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Derivation of a Vector Model for a Brushless
Doubly-Fed Machine with Multiple Loops per Nest
Farhad Barati† (barati@ee.sharif.edu), Hashem Oraee†, Ehsan Abdi‡ (ea257@cam.ac.uk) and Richard McMahon‡
† Electrical Engineering Department, Sharif University of Technology, Tehran, Iran
‡ Electrical Engineering Division, University of Cambridge, Cambridge, UK
Abstract - The paper presents a vector model for a Brushless
Doubly-Fed Machine (BDFM). The BDFM has 4 and 8 pole
stator windings and a nested-loop rotor cage. The rotor cage has
six nests equally spaced around the circumference and each nest
comprises three loops. All the rotor loops are short circuited via
a common end-ring at one end. The vector model is derived
based on the electrical equations of the machine and
appropriate vector transformations. In contrast to the stator,
there is no three phase circuit in the rotor. Therefore, the vector
transformations suitable for three phase circuits can not be
utilised for the rotor circuit. A new vector transformation is
employed for the rotor circuit quantities. The approach
presented in this paper can be extended for a BDFM with any
stator poles combination and any number of loops per nest.
Simulation results from the model implemented in Simulink are
presented.
Keyword: Brushless Doubly-Fed Machine, Vector
Transformation, Vector Model
NOMENCLATURE
111 ,, cba
ν
ν
ν
stator1 phase voltages
222 ,, cba
ν
ν
ν
stator2 phase voltages
21 ,ss RR phase resistances of stator1 and stator2
omi RRR ,, inner, middle and outer loops resistances
21 ,lsls LL leakage inductances of stator1 and stator2
21 ,ss LL self inductances of stator1 and stator2
lolmli LLL ,, inner, middle and outer loops leakage
inductances
omi LLL ,, inner, middle and outer loops self
inductances
iomoim LLL ,, mutual inductances between loops of a nest
oommii MMM ,, mutual inductance between identical loops
in 2 nests
iomoim MMM ,, mutual inductance between nonidentical
loops in 2 nests
omi rsrsrs MMM 111 ,,
mutual inductance between stator1 phase
winding and inner, middle and outer loops
in a nest
omi rsrsrs MMM 222 ,,
mutual inductance between stator2 phase
winding and inner, middle and outer loops
in a nest
BJ , rotor moment of inertia and friction
coefficient
p
dtd operator
21 ,PP pole numbers of stator1 and stator2
rss
ϕ
θ
θ
,, 21 arbitrary functions of time utilized in
stator1, stator2 and rotor transformations
rss ',, 21
ϕ
ω
ω
time-derivatives of rss
ϕ
θ
θ
,, 21
111 ,, sss IV
λ
stator1 voltage, current and flux in the
vector model
222 ,, sss IV
λ
stator2 voltage, current and flux in the
vector model
rrr IV
λ
,, rotor voltage, current and flux in the vector
model
{
}
{
}
imgreal , real and imaginary parts of a complex
number
*
Z
complex conjugate of
Z
t transpose of a matrix or vector
I. INTRODUCTION
The Brushless Doubly-Fed Machine (BDFM) shows
commercial promise as both a variable speed drive and
generator. The contemporary BDFM is a single frame
induction machine with two 3-phase stator windings of
different pole numbers, and a special rotor design. Typically
one stator winding (the power winding) is connected to the
mains and the other (the control winding) is supplied with
variable voltage at variable frequency from a converter [1].
The nested-loop rotor comprises a number of nests which
are equally spaced around the circumference. The number of
nests is equal to the sum of the stator windings pole pairs [2].
Each nest has one or more loops. Fig. 1 shows the nested-
loop rotor. This machine has 4 and 8 pole stator windings
which are used as control and power windings respectively.
Therefore, the number of rotor nests is 6. There are 3 loops in
each nest: inner, middle, and outer loops. All the loops are
short circuited through a common end ring at one end of the
rotor (figure 1). The machine details are provided in table 1.
606978-1-4244-1666-0/08/$25.00 '2008 IEEE
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 11:37:50 UTC from IEEE Xplore. Restrictions apply.
