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.
ICSET 2008
Vector Control of the Brushless Doubly-Fed
Machine for Wind Power Generation
Shiyi Shao, Ehsan Abdi and Richard McMahon
Electrical Engineering Division, Cambridge University, 9 JJ Thomson Avenue, Cambridge, CB3 0FA, United Kingdom
Phone: +44-01223-748316, Email: ea257@cam.ac.uk
Abstract—The Brushless Doubly-Fed Machine (BDFM) shows
commercial benefits in the wind power generation. This paper
presents a vector control scheme for the BDFM operating as a
variable speed generator (VSG). The proposed vector controller
is developed on the power winding stator flux frame, and can be
used to control both speed and reactive power. The machine
model and the control system are developed in MATLAB.
Simulation and experimental results show that the proposed
controller can stabilize the BDFM when changes in speed and
reactive power are applied.
LIST OF SYMBOLS
v, i, ψ voltage, current and flux
ω1,ω
2the angular frequency of the power
winding and control winding excita-
tion
ωrthe angular velocity of the rotor
θ1,θ
rthe angular position of the power
winding flux frame and the rotor
s1,s
2the slips of both stator windings
p1,p
2the pole pair numbers of power
winding and control winding
Te,T
lthe electrical torque output and the
exerted load torque
Rs1,R
s2,R
rthe resistance of the power winding,
the control winding and the rotor
Ls1,L
s2,L
rself-inductance of stator winding
and rotor [18]
Ls1r,L
s2rcoupling inductance between the sta-
tor winding and the rotor [18]
Im[] imaginary part
∗complex conjugate
|| magnitude of the vector
subscripts
1,2,r power winding, control winding and
rotor
d, q the direct and quadrant component
on the power winding flux frame
I. INTRODUCTION
The Brushless Doubly-Fed Machine (BDFM) promises sig-
nificant advantages as a variable speed generator [1][2] as
it offers high reliability and low maintenance requirements
by virtue of the absence of brush gear. This is particularly
important as more and more installations are being constructed
offshore and in difficult-to-reach places. In order to progress
the BDFM towards commercial wind power applications, the
machine must be fully controllable so that it can operate at
a specific shaft speed set by wind conditions to gain the
maximum power output.
A number of scalar control algorithms have been developed
for the BDFM, such as open-loop current control [3], closed-
loop frequency control [4] and phase angle control [5][6],
and they were proved to stabilize the BDFM in a wide speed
range. However, vector control (VC) methods, also called field
oriented control (FOC), are known to give better dynamic
performance [7].
A research group at Oregon State University first repre-
sented a vector control system for the BDFM. The controller
is oriented with the rotor flux and aims to control the angle
between the power and control windings’ synchronous frames
[7][8], which increases the complexity and computational
burden. Furthermore, Oregon did not investigate methods of
controlling the reactive power, which is, in fact, important for
power generation applications. Noting that vector controllers
designed for the Doubly-Fed Induction Generator (DFIG)
[9][10] have been widely used in wind generators regulating
both speed and reactive power, Hopfensperger developed a
controller, similar to the one used for the DFIG, for the
Cascade Doubly-Fed Machine (CDFM), whose refrence frame
is aligned with the “side 1 stator flux” [11][12]. The cross
coupling effect was also mentioned. Later, a vector controller
based on the power winding flux was investigated for the
BDFM by Poza [13] with some experimental results presented
on light load. However, the two controllers on the stator
winding flux frame presented in [12][13] were not fully
described. In fact, the development of such control algorithms
is not trivial because of the existence of the cross coupling
compensator.
This paper presents a simplified controller oriented with
the power winding stator flux with a complete mathemati-
cal derivation. Both simulation and experimental results are
provided to demonstrate the performance of the controller.
Experimental tests have been carried out on a 180 frame size
BDFM with a nested-loop rotor. The results show the promise
of using the BDFM for the wind power generation.
II. BDFM OPERATION
The stator of the BDFM is furnished with two separate stator
windings which differ in pole pair numbers to avoid the direct
coupling between the windings. The rotor is specially designed
to couple both of the two stator windings [16]. Generally
322
978-1-4244-1888-6/08/$25.00 c
2008 IEEE
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:12:02 UTC from IEEE Xplore. Restrictions apply.
