similar to the rotor leakage inductance matrix of equation (15).
The rotor resistance matrix for each nest of the nested loop
rotor is formed using (16):
Rr=
Rloop 0!0
0Rloop " #
# " " 0
0!0Rloop
!
"
#
#
#
#
#
$
%
&
&
&
&
&
(16)
where Rloop is the diagonal matrix of nest resistances. It is
assumed that the resistance matrix has a constant value
throughout the machine operation. For example for nested
loop rotor of the D400 BDFM with 5 rotor loops and 6 nests,
M!!!"#$ and Rr matrices have dimensions of 30×30.
In both nested loop and p1+p2 bar cage rotors the
magnetizing and mutual leakage components dominate over
conventional leakage inductance due to the higher harmonic
components of the magnetic field. Since the p1+p2 bar cage
rotor must also produce the same mmf pattern as the nested
loop rotor, the magnetizing and mutual inductance component
of the rotor leakage inductance remains the same. However,
the resistance and conventional leakage inductance matrices
will change to have the following form:
Mrr
leak =
2Lbar +2Lend Lbar 0Lbar
Lbar 2Lbar +2Lend Lbar 0
0Lbar 2Lbar +2Lend Lbar
Lbar 0Lbar 2Lbar +2Lend
"
#
$
$
$
$
$
%
&
'
'
'
'
'
(17)
Rr=
2Rbar +2Rend Rbar 0Rbar
Rbar 2Rbar +2Rend Rbar 0
0Rbar 2Rbar +2Rend Rbar
Rbar 0Rbar 2Rbar +2Rend
"
#
$
$
$
$
$
%
&
'
'
'
'
'
(18)
where Rbar and Rend are the resistances and Lbar and Lend are
the inductances due to bar and end ring respectively.
The effective turns ratio is defined as the square root of the
fundamental space harmonics leakage terms for the rotor and
the stator. Using the p1+p2 bar cage rotor instead of a nested
loop results in changes in the rotor inductance matrix Mrr,
hence, the turns ratio of a cage rotor is different to a nested
loop rotor. This difference is due to the change in the ratio of
the number of turns of the rotor to stator.
IV. PARAMETER EXTRACTION AND EXPERIMENTS
The two previously mentioned methods of obtaining rotor
parameters are applied to the nested loop rotor of the 4/8 D400
BDFM prototype and the p1+p2 bar cage rotor of the 4/8 D160
BDFM, shown in Figure 4. The D400 nested loop rotor has 5
loops per nest with spans of 54, 42, 30, 18 and 6. The
D160 p1+p2 bar cage rotor has a cage and a central loop with
spans of 60 and 30.
A. Comparison of parameter values
The winding factor method includes coupling via harmonic
fields. However, the rotor-stator coupling has a very small
effect on the rotor parameters and therefore it can be
neglected.
For the coupled circuit model, the stator to rotor couplings
is assumed to take place only via the principal p1+p2 fields but
couplings between rotor loops via harmonic fields are
included. The values obtained from the two models are
compared to those determined experimentally.
The equivalent circuit parameters are measured for each
rotor using the procedure proposed by [11]. The experimental
arrangement and procedure for the extraction of parameters
has been described in [7]. A comparison between coupled
circuit and winding factor method is carried out on the D400
[3] and the D160 BDFMs. The parameters are calculated from
the machine geometry and are given in Table IV.
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Table IV: D160 and D400 experimental parameters
Parameter
D160
D400
R1 (Ω)
2.1
0.0907
R2 (Ω)
1.3
0.667
L1 (mH)
34.7
0.576
Lm1 (mH)
329
89.3
L2 (mH)
27.4
13.1
Lm2 (mH)
83
286
For the speed dependent winding factor method, the
parameters are evaluated at natural speed, defined as the
synchronous speed when the CW is fed with DC. Table V
gives rotor parameters calculated from the coupled circuit
model and the winding factor method. It can be seen that both
methods produce similar rotor equivalent circuit parameters
for the D160 and D400 prototype BDFMs.
Table V: Comparison of complete equivalent circuit rotor
parameters
Machine
Parameter
Coupled Circuit
Winding Factor
D160
Rr (Ω)
xxx
xxx
Lr (mH)
xxx
xxx
nr
xxx
xxx
D400
Rr (Ω)
0.112
0.110
Lr (mH)
5.94
5.86
nr
0.401
0.409
As mentioned previously in section III, rotor parameters
obtained from the winding factor method are speed dependent.
Table VI shows the variance in rotor parameters at different
rotational speeds. It is evident from the table that the effect on
rotor parameters is negligible.
Table VI: Comparison of rotor parameters for different
operating speeds
Speed
(rev/min)
D160
D400
Lr (mH)
Rr (Ω)
Lr (mH)
Rr (Ω)
350
xxx
xxx
0.5863
0.1097
500
xxx
xxx
0.5859
0.1103
650
xxx
xxx
0.5857
0.1109
B. Experimental parameter extraction
Machine parameters were extracted from the D160 and
D400 frame BDFMs by cascade tests using a procedure
described by Roberts et al. [11] in which parameters were
obtained from experimentally determined torque-speed
characteristics in the cascade mode.