Fig.1: Nested loop rotor cage of Cambridge prototype
TABLE 1
PROTOTYPE MACHINE SPECIFICATIONS
Parameter Value
Frame size
Stator 1 pole-pairs
Stator 2 pole-pairs
Stator 1 rated current
Stator 2 rated current
Rotor design
D180
2
4
8A
8A
‘Nested-loop’ design consisting of 6 ‘nests’ of 3
concentric loops of pitch 5/36, 3/36 and 1/36 of
the rotor circumference. Each nest offset by 1/6
of the circumference, for details see [11].
Considerable research activities have been carried out on
BDFM modeling and analysis, and a spectrum of analytical
modeling techniques has been successfully employed in the
study of the BDFM. A team at Oregon State University
developed a detailed mathematical model for a prototype
BDFM for the first time. Based on this model, a d-q model
was derived which was then employed to analyse the
machine operation [3-5]. A generalised harmonic model was
developed by Williamson et al at Cambridge University [2,6]
which is capable of predicting the steady-state performance
of the BDFM with a nested-loop rotor with any stator
winding, allowing the harmonic contributions to be
specifically analysed.
Roberts in [7] developed a generalised dynamic and
steady-state modeling framework which can be applied to a
wide class of BDFMs, including, but not limited to, those
with a nested-loop design rotor. A rigorous derivation of
parameter values for the coupled-circuit, d-q axis and
equivalent circuit models was presented. The coherence
between these models allows parameters calculated for the
coupled-circuit model to provide parameter values for the
other models.
Abdi in [8] developed a simplified approach to derive the
equivalent circuit parameters from the machine geometry. A
curve fitting approach to estimate the machine parameter
values was also proposed in [8]. An analytical method based
on the coupled circuit model was presented taking into
account the effects of iron saturation.
Poza et al at Mondragon University developed a vector
model for a BDFM with a nested-loop rotor with one loop in
each nest [9, 10]. They also developed a coupled-circuit
model [11] and d-q axis model for their machine [12]. Their
machine is a 2/6 BDFM with a nested-loop rotor with 4 nests
and one loop per nest.
Muñoz et al in [13] presented a vector model for a Dual
Stator winding Induction Machine (DSIM) with squirrel cage
rotor. Since the real structure of the rotor is considered in the
modeling proposed in [13], it is considered relevant to the
discussions of this paper.
This paper will present a vector model for the BDFM with
a nested-loop rotor with multiple loops in each nest. The
approach is therefore more general than previous vector
models presented for the BDFM [3-5,9,10]. The model is
developed for a BDFM with 4 and 8 pole stator windings and
a nested loop rotor with six nests and three loops in each
nest. The rotor is shown in figure 1. The machine details are
provided in table 1. Although the model is presented for a
particular machine, it can be generalised for a BDFM with
any combination of stator pole numbers and the nested-loop
rotor design with any number of loops in each nest.
II. BDFM MATHEMATICAL MODEL
The BDFM considered in this paper has two stator
windings with 4 and 8 poles represented by S1 and S2,
respectively. The rotor cage has six nests with three loops in
each nest. All the rotor loops are shorted via a common end-
ring at one end of the rotor. Figure 2 shows the expanded
view of the rotor cage.
The axis of phase “a” of the control winding, S1, is set as
the reference axis as shown in figure 3. It can be depicted
from the figure that
ξ
, the displacement angle, is the angle
of the axis of phase “a” of the power winding, S2, with
respect to the reference axis. Moreover, r
θ
is the angle of
nest1 axis with respect to the reference.
Fig.2: Expanded view of the rotor cage
607
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Figure 3: Reference axis and relative angles
Inner loop Stator 1
Middle loop Stator 2
Outer loop
The machine equation set consists of the voltage, flux and
torque equations. The following equations represent the
voltage equations of the BDFM:
1111 ssss piR
λ
ν
+= (1-a)
2222 ssss piR
λ
ν
+= (1-b)
rrrr piR
λ
ν
+= (1-c)
where
[]
t
cbas 1111 ,,
ννν
ν
= and
[]
t
cbas 2222 ,,
ννν
ν
= are the
stator1 and stator2 voltage vectors , respectively.