Fig. 1: BDFM operation
speaking, the stator winding 1 is connected to the power
grid directly, and therefore known as the power winding. In
contrast, the stator winding 2, named the control winding, is
supplied with a converter to handle only fractional power. This
configuration, shown in Fig.1, has the advantage of reducing
the rating of the power electronics.
The BDFM can be operated in several modes including the
synchronous (doubly-fed) mode, cascade mode and induction
mode [15]. The synchronous mode is the most desirable mode,
where the shaft speed is independent of the torque exerted on
the machine, and can be expressed as:
ωr=ω1+ω2
p1+p2
(1)
where ω1and ω2are the excitation angular frequencies sup-
plied to the two stator windings.
III. MATHEMATICAL REPRESENTATION OF VECTOR
MODEL
The proposed controller is aligned with the power winding
flux frame. Therefore, ψ1d=|ψ1|and ψ1q=0. The model in
the power winding flux frame is expressed by Equation (2) to
Equation (7) [13][14]:
v1=Rs1i1+dψ1
dt +jω1ψ1(2)
ψ1=Ls1i1+Ls1rir(3)
v2=Rs2i2+dψ2
dt +j(ω1−(p1+p2)ωr)ψ2(4)
ψ2=Ls2i2+Ls2rir(5)
vr=Rrir+dψr
dt +j(ω1−p1ωr)ψr(6)
ψr=Lrir+Ls1ri1+Ls2ri2(7)
and the electric torque is:
Te=3
2p1Im[
∗
ψ1i1]+3
2p2Im[ψ2
∗
i2](8)
Suppose that the BDFM is running in steady state, then the
dynamic model can be transferred to the steady state model,
which is equivalent to the coupled coils model proposed by
Roberts in [18] for a BDFM with a single rotor circuit:
v1=Rs1i1+jω1Ls1i1+jω1Ls1rir(9)
s2
s1
v2=s2
s1
Rs2i2+jω1Ls2i2+jω1Ls2rir(10)
1
s1
vr=1
s1
Rrir+jω1Lrir+jω1Ls1ri1+jω1Ls2ri2(11)
s1and s2are the slips, which are defined as:
s1
Δ
=ω1−p1ωr
ω1
(12)
s2
Δ
=ω2−p2ωr
ω2
(13)
IV. CONTROLLER DESIGN
A. Power winding flux estimator
In order to orientate all the quantities in the reference frame,
the angle of the power winding flux, θ1, has to be known.
A common way is using Equation (2) and measuring the
power winding voltage v1while neglecting the power winding
stator resistance Rs1[17]. As a result, the power winding
flux vector lags 90obehind the power winding voltage vector,
and therefore, a subtraction of π/2 from the voltage vector
angle leads to the power winding flux angle θ1, as is shown
in Equation (14). The derivations on the dq frame are given
in Equations (15) and (16).
v1=jω1ψ1(14)
v1d=−ω1ψ1q=0 (15)
v1q=ω1ψ1d=ω1|ψ1|(16)
Since the power winding voltage is connected to the 50 Hz,
240 V grid, the power winding flux has also a constant
magnitude and rotates at fixed 50 Hz and independent to the
machine speed.
B. Control of power winding current
Equation (17) can be obtained by combining Equation (9)
with Equation (11):
v1=(Rs1+s1ω2
1L2
s1r
Rr+js1ω1Lr
+jω1Ls1)i1+s1ω2
1Ls1rLs2r
Rr+js1ω1Lr
i2
(17)
Spliting Equation (17) into dq components, considering Equa-
tions (15) and (16) and neglecting the power winding stator
resistance Rs1, yields
i2d=Ls1Lr−L2
s1r
Ls1rLs2r
i1d−|ψ1|Lr
Ls1rLs2r
+RrLs1
s1ω1Ls1rLs2r
i1q
(18)
i2q=Ls1Lr−L2
s1r
Ls1rLs2r
i1q+|ψ1|Rr
s1ω1Ls1rLs2r
−RrLs1
s1ω1Ls1rLs2r
i1d
(19)
323
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:12:02 UTC from IEEE Xplore. Restrictions apply.
