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Published in IET Electric Power Applications
Received on 28th March 2012
Revised on 7th July 2012
doi: 10.1049/iet-epa.2012.0101

ISSN 1751-8660

Performance of the brushless doubly-fed machine

under normal and fault conditions
S. Tohidi1 M.R. Zolghadri1 H. Oraee1 P. Tavner2 E. Abdi3 T. Logan3
1
Department of Electrical Engineering, Sharif University of Technology, Azadi Street, Tehran 11365-8639, Iran
2
School of Engineering and Computing Sciences, Durham University, Durham DH1 3LE, UK
3
Electrical Engineering Division, University of Cambridge, 9 JJ Thomson Avenue, Cambridge CB3 0FA, UK
E-mail: stohidi@ee.sharif.ir

Abstract: In this study, the steady-state operation of the brushless doubly-fed machine (BDFM) in various modes is physically
elaborated and the active power flow and torque analysis are presented for each operating mode alongside confirmatory
experimental results on a 4/8 pole D160 size machine. The machine behaviour in asynchronous operating modes is described
similar to the conventional induction machines with corresponding pole numbers. Moreover, its performance in synchronous
mode is shown to be similar to synchronous machines. On the basis of the above, the BDFM performance is further analysed
under two possible fault conditions: first, a controller or converter fault leading to loss of synchronism is considered, where the
asynchronous, double-cascade mode must be introduced to describe the machine performance. Second, a voltage dip at the
mains terminal is considered. The simplistic behaviour of the BDFM during this fault, including the operation of a converter
protective crowbar, is investigated using steady-state torque–speed curves to obtain a general view of the machine performance.

Nomenclature of the DFIG is its use of slip-rings and brush-gear, which

reduces reliability and increases maintenance costs [1].
p number of pole pairs To overcome this problem, the brushless doubly-fed
Nr number of rotor circuits (nests) machine (BDFM) has been proposed as an alternative to the
v angular speed DFIG. The BDFM has all the DFIG benefits, but without
slip-rings and brush-gear. Hence, it has potential future use
R winding resistance in wind generation [2], particularly offshore and pump
L winding inductance drives [3].
n efficient number of winding turns Several papers have been published recently on BDFM
steady-state and dynamic modelling. In [4], a dq equivalent
s rotor slip circuit for a 6/2 pole BDFM is presented in the rotor
cosw power factor reference frame model. In [5], the authors develop a
Z, z synchronous impedance amplitude and phase steady-state rotor reference frame model. Steady-state
1, 2, r subscripts of the PW, CW and rotor modelling and a brief power balance analysis are given in
[6]. In [7], a unified reference frame dq model for the
l, m subscripts of leakage and magnetising BDFM is proposed and experimentally verified. Another
p superscript of the parameters referred to the PW steady-state model is suggested and experimentally
side validated in [8]. Using this model, BDFM performance is
analysed and some relations for the active and reactive
powers of BDFM are extracted in [9].
1 Introduction The above studies investigate the modelling of the BDFM,
but there is a need for detailed understanding of the machine’s
The penetration of wind power into electrical grids worldwide different operating modes. This paper fills this gap
has increased significantly in the last five years. Variable by physical explanation of machine performance in all
speed wind turbines (VSWT) are used in most large its different modes, especially the synchronous mode.
modern wind farms, because of their high wind energy In addition, a new mode, called double-cascade, is
transfer characteristics. Among the various types of introduced to account for when synchronism is lost.
technologies used for VSWTs, the doubly fed induction This detailed steady-state understanding can then be used
generator (DFIG) is the most common. With its to rationalise the BDFM’s behaviour under terminal voltage
fractionally-rated converter, it delivers flexible power factor dip conditions, which leads to the operation of the converter
and rotor speed control at relatively low cost. The drawback protection crowbar. Although steady-state models cannot