211 ,, sss ii
λ
and 2
s
λ
have the same format.
[]
t
rrrrr 6321 ,...,,,
νννν
ν
= is the rotor voltage vector and
comprises the voltage vectors of the rotor nests. The voltage
vector of the kth rotor nest is defined as
[]
6,...,2,1,,, == k
t
omir kkkk
ννν
ν
. vik, vmk and vok are the inner,
middle and outer loop voltages within the kth nest. r
i
and r
λ
have the same format as r
v. Rs1, Rs2 and Rr are diagonal
matrices representing the stator1, stator2, and rotor
resistances.
{
}
r
r
r
r
r
r
r
r
r
r
r
rdiag
r
R,,,,,= is the rotor
resistance matrix in which r
ris the nest resistance matrix.
r
rconsists of the inner, middle, and outer loops resistances
diagonally i.e.
{
}
o
R
m
R
i
Rdiag
r
r,,=.
The following represents the machine flux equations:
rssss iMiM r1111 +=
λ
(1-d)
rssss iMiM r2222 +=
λ
(1-e)
rrs
t
rss
t
rsr iMiMiM ++= 2211
λ
(1-f)
where 21 ,ss MM , and r
M are the self inductance matrixes of
stator1, stator2 and rotor, respectively. 2,1,
=
iM rsi is the
mutual inductance vector between the rotor circuit and the ith
stator winding. The self and mutual inductance matrices of
the machine winding can be calculated from the machine
specifications as described in [7, 8].
The machine electromagnetic torque and torque balance
equations can be calculated as:
[]
r
r
rs
r
rs
t
s
t
se i
d
dM
d
dM
iiT
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
θ
θ
2
1
21
(1-g)
rrlrre pTBJpT
θ
ω
ω
ω
=
+
+
=
, (1-h)
where Te and Tl are the machine and load torques,
respectively. r
ω
and r
θ
are the rotor speed and position.
III. VECTOR TRANSFORMATIONS
Suitable vector transformations for the machine vector
quantities are presented in this section. Applying a vector
transformation to a machine vector quantity yields a complex
quantity. The vector quantities for the stator1 and stator2
include the voltage, current and flux vectors. The following
is the transformation vector for the stator1 voltage [15,16].
1
1
1
3
4
3
2
1
3
2
s
jj
j
see
e
V
s
ν
ππ
θ
⎥
⎦
⎤
⎢
⎣
⎡
= (2-a)
Similar transformations are used for 1
s
i and 1
s
λ
to derive
1
s
I and 1
s
λ
. The transformations for stator2 quantities
including 222 ,, sss i
λ
ν
are similar to stator 1. The following is
the transformation for 2
s
i:
2
2
2
3
4
3
2
1
3
2
s
jj
j
siee
e
I
s
⎥
⎦
⎤
⎢
⎣
⎡
=
ππ
θ
(2-b)
In order to derive a vector model from the electrical
equations of the machine, the reverse transformations
corresponding to the vector transformations must be
available. The followings are the stator1 and stator2 reverse
vector transformations:
}1.{ 1
1
1
3
4
3
2
s
t
jj
j
sVeeereal s⎥
⎦
⎤
⎢
⎣
⎡
=−−
−
ππ
θ
ν
(3-a)
}1.{ 2
2
2
3
4
3
2
s
t
jj
j
sIeeereali s⎥
⎦
⎤
⎢
⎣
⎡
=−−
−
ππ
θ
(3-b)
r
ω
r
θ
ζ
Stato
r
Roto
r
608
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,
r
i
P
j
eI
P
j
eI
P
j
eI
P
j
eI
P
j
eII
r
P
j
e
Ir
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
××××××
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
3
5
2
1
.
33
,
3
4
2
1
.
33
,
2
1
.
33
,
3
2
2
1
.
33
,
32
1
.
33
,
33
3
2
1
ππ
π
ππϕ
(4-a)
}.