The first term of Equation (18),
Ls1Lr−L2
s1r
Ls1rLs2r
i1d
defines the direct coupling between i2dand i1d, and the
coefficient is constant. The second term,
−|ψ1|Lr
Ls1rLs2r
performs as a constant offset. Finally, the third term,
RrLs1
s1ω1Ls1rLs2r
i1q
reflects the cross coupling. Obviously, the only variable is
s1, and therefore this cross coupling term varies with the
shaft speed. The coefficient of the cross coupling term can
be neglected as compared to the direct coupling term if the
operating speed of the rotor is within ±50% of the machine
natural speed.
As a conclusion, i2dis linear with i1dif the effect of the
cross coupling term is neglected. Similar analysis applies to
Equation (19), with a conclusion that i2qis linear with i1q.
C. Control of control winding current
If Equations (4) and (5) are split into dq components, then:
v2d=Rs2i2d+dψ2d
dt −(ω1−(p1+p2)ωr)ψ2q(20)
v2q=Rs2i2q+dψ2q
dt +(ω1−(p1+p2)ωr)ψ2d(21)
ψ2d=Ls2i2d+Ls2rird (22)
ψ2q=Ls2i2q+Ls2rirq (23)
From Equation (3), rotor current ircan be expressed as
ir=ψ1−Ls1i1
Ls1r
(24)
Splitting (24) to dq components and substituting Equations
(15) and (16):
ird =ψ1d−Ls1i1d
Ls1r
=|ψ1|−Ls1i1d
Ls1r
(25)
irq =ψ1q−Ls1i1q
Ls1r
=−Ls1i1q
Ls1r
(26)
Combining with Equations (20), (21), (22), (23), (25) and (26),
and neglecting Rs1, the control winding voltage in the dq
frame can be derived as:
v2d=Rs2i2d+Ls1Ls2Lr−Ls2L2
s1r−Ls1L2
s2r
Ls1Lr−L2
s1r
di2d
dt
−L2
s1Ls2rRr
s1ω1Ls1r(Ls1Lr−L2
s1r)
di1q
dt
−(ω1−(p1+p2)ωr)(Ls2i2q−Ls2rLs1
Ls1r
i1q)
(27)
v2q=Rs2i2q+Ls1Ls2Lr−Ls2L2
s1r−Ls1L2
s2r
Ls1Lr−L2
s1r
di2q
dt
+L2
s1Ls2rRr
s1ω1Ls1r(Ls1Lr−L2
s1r)
di1d
dt
+(ω1−(p1+p2)ωr)(Ls2i2d+Ls2r
|ψ1|−Ls1i1d
Ls1r
)
(28)
Similar analysis as for the control of the power winding
current can be applied to Equation (27). The first term,
Rs2i2d+Ls1Ls2Lr−Ls2L2
s1r−Ls1L2
s2r
Ls1Lr−L2
s1r
di2d
dt
shows direct relation between v2dwith i2d. The transfer
function has first order and constant components. The second
term,
−L2
s1Ls2rRr
s1ω1Ls1r(Ls1Lr−L2
s1r)
di1q
dt
is essentially a first order cross coupling, which is also
dependent on the shaft speed. Generally speaking, it can be
neglected compared with the direct coupling term for the same
reason discussed above. The third term,
−(ω1−(p1+p2)ωr)(Ls2i2q−Ls2rLs1
Ls1r
i1q)
shows another cross coupling with lower order compared to
both of the first two terms, and therefore can be neglected.
Therefore, if cross coupling is neglected, v2dand i2dhave
a constant and a first order relation. A similar derivation can
be applied to the analysis of Equation (28), concluding that
the v2qand i2qalso have a constant and a first order relation.
D. Control of torque
Considering Equations (8), (15), (16), (25), (26), (27) and
(28), the torque can be expressed as:
Te=3
2(p1+p2)|ψ1|i1q−3
2
p2|ψ1|2Rr
s1ω1L2
s1r
−3
2
p2RrL2
s1
s1ω1L2
s1r
(i2
1d+i2
1q)
(29)
The first term, 3
2(p1+p2)|ψ1|i1q
shows the fact that the torque of the BDFM is directly related
to i1q. The other term,
−3
2
p2|ψ1|2Rr
s1ω1L2
s1r
−3
2
p2RrL2
s1
s1ω1L2
s1r
(i2
1d+i2
1q)
shows time-varying relation and can be analyzed like the
control of power winding and control winding current. Con-
sequently, Teis almost linear with i1q.