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predict machine’s transient behaviour precisely, they can be loops in the nested loop rotor (Fig. 2). However, other
used to obtain a general view demonstrating results of possible rotor arrangements are currently under investigation.
crowbar operation. In this situation, the p1 or p2 pole pair components of the
RW MMF will rotate at identical speeds but in opposite
directions with respect to the rotor [14]. The p1 and p2 pole
2 The BDFM pair MMFs produced in the air gap with Nr rotor circuits is
The BDFM has two, three-phase windings in its stator. Their tabulated in Table 1.
pole numbers are arranged to be different to avoid Owing to its special design, the RW cannot be considered
direct-coupling [10] purely sinusoidally space-distributed, and therefore contains
high-order space harmonics. Nevertheless, as PW and CW
are assumed to be sinusoidally space-distributed, the
p1 = p2 (1) resulting space harmonics of the air-gap flux cannot induce
electro-motive force (EMF). However, the considerable
Furthermore, in order to reduce the asymmetrical harmonic content of the rotor MMF leads to increased rotor
electromagnetic force on the rotor, their pole pair difference leakage inductance.
should be greater than one [10]
3 Steady-state model
|p1 − p2 |l1 (2)
The PW and CW can each be modelled in a similar way to the
As can be seen in Fig. 1, the power winding (PW) is stator winding of a conventional asynchronous machine.
connected directly to the grid and therefore works at grid However, because of the special RW design of the BDFM,
frequency. Most of the power is transferred between the it reacts to both p1 and p2 pole pair stator MMFs. Hence,
BDFM and grid through this winding. The PW produces an the rotor model could be drawn as shown in Fig. 3.
magneto-motive force (MMF) in the air gap rotating at grid Saturation has been neglected and the BDFM core loss is
frequency. The control winding (CW) is connected to the not taken into account here.
grid through a bidirectional, fractionally-rated frequency
converter. The frequency converter usually consists of two 3.1 Simple induction mode
back-to-back voltage source inverters. The inverter
connected to the CW, the machine side inverter (MSI), In this asynchronous mode, one of the stator windings, for
controls the rotor speed and the machine reactive power example the CW, is opened and the other is supplied.
[11, 12]. The grid side inverter controls the DC-link voltage The PW produces a p1 pole pair rotating field at speed
and can assist in controlling the BDFM terminal voltage by of v1. The rotor slip due to the field, s1 , can be determined
supplying or absorbing reactive power. The CW produces from
an MMF in the air gap rotating at MSI frequency. v1 − p1 vr
To establish cross-coupling between the PW and CW, the s1 = (3)
rotor winding (RW) must produce both p1 and p2 pole pair v1
MMFs in response to either p1 or p2 pole pair MMFs
produced by the PW and CW, respectively. To fulfil this Owing to the special RW design, a p2 pole pair MMFr1 – 2 and
requirement, the number of sets of rotor circuits, Nr , should consequently rotating air-gap flux component is produced, in
be equal to p1 + p2 [13], each set comprising one or more addition to the explicit p1 pole pair MMFr1 – 1. As shown in
Table 1, MMFr1 – 2 rotates with respect to the CW at the
speed of v′2 (4) and induces an EMF in the CW.

v′2 = Nr vr − v1 (4)

Now, the rotor slip because of this field can be calculated, that
is
v1 − p1 vr
s′2 = (5)
v1 − Nr vr

Fig. 1 BDFM structure The machine model in the simple induction mode can
therefore be drawn as Fig. 4a. The parameters of the CW
and the rotor are divided by s1 /s′2 ( = −v′2 /v1 ) and s1 ,
respectively, to translate them to the PW frequency. In this
way, the amplitudes and phases of the CW and rotor
currents remain unchanged. If all the parameters are referred
to the PW side and the Thevenin equivalent circuit of the
PW is used, the equivalent circuit can be transformed to
Fig. 4b.
The p2 pole pair component of flux produced by the RW
can be considered as a flux leakage. Hence, as shown in
Fig. 4b, there is a high RW leakage reactance that decreases
the currents. Consequently, the BDFM active power will be
significantly lower than that of a similar conventional
Fig. 2 Nested loop rotor of the prototype D160 BDFM induction machines. Thus, the active power flow can be

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Table 1 MMFs in the BDFM air gap

Produced by Number of Electrical speed with Electrical speed with

pole pairs respect to rotor respect to stator

MMF1 PW p1 v1 2 p1vr v1
MMFr121 RW in response to MMF1 p1 v1 2 p1vr v1
MMFr122 RW in response to MMF1 p2 2(v1 2 p1vr) −v1 + p1 vr + p2 vr
= −v1 + Nr vr
MMF2 CW p2 v2 2 p2vr v2
MMFr221 RW in response to MMF2 p1 2(v2 2 p2vr) −v2 + p2 vr + p1 vr
= −v2 + Nr vr
MMFr222 RW in response to MMF2 p2 v2 2 p2vr v2