3
5
2
1
.
33
,
3
4
2
1
.
33
,
2
1
.
33
,
3
2
2
1
.
33
,
32
1
.
33
,
33
2
1
{r
I
t
P
j
eI
P
j
eI
P
j
eI
P
j
eI
P
j
eII
r
P
j
ereal
r
i
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡−
×
−
×
−
×
−
×
−
××
−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
ππ
π
ππ
ϕ
(4-b)
The above transformations are suitable for three phase
circuits. Since the rotor circuit can not be simply regarded as
a three phase circuit, appropriate transformations must be
derived for the rotor circuit. For a nested-loop rotor with
three loops in each nest, three sets of circuits may be
considered including inner, middle and outer loops of the
rotor cage. In the vector transformation of the rotor circuit
the quantities of each set are transformed to a complex
quantity. Therefore, each quantity (e.g. voltage) will have a
vector of three complex elements after transformation. The
proposed rotor vector transformation and its corresponding
reverse transformation are given in equations (4-a, 4-b).
Similar transformation can be used for the rotor circuit by
replacing P1 with P2 in equations (4-a, 4-b).
21 ,ss
θ
θ
and r
ϕ
are the free parameters in the vector
transformation of the stator1, stator2 and rotor quantities
respectively and are time-varying in general [15, 16].
The vector model of the machine can be derived by applying
the above transformations to the mathematical equations of
the machine.
IV. VECTOR MODEL OF THE BDFM
Based on the basic mathematical model of the BDFM
presented in section II and the vector transformations
presented in section III, the vector model for the BDFM can
be derived. These equations include the voltage, flux and
torque equations:
111111 ssssss jpIRV
λωλ
−+= (5-a)
222222 ssssss jpIRV
λωλ
−+= (5-b)
r
r
j
r
p
r
I
r
r
r
V
λϕλ
'
2.0 −+== (5-c)
The above are the voltage equations of stator1, stator2 and
rotor circuit in the BDFM vector model. The rotor circuit is
shorted and this is represented in (5-c).
The flux equations of the BDFM vector model are as
follows:
r
j
rssslss IeMILL
η
λ
−
++= 11111 3)
2
3
( (5-d)
*
22222 3)
2
3
(r
j
rssslss IeMILL
γ
λ
++= (5-e)
*
2211 2
3
2
3
)( s
j
t
rss
j
t
rsrNNNr IeMIeMIML
γη
λ
++−= (5-f)
where:
],,[ 1111 omi rsrsrsrs MMMM = is the stator1-rotor mutual
inductance vector;
],,[ 2222 omi rsrsrsrs MMMM = is the stator2-rotor mutual
inductance vector;
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
+
+
=
olomoio
momlmim
ioimili
N
LLLL
LLLL
LLLL
L is the nest self
inductance matrix;
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
oomoio
mommim
ioimii
NN
MMM
MMM
MMM
M is the nest-nest mutual
inductance matrix;
rsr
θ
θ
ϕ
η
22 1
−
−
=
and )(42 2
ξ
θ
θ
ϕ
γ
−
++= rsr .
The machine torque can be calculated from the vector
model quantities as [15]:
}..{18}..{.9 )
2
(
*
)
2
(
*
2211
γ
π
η
π
+− += j
srrs
j
srrse eIIrealMeIIrealMT (5-g)
The stator windings input power may be calculated as:
609
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).{
2
3
)( *
111 ssin IVrealtP = (6-a)
).{
2
3
)( *
222 ssin IVrealtP = (6-b)
where )(
1tP
in and )(
2tP
in denote the instantaneous input
active power to stator1 and stator2, respectively. The
instantaneous input reactive powers are calculated as follows:
).{
2
3
)( *
111 ssin IVimgtQ = (6-c)
).{
2
3
)( *
222 ssin IVimgtQ = (6-d)
Equations (5), (6) and (1-h) form the BDFM vector model.
This vector model is a suitable tool for the analysis of the
machine behaviour in both dynamic and steady-state
operations. It can also be used to develop a vector control
algorithm