Experimentally extracted parameters are for the reduced
form of the equivalent circuit, whereas the parameters
obtained in the previous section are for the full model.
Converting the full model to the reduced model using the
method described in [14] enables a comparison to be made
with the experimentally determined values and those using the
coupled circuit and winding factor methods. The results are
shown in Table VII. Comparison of the results from the
coupled circuit model and the winding factor method given in
Table VII shows good agreement.
Table VII: Reduced form of extracted rotor parameters
Machine
Parameter
Coupled
Circuit
Winding
Factor
Experiment
D160
Rr (Ω)
xxx
xxx
4.05
Lr (mH)
xxx
xx
127.5
nr
xxx
xxx
1.17
D400
Rr (Ω)
0.114
0.114
0.114
Lr (mH)
8.64
8.75
12.5
nr
0.386
0.393
0.380
V. ROTOR DESIGN OPTIMIZATION
A. Optimization parameters and constraints
The rotor design theory is developed with a view to
obtaining equivalent circuit parameters, enabling the
assessment of the overall machine design and performance.
The rotor optimization aims to achieve a rotor with near to
optimum turns ratio while keeping Lr and Rr within the desired
range for maximum efficiency.
As mentioned previously, the rotor leakage inductance
consists of a conventional leakage component as well as space
harmonics manifested as magnetizing and mutual inductances.
The magnetizing inductance is significantly affected by the
variation of loops span as well as the ratio of PW and CW pole
pairs, as can be seen from (19):
Lm=
µ
0
g
ldq
π
Neff
p
!
"
#$
%
&
2
(19)
evaluated for p pole-pair space harmonics. The effective
turns corresponds to each pole and g is the airgap length.
To achieve optimum rotor Lr and Rr in a design, loops are
added progressively to the nested-loop design and loop spans
are adjusted for equal slot spacing. The optimization
procedure is then similarly performed on the p1+p2 bar cage
rotor with arbitrary pitch and placement. This can be achieved
by adding concentric loops progressively adjusting loop spans
by increments of one degree.
The rotor optimization is determined by the design
IEEE TRANSACTION ON ENERGY CONVERSION
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specifications such as the requirements on pull-out torque,
reactive power management and low voltage ride-through
capability. In the optimization problem, turns ratio and skin
effect were taken as design constraints.
B. Effect of loop span
The influence of loop span on the rotor parameters for the
nested loop and the p1+p2 bar cage rotors of Figure 4 is
investigated in this section.
The rotor resistance and conventional leakage component of
the rotor inductance do not vary significantly with changes in
the loop span. However, the harmonic component of the rotor
inductance changes considerably. According to equation (19),
the magnetizing inductance in a machine with uniform airgap
is inversely proportional to the square of the harmonic pole
pairs of the magnetized field. The mmf, magnetizing each
field is determined by the harmonic winding factor of the
winding. The winding factor, 𝑘!, for a single rotor loop in a
nest is given by:
kw=sin(
β
p
2)
sin(wsr p
2)
wsr p
2
(20)
where 𝛽 is the coil span of the loop, p is the harmonic pole
pair and w!" is the slot mouth in radians.
It can be seen from Figure 5 that adding loops to the rotor
nests corresponds to a lower harmonic leakage inductance due
to the addition of steps in the mmf pattern. Furthermore, loops
with larger spans have a much lower referred rotor leakage
inductance and resistance, hence give better results on the
chosen measure of the machine performance. The spans have
been optimized for 5 loops to give minimum rotor parameters,
given in Table VIII. Similar procedure can be applied if a
p1+p2 bar cage rotor is employed in the D400 BDFM to further
optimize rotor parameters. Figure 6 shows the effect of adding
loops to the p1+p2 bar cage resulting in a reduced referred
rotor leakage inductance and resistance. The optimized rotor
loop spans for the p1+p2 bar cage rotor is given in Table IX.
a. Rotor leakage inductance
b. Rotor resistance
Figure 5: D400 nested loop rotor parameter variation with
number of loops
xxx
a. Rotor leakage inductance
xxx
b. Rotor resistance
Figure 6: D400 p1+p2 bar cage rotor parameter variation with
number of loops
12345
0
0.01
0.02
0.03
0.04
Number of Loops
Lrprime (H)
12345
0
0.05
0.1
0.15
0.2
0.25
Number of Loops
Rrprime (ohms)
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Table VIII: Minimum rotor parameters for optimized spans of
nested loop rotor
Nested loop spans
59
53
46
39
28
𝐿!
! (mH)
4.5
𝑅!
!
(Ω)
0.05
𝑛!
xxx
Table IX: Minimum rotor parameters for optimized spans of
p1+p2 bar cage rotor
Nested loop spans
𝑥
𝑥
𝑥
𝑥
𝑥
𝐿!
! (mH)
xxx
𝑅!