Table 2 Active power in the simple asynchronous mode

Voltage source power Copper loss Mechanical power

 
p1 vr
3V1thI1cosw1 3(R1th + RrP )I12 3 RPI2
v1 − p1 vr r 1

Fig. 3 Rotor model induces an EMF in the rotor with the slip s1 . The rotor
produces an MMF, which contains both p1 and p2 order
harmonics, similar to the simple asynchronous mode.
calculated, as illustrated in Table 2. As observed, the Repeating the procedure of the previous section and also,
mechanical power has the same form as an induction replacing the CW circuit by its Thevenin equivalent circuit
machine with p1 pole pairs. leads to the equivalent circuit shown in Fig. 6.
Then, the electromagnetic torque produced in the simple It is noteworthy that this mode of operation was the
asynchronous mode can be obtained as intended mode of the self-cascaded machine [14 – 16].
  The mechanical power generated by the machine is divided
p1 into two terms as shown in Table 3. The first term (Pm21)
T =3 RP I 2 (6)
v1 − p1 vr r 1 is similar to the mechanical power of an asynchronous
machine with p1 pole pairs. It is similar to the mechanical
Using this relationship, the torque – speed curve for the D160 power in the simple asynchronous mode, but much greater
BDFM with a 400 V, 50 Hz supply for the PW and opened because of the higher currents in the windings. The second
CW are plotted in Fig. 5 alongside the experimental results. term (Pm22) in the cascade mode is similar to
As predicted, the machine torque is small in this mode. The the mechanical power of an asynchronous machine with Nr
parameters of machine are included in Table 5 in Appendix 1. pole pairs.
Considering Pm21 and Pm22 , the electromagnetic torque
produced in the cascade mode is obtained as (7).
3.2 Cascade induction mode

In this further asynchronous mode, one of the stator windings,    

p1 Nr
for example the PW, is supplied and the other is shorted. T =3 RPr I12 + 3 RP I 2 (7)
The PW produces MMF1 and its corresponding flux v1 − p1 vr v1 − Nr vr 2th 1

Fig. 4 BDFM model in the simple induction mode

a Detailed model
b Simplified model referred to the PW side

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Fig. 5 Torque– speed curve of the BDFM in the simple induction Fig. 7 Torque– speed curve of the BDFM in the cascade induction
mode when CW is opened; 2 steady-state model, +++ mode when CW is shorted; 2 steady-state model, +++
experimental experimental

The synchronous mode of operation can be considered to

be the truly attractive BDFM mode. In this mode, in
addition to the couplings described in the previous section,
there are cross-couplings between MMFr1 – 2 and MMF2 as
well as between MMFr2 – 1 and MMF1 , because of their
identical rotational speeds. In this mode, the frequencies of
Fig. 6 BDFM model in the cascade induction mode referred to the EMFs induced in the rotor, because of the p1 and p2 pole
PW side pair components of its flux linkage, are equal. Therefore the
frequency of EMFs induced in the PW because of the CW
equals the frequency of the PW voltage source and vice
Using (7), the predicted and experimental torque – speed versa. This necessitates the production of two fields by the
curves of the D160 BDFM in the cascade mode with PW and CW rotating at the same electrical speed but in
50 Hz, 400 V supply for the PW are illustrated in Fig. 7. opposite directions with respect to the rotor. In this
As can be observed in Fig. 7, the BDFM has two stable situation, the synchronous rotor speed is
operating regions in the cascade mode
v1 + v2
† Around the synchronous speed of the PW (1500 rpm), v1 − p1 vr = −(v2 − p2 vr ) ⇒ vr = (8)
p1 + p2
corresponding to Pm21;
† Around the natural speed (500 rpm), corresponding to
Pm22 . Indeed, the CW frequency v2 should be regulated so that (8)
is fulfilled. Therefore the rotor slips for the p1 and p2 pole pair
In fact, the BDFM in cascade mode can be considered as air-gap flux components are equal to s1 and s2 , respectively,
two induction machines, the first explicit, because of as (9).
the energised stator winding and rotor as mentioned before, ⎧ v1 − p 1 vr
the second implicit, because of the rotor and shorted stator ⎪
⎨ s1 =
winding. v1
v2 − p2 vr v1 − p1 vr (9)
For instance, in the case of cascade BDFM with shorted ⎪
⎩ s2 = =
CW, the rotor produces MMFr1 – 2 , which rotates at v2 v1 − Nr vr
2(v1 2 p1vr) with respect to the rotor. On the other hand,
the CW rotates with respect to the rotor at 2p2vr . Hence, In this mode, the p1 (or p2) pole pair MMFs produced by the
the synchronous speed of this machine will be equal to PW (or CW) and RW have the same electrical speed and
natural speed (v1/( p1 + p2)). direction, therefore they can establish a non-zero-mean
electromagnetic torque. The machine equivalent circuit can
then be drawn including both the voltage sources
3.3 Synchronous mode simultaneously, as shown in Fig. 8a. It should be noted that
from (9), s2/s1 is equal to 2 v1/v2 .
The BDFM performance in the asynchronous modes is Now, if the machine Thevenin equivalent circuit seen from
weaker than the conventional asynchronous machines with the PW side is replaced in Fig. 8a, the simpler model of
similar pole numbers, because of higher flux leakage arising Fig. 8b will be obtained. It can be observed that the BDFM
from special rotor form. model is quite similar to that for a synchronous machine,