E. Control of speed
The mechanical differential equation is:
Te=Jdωr
dt +Tl(30)
where friction force is neglected. Therefore, ωrcan be regu-
lated by controlling Tesince they have a first order relation.
324
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:12:02 UTC from IEEE Xplore. Restrictions apply.
PI
*
r
ω
+-
r
ω
PI
Var controller
+-
*
1
Q
1
Q
v
2q
v
2d
v
2d
v
2q
r
θ
1
θ
23v
2b
v
2c
PWM
generator PWM
signal
*
2c
v
*
2b
v
*
2a
v
v
2a
Flux
estimator
BDFM
v
1a
v
1b
v
1c
1
θ
Reactive
power
estimator
23
v
1a
v
1b
v
1c
i
1a
i
1b
i
1c
i
1a
i
1b
i
1c
v
1d
v
1q
i
1d
i
1q
Encoder
r
ω
r
θ
Speed controller
Fig. 2: Schematic of the proposed vector system
F. Control of reactive power
The reactive power of the power stator winding is expressed
as [9][10]:
Q1=3
2(v1qi1d−v1di1q)(31)
Considering Equations (15) and (16),
Q1=3
2|ψ1|ω1i1d(32)
Therefore, a linear relation exists between Q1and i1d.
G. Proposed vector control scheme
Based on the analysis discussed above, the shaft speed ωr
and reactive power Q1can be controlled by controlling v2q
and v2drespectively:
ωr⇒Te⇒i1q⇒i2q⇒v2q
and
Q1⇒i1d⇒i2d⇒v2d
Fig.2 shows the speed and reactive power control system
using the proposed control algorithm. The 3→2 module is
realized by well-known Clark Transformation and Park Trans-
formation [20].
V. E XPERIMENTAL AND SIMULATION RESULTS
A. Experimental rig and simulation setup
An experimental test rig is established in order to validate
the control algorithm, experimental tests have been carried
out on a 180 frame BDFM. Table I gives the physical data
for the machine. A DC machine is mechanically coupled
to the BDFM in order to provide the required torque. An
HBM T30FN torque transducer is used to monitor the torque
data. The speed and position signals are obtained from an
incremental encoder, RP442-z, with the resolution of 2500,
provided by ONO SOKKI. The voltage and the current of each
stator phase are measured by LEM LV 25-p and LEM LTA
100-p respectively. The control system is implemented based
on the xPC Target which receives all the signals mentioned
above. The control loop time delay is 0.2 ms, updating the
three phase voltage waveforms for the control winding. The
outputs of the xPC target are connected to an FPGA-based
PWM generator driving the converter for the BDFM control
winding.
Simulations are also implemented using Simulink, and a
coupled-circuit model for the BDFM [18][19] is developed
to simulate the dynamic performance of the proposed vector
controller. A fixed-step ode5 solver is used, and the step size
is set to 0.4 ms.
B. Experimental and simulation results
In wind power applications, the variable speed generation is
mostly used to enhance the wind turbine efficiency. Therefore,
the control of the rotor speed is critically important in such
applications. Fig.3 illustrates the speed response from 550 rpm
to 720 rpm. Since the DC machine is not equipped with a
suitable controller, in the experiment, the torque varies with
the speed, from 45 Nm to 60 Nm, as shown in Fig.3b. The
rise time of the speed, shown in Fig.3a, is less than 5 s. The
reactive power is set to be constant 1 kVAR. As is shown in
Fig.3c, there is an overshoot of about 2 kVAR in the reactive
power, but it quickly settles down to the reference value. The
simulation adopts the same setup applied in the experiment,
and produces similar results as from Fig.3d to Fig.3f.
The proposed vector controller also has the ability to reg-
ulate the reactive power of the BDFM power winding, which
is paramountly attractive in wind power generation. Fig.4a
shows the experimental results of reactive power changing
from 2 kVAR to 0.5 kVAR and then back to 2 kVAR, while
325
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:12:02 UTC from IEEE Xplore. Restrictions apply.