!
(Ω)
xxx
𝑛!
xxx
C. Effect of pole pair
Important performance measures such as output torque,
speed and magnetization considerations determine the most
appropriate BDFM pole pair combination. To achieve the
most appropriate BDFM pole combination with minimum
rotor inductance and resistance a comparison is conducted.
The complete equivalent circuit rotor parameters of the nested
loop and p1+p2 bar cage rotor for common BDFM pole pair
combinations are given in Tables X and XI, respectively.
Figure 7 shows that for the same physical dimensions, the 2/6
pole machine gives lowest referred rotor inductance and
resistance for the nested loop rotor. Figure 8 shows the p1+p2
bar cage rotor arrangement giving lower rotor inductance and
resistance on machines with further away poles.
Table X: Referred rotor parameters for nested loop rotor
Machine
2/6
4/8
4/12
8/12
𝐿!
! (mH)
4.2
5.6
8.3
8.6
𝑅!
!
(Ω)
0.07
0.11
0.25
0.29
𝑛!
0.53
0.68
0.52
0.80
Table XI: Referred rotor parameters for p1+p2 bar cage rotor
Machine
2/6
4/8
4/12
8/12
𝐿!
! (mH)
xxx
xxx
xxx
xxx
𝑅!
!
(Ω)
xxx
xxx
xxx
xxx
𝑛!
xxx
xxx
xxx
xxx
a. Rotor leakage inductance
b. Rotor resistance
Figure 7: D400 BDFM nested loop rotor parameter variation
with pole number (p1/p2)
xxx
a. Rotor leakage inductance
xxx
b. Rotor resistance
Figure 8: D400 BDFM p1+p2 bar cage rotor parameter
variation with pole number (p1/p2)
BDFM pole (p1/p2)
8/12 4/12 4/8 2/6
Lrprime (mH)
0
2
4
6
8
10
BDFM pole (p1/p2)
8/12 4/12 4/8 2/6
Rrprime (ohms)
0
0.1
0.2
0.3
0.4
IEEE TRANSACTION ON ENERGY CONVERSION
9
VI. CONCLUSIONS AND DISCUSSION
Modeling analysis and experiments on rotor winding
arrangements of the D160 and D400 BDFM prototypes have
been used to verify different methods of obtaining rotor
equivalent circuit parameters in this investigation. A
comparison between rotor parameters obtained from the
coupled circuit model and the winding factor method was
conducted to prove good agreement with experimental
parameters.
Furthermore, optimization with the aim of minimizing rotor
parameters on the nested loop rotor and the p1+p2 bar cage
rotor was carried out. The effect of loop span and number of
loops was studied for different rotors and it was shown that
increasing the loop span results in lower referred inductance
and resistance. Comparing results from the nested loop and the
cage rotor shows that the cage rotor has significantly lower
referred resistance for the same filled slots. Design studies
were performed to find optimum number of loops per nest for
nested loop and bar cage rotor and parameters were extracted
and compared using different methods. It was shown that
using a cage configuration offers dramatic reduction in rotor
impedance due to highest possible loop span and phasor
addition of currents in rotor bars. Therefore, for a MW scale
BDFM this suggests rotor design of p1+p2 bar cage with loops
offering better machine performance.
In addition, the effect of pole pair combination on rotor
parameters was investigated for the nested loop and p1+p2 bar
cage rotor. It was shown that for machines with further away
pole pairs, the cage rotor offers lower inductance and
resistance, however, as the pole numbers become closer nested
loop type rotors are more favorable.
ACKNOWLEDGMENT
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.
REFERENCES
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[2] Broadway, A.R.W., Burbridge, L.: Self-cascaded machine: a low speed
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... While both NL and cage+NL rotors offer similar advantages, NL rotors are more commonly utilized in BDFMs [4], [15]. The cage+NL rotor is more advantageous when there is a significant disparity between p 1 and p 2 , a situation not observed in the 4/6 BDFM [16]. ...
... These factors are crucial considerations, especially when accounting for mechanical limitations, as they serve as reliable indicators of the machine's power density and harmonic characteristics. According to [16], for the 4/6 BDFM, the rotor referred inductance for the cage+NL was 44.2% higher than for the NL rotor, while the referred rotor resistance increased by only 11.5%. In this pole pair combination, the NL rotor appears better suited for high torque production, corroborated by its selection in [5]. ...
... The loops are often made of solid bars which are shorted at one end of the rotor with a common end ring. Alternative rotor designs, such as the bar cage rotor with internal loops [32], [38], [39], hybrid rotor [40] and wound rotor [41], [42] have also been studied for the BDFM. ...
... This transformation enables the utilisation of rotor current measurements in the parameter estimation process, ensuring unambiguous derivation of all inductances in the 'full' equivalent circuit. While the approach is presented specifically for the nested-loop rotor, it can be extended to other rotor designs, such as the bar cage rotor with internal loops [32], [38]. Table II 8 70 80 90 100 110 120 130 140 150 160 ...
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