Table 3 Active powers in the cascade mode

Voltage source power Copper loss Mechanical power

Pm21 Pm22
   
p1 vr Nr vr
3V1thI1cosw1 3(R1th + RrP + R2th
P
)I12 3 RrP I12 3 RP I2
v1 − p1 vr v1 − Nr vr 2th 1

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Fig. 8 BDFM model in the synchronous mode referred to the PW side

a detailed model
b simplified model

but its internal voltage and synchronous impedance depend frequency does not fulfil (8), synchronism may be lost and
upon rotor speed. In fact, the BDFM in this mode acts as a consequently, ‘double-cascade mode’ can take place. In this
synchronous machine and its speed can be controlled at the situation, the fields produced by the PW and CW will rotate
grid frequency without using brushes and slip-rings. at different electrical speeds with respect to the rotor.
Regarding Fig. 8b, the PW active power, reactive power Thereupon, MMFr1 – 2 and MMF2 as well as MMFr2 – 1 and
and electromagnetic torque are obtained as (10) – (12), MMF1 , which have the same pole pairs, will rotate at
respectively. unequal speeds, hence generating oscillatory torques about
a zero mean.
V12 V E In this asynchronous mode, the cross-coupling between the
P1 = 3 cos(z) − 3 1 cos (z − d) (10) two stator windings cannot exist. Therefore in the double-
Z Z
cascade mode, the BDFM can be modelled as two separate
V12 V E cascade machines with coupled rotors, one with a shorted
Q1 = 3 sin(z) − 3 1 sin (z − d) (11) CW and the other with a shorted PW. Hence, two
Z Z
equivalent circuits may be drawn, one from the PW and
3 V1 E 3 E2 another from the CW side. The machine’s performance
Te = cos(z + d) − cos (z) (12) would then be the result of the superposition of these two
vr Z vr Z
circuits. For instance, if both stator windings are supplied
However, because of different PW and CW frequencies, it is
hard to find an analytical relationship for d. Hence, the PW
active and reactive power can be regulated by closed-loop
control of the CW voltage amplitude and phase. To validate
(10) – (12), tests have been performed on D160 BDFM in
synchronous mode. The active power at three rotor speeds
and different CW voltages is measured and using (10), its
corresponding d is obtained. As can be observed in Table 4,
the calculated and measured values show good agreement.

3.4 Double-cascade mode

As stated earlier, if both stator windings are supplied, MMF1

and MMFr1 – 1 with p1 pole pairs and also, MMF2 and
MMFr2 – 2 with p2 pole pairs will produce direct-coupling,
similar to two asynchronous machines in tandem. However, Fig. 9 Torque– speed curve of the BDFM in the double-cascade
if the CW converter or controller fails and the CW mode

Table 4 Calculated and experimental results for synchronous operation of D160 BDFM

N, V1 , V V2 , V measured d, electrical measured calculated measured calculated

rpm P1 , W degree cosw1 cosw1 Te , Nm Te , Nm

650 400 137 4731 17.4 0.881 0.911 67 66.9

698 400 159 4736 217.8 0.907 1 69 63.7
745 400 182 4774 218.9 0.949 0.984 64 59.1