0 5 10 15 20
500
550
600
650
700
750
Time (s)
Speed (rpm)
Ref Speed
Exp Speed
(a) Experimental speed
0 5 10 15 20
30
40
50
60
70
Time (s)
Torque (Nm)
(b) Experimental torque
0 5 10 15 20
−1000
−500
0
500
1000
1500
2000
2500
Time (s)
Reactive Power (VAR)
(c) Experimental reactive power
0 5 10 15 20 25
500
550
600
650
700
750
Time (s)
Speed (rpm)
Ref Speed
Sim Speed
(d) Simulated speed
0 5 10 15 20 25
30
40
50
60
70
Time (s)
Torque (Nm)
(e) Simulated torque
0 5 10 15 20 25
−500
0
500
1000
1500
2000
2500
3000
Time (s)
Reactive Power (VAR)
(f) Simulated power
Fig. 3: Simulation and experimental results of speed change response, 40 to 60 Nm, 1 kVAR
010 20 30 40
−500
0
500
1000
1500
2000
2500
3000
Time (s)
Reactive Power (VAR)
Ref Reactive Power
Exp Reactive Power
(a) Experimental reactive power
010 20 30 40
500
520
540
560
580
600
Time (s)
Speed (rpm)
(b) Experimental speed
0 5 10 15 20 25
−500
0
500
1000
1500
2000
2500
3000
Time (s)
Reactive power (VAR)
Ref Reactive Power
Sim Reactive Power
(c) Simulated reactive power
0 5 10 15 20 25
500
520
540
560
580
600
Time (s)
Speed (rpm)
(d) Simulated speed
Fig. 4: Simulation and experimental results of reactive power change, 550 rpm, no load
0 5 10 15 20 25 30
−20
0
20
40
60
80
Time (s)
Torque (Nm)
(a) Experimental torque
0 5 10 15 20 25 30
400
450
500
550
600
650
700
Time (s)
Speed (rpm)
(b) Experimental speed
010 20 30
−1000
0
1000
2000
3000
Time (s)
Reactive Power (VAR)
(c) Experimental reactive power
0 5 10 15 20 25 30
−20
0
20
40
60
80
Time (s)
Torque (Nm)
(d) Simulated torque
0 5 10 15 20 25 30
400
450
500
550
600
650
700
Time (s)
Speed (rpm)
(e) Simulated speed
010 20 30
−500
0
500
1000
1500
2000
2500
3000
Time (s)
Reactive Power (VAR)
(f) Simulated reactive power
Fig. 5: Simulation and experimental results of torque change, 550 rpm, 1 kVAR
326
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:12:02 UTC from IEEE Xplore. Restrictions apply.
TABLE I: Prototype Machine Specifications
Parameter Value
Frame size D180
Stator 1 pole-pairs 2
Stator 2 pole-pairs 4
Stator slots 48
Rotor slots 36
Rotor design Nested-loop design [18]
the speed keeps constant at 550 rpm as shown in Fig.4b. The
experiment is done at no load. The simulation results in Fig.4c
and Fig.4d validate the experimental results.
Finally, the torque change from 0 to 60 Nm is supplied to
the BDFM to test the stability of the controller in Fig.5a. Both
speed and reactive power come back to the reference values
with an overshoot shown in Fig.5b and Fig.5c respectively,
which displays the stability under the applied vector controller.
Again, similar simulation results have been obtained from
Fig.5d to Fig.5f.
VI. CONCLUSIONS
This paper proposes a simplified power winding flux ori-
ented vector control scheme without the cross coupling com-
pensator. Detailed theoretical analysis is done to represent
the decoupled control structure of the speed and reactive
power control. The controller is able to regulate both the
rotor speed and reactive power. The machine model and the
controller have been implemented in MATLAB and verified
experimentally. The experimental results have shown stable
operation of the machine under various controlling commands.
The performance obtained further reinforces the suitability of
the BDFM for wind power generation.