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Fig. 12 Wind turbine load and electromagnetic torques for a

Fig. 10 Torque– d curve of the BDFM in the synchronous mode
BDFIG during a voltage dip and recovery
when both the PW and CW are supplied with 400 V, 50 Hz

with 400 V, +50 Hz, the BDFM torque – speed curve in and temperature damaging the converter power electronic
double-cascade mode will be obtained, as shown in Fig. 9. switches or its DC-link capacitor. To protect the converter
From this figure, there are three stable operating regions: against this current, the CW could be shorted, through a
series of parallel resistors, named crowbar, as shown in
† Around natural speed (500 rpm); Fig. 11 and in the mean time, the converter is turned off.
† Around synchronous speed of the CW (750 rpm); This has been the approach extensively adopted to protect
† Around synchronous speed of the PW (1500 rpm). DFIG converters [18]. However, when the crowbar is
active, the BDFM operates in the cascade mode.
In order to further investigate the impact of cross-coupling Obviously, a terminal voltage dip leads to a reduction of
between the PW and CW, the BDFM torque is plotted in torque, proportional to the square of voltage, but adding
Fig. 10 against d in the synchronous mode. Both stator crowbar resistance does not affect the maximum torque and
windings are supplied with 400 V, +50 Hz and hence the only shifts its corresponding speed.
rotor speed is 1000 rpm. As observed, the BDFM can For instance, the performance of a BDFM fitted to a VSWT
develop any desirable torque between 2197 to +87 Nm by during and after a 70% grid voltage dip is now investigated
regulating the CW voltage phase. Whereas, according to using the torque– speed curves shown in Fig. 12 as follows:
Fig. 9, the machine in double-cascade mode is unstable at
1000 rpm. Its operating point will be determined by † Curves 1A and 1B, the input mechanical torque captured
torque –speed curve of the mechanical load or prime mover. through turbine in two different wind speeds;
Another problem associated with this mode is that it † Curve 2, the during-fault cascade-mode torque, at reduced
produces oscillatory torques that can damage the rotating terminal voltage;
parts or reduce their life. As mentioned earlier, these † Curve 3, the post-fault cascade-mode torque, at full
torques arise because of the different speeds of MMFr1 – 2 terminal voltage.
and MMF2 as well as MMFr2 – 1and MMF1 .

The initial pre-fault operating point is considered at A1 or

4 Voltage dip analysis A2 in Curve 1A or Curve 1B, respectively. When a 70%
The principles developed in the previous sections are used to voltage dip occurs and crowbar acts, the machine torque
understand the BDFM behaviour during grid faults, which shifts instantaneously from point A1 or A2 to the during-
result in voltage dips at the BDFM terminals. fault Curve 2. In this situation, because of a mismatch
When the terminal voltage dips, a high EMF is induced in between mechanical and electrical torques, rotor starts to
the RW because of the p1 pole pair flux linkage variations, accelerate on Curve 2 towards point B1 or B2 at which the
and as a result of coupling between the RW and CW, an terminal voltage recovers to its pre-fault value.
EMF is also induced in the CW. Depending on the voltage After voltage recovery, the crowbar remains closed for a
dip level, this EMF can be very high and lead to high CW short time, to ensure converter safety. Hence, the rotor
current [17] with consequent high electromagnetic forces accelerates or decelerates along Curve 3 until point C1 or
C2, where the crowbar turns off and MSI becomes active.
Then, the machine can produce the required active power
and the MSI controls the rotor speed down to its initial value.
As can be seen from Fig. 12, at sub-natural speeds
the BDFM operates as motor while the crowbar is on. On
the other hand, at super-natural speeds, the machine always
is in the generating mode. The machine may even
decelerate after voltage recovery and before crowbar
deactivation, for example, between B2 and C2. It is
noteworthy that the BDFM absorbs reactive power when
crowbar is on and the machine is in cascade induction
mode. Either active power consumption in sub-natural
speeds or reactive power absorption is against the most grid
codes, such as the German Transpower [19]. In [17], the
Fig. 11 Proposed crowbar in BDFM use of dynamic resistors in series with the CW has been

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& The Institution of Engineering and Technology 2012 doi: 10.1049/iet-epa.2012.0101
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proposed under voltage dip conditions. They demonstrate 7 Poza, J., Oyarbide, E., Roye, D., Rodriguez, M.: ‘Unified reference
better performance towards satisfying low-voltage ride- frame dq model of the brushless doubly fed machine’, IEE Proc. –
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through requirements of grid codes. 8 Roberts, P.C., McMahon, R.A., Tavner, P.J., Maciejowski, J.M.,
Flack, T.J.: ‘Equivalent circuit for the brushless doubly fed machine
5 Conclusions (BDFM) including parameter estimation and experimental
verification’, IEE Proc. – Electr. Power Appl., 2005, 152, (4),
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The steady-state performance of the BDFM has been 9 McMahon, R.A., Roberts, P.C., Wang, X., Tavner, P.J.: ‘Performance of
elaborated for asynchronous and synchronous modes of BDFM as generator and motor’, IEE Proc. – Electr. Power Appl., 2006,
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the asynchronous modes and their corresponding torque – 10 Roberts, P.C.: ‘A study of brushless doubly fed (induction) machines’.
speed curves, obtained from theory and experiment, are PhD thesis, Emanuel College, University of Cambridge, 2004
11 Poza, J., Oyarbide, E., Sarasola, I., Rodriguez, M.: ‘Vector control
physically justified. Moreover, straightforward relationships design and experimental evaluation for the brushless doubly fed
for the active and reactive power of the machine in machine’, IET Electr. Power Appl., 2009, 3, (4), pp. 247– 256
synchronous mode have also been proposed, showing that 12 Shao, S., Abdi, E., Barati, F., McMahon, R.: ‘Stator-flux-oriented vector
although its speed can be variable at constant grid control for brushless doubly fed induction generator’, IEEE Trans. Ind.
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13 Williamson, S., Ferreira, A.C., Wallace, A.K.: ‘Generalized theory of the
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or converter fault. 17 Tohidi, S., Oraee, H., Zolghadri, M., Shao, S., Tavner, P.: ‘Analysis and
enhancement of low voltage ride-through capability of brushless doubly
Furthermore, the operation of a converter protective crowbar fed induction generator’, IEEE Trans. Ind. Electron., 2012 (early access
in a VSWT BDFM during voltage dips is explained. articles)
The results show that although the crowbar protects the 18 Morren, J., de Haan, S.W.H.: ‘Ride-through of wind turbines with
converter against high CW currents during voltage dips, it doubly-fed induction generator during a voltage dip’, IEEE Trans.
cannot fulfil the active and reactive power requirements of Energy Conv., 2005, 20, (2), pp. 435– 441
19 Grid Connection Code – extra high voltage – Transpower
grid codes and hence, other approaches are needed to ensure stromübertragungs gmbh, April 2009, http: //www.tennettso.de
BDFM low voltage ride-through capability.

6 References 7 Appendix
1 Arabian-Hoseynabadi, H., Oraee, H., Tavner, P.J.: ‘Failure modes and
effects analysis (FMEA) for wind turbines’, Int. J. Electr. Power See Table 5
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machine: its advantages, applications and design methods’. Sixth Int. Table 5 Prototype D-160 BDFM specifications
Conf. on Electrical Machines and Drives (Conf. Publ. no. 376),
Oxford, UK, September 1993, pp. 511–517 Parameter Value Parameter Value
3 Boger, M.S., Wallace, A.K., Spee, R.: ‘Investigation of appropriate pole p1 2 Lm1 1.125 H
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pump drives’, IEEE Trans. Ind. Appl., 1996, 32, (1), pp. 189 –194 p2 4 Lpm2 0.461 H
4 Li, R., Wallance, A., Spee, R., Wang, Y.: ‘Two-axis model development Nr 6 Ll1 0.017 H
of cage-rotor brushless doubly-fed machines’, IEEE Trans. Energy stator slots 36 Lpl2 0.021 H
Conv., 1991, 6, (3), pp. 453– 460 rotor slots 24 Lplr 0.067 H
5 Boger, M.S., Wallace, A.K., Spee, R., Li, R.: ‘General pole number
PW nominal voltage 400 V R1 7.28 V
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for brushless doubly-fed machines’, IEEE Trans. Energy Conv., 1996, CW nominal current 7A n1/n2 1.176
11, (4), pp. 687–692

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