REFERENCES
[1] McMahon R.A. Wang X. Abdi E. Tavner P.J. Roberts P.C. Jagiela M. The
BDFM as a Generator in Wind Turbines, Power Electronics and Motion
Control Conference, 12th International, Aug. 2006, Page(s): 1859-1865
[2] Wang X. Roberts P.C. McMahon R.A. Studies of inverter ratings of
BDFM adjustable speed drive or generator systems, Power Electronics
and Drives Systems, Jan. 2006, Volume: 1, Page(s): 337-342
[3] Zhou D. Spee R. Wallace A.K. Laboratory control implementations for
doubly-fed machines, Industrial Electronics, Control, and Instrumenta-
tion, 1993. Proceedings of the IECON apos;93, Nov 1993, Volume: 1,
Issue: 15-19 Page(s): 1181-1186
[4] Li R. Spee R. Wallace A.K. Alexander G.C. Synchronous drive per-
formance of brushless doubly-fed motors, Industry Applications, IEEE
Transactions, July-Aug. 1994, Volume: 30, Issue: 4, Page(s):963-970
[5] Shao S. Abdi E. McMahon R.A. Stable Operation of the Brushless
Doubly-Fed Machine (BDFM), Power Electronics and Drive Systems,
2007, Nov. 2007, Page(s): 897-902
[6] Shao S. Abdi E. McMahon R.A. Operation of Brushless Doubly-Fed
Machine for Drive Applications, Power Electronics, Machines and
Drives, 2008, April 2008, Page(s):340-344
[7] Zhou D. Spee R. Synchronous frame model and decoupled control
development for doubly-fed machines, Power Electronics Specialists
Conference, PESC ’94 Record., 25th Annual IEEE , June 1994, Volume:
2, Page(s):1229-1236
[8] Zhou D. Spee R. Alexander G.C. Wallace A.K.; A simplified method for
dynamic control of brushless doubly-fed machines, Industrial Electron-
ics, Control, and Instrumentation, 1996., Proceedings of the 1996 IEEE
IECON, 22nd International Conference, Aug. 1996, Volume: 2, Page(s):
946-951
[9] Tang Y. Xu L. A flexible active and reactive power control strategy for a
variable speed constant frequency generating system, Power Electronics,
IEEE Transactions, July 1995, Volume: 10, Issue 4, Page(s): 472 - 478
[10] Yamamoto M. Motoyoshi O. Active and reactive power control for
doubly-fed wound rotor induction generator , Power Electronics, IEEE
Transactions, Oct 1991, Volume: 6, Issue 4, Page(s): 624 - 629
[11] Hopfensperger B. Atkinson D.J. Lakin R.A. Stator flux oriented
control of a cascaded doubly-fed induction machine, Electric Power
Applications, IEE proceedings, Sep. 2006, Volume: 153, Issue 5, Page(s):
1138-1143
[12] Hopfensperger B. Atkinson D.J. Lakin R.A. The application of field
oriented control to a cascaded doubly-fed induction machine, Power
Electronics and Variable Speed Drives, 2000, Sep. 2000, Page(s): 262-267
[13] Poza J. Oyarbide E. Roye D. New vector control algorithm for brushless
doubly-fed machines, IECON 02 [Industrial Electronics Society, IEEE
2002 28th Annual Conference of IEEE], Nov. 2002, Volume: 2, Page(s):
1138-1143
[14] Poza J. Oyarbide E. Roye D. Rodriguez M. Unified reference frame dq
model of the brushless doubly fed machine, Electric Power Applications,
IEE proceedings, Nov. 1999, Volume: 146, Issue 6, Page(s): 597-605
[15] Broadway A.R.W. Burbridge L. Self-cascade machine: a low-speed
motor or high frequency brushless alternator, IEE Proceedings, 1974,
Pare(s): 1529-1535
[16] McMahon R.A. Roberts P.C. Wang X. Tavner P.J. Performance of BDFM
as generator and motor, Electrical Power Applications, IEE Proceedings,
Mar 2006, Volume: 153, Issue 2, Pare(s): 289-299
[17] Leonhard W. Control of electric drives, 2nd den, Springer-Verlag,
Berlin, 1996
[18] Poberts P. C. A study of Brushless Doubly-Fed (Induction) Machines,
PhD Thesis, University of Cambridge, Sep. 2004
[19] Abdi E. Modelling and Instrumentation of Brushless Doubly-Fed
(Induction) Machines, PhD Thesis, University of Cambridge, Sep. 2006
[20] Vas P. Sensorless Vector and Direct Torque Control, Oxford Univer-
sity Press, Oxford New York Tokyo, 2004
327
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:12:02 UTC from IEEE Xplore. Restrictions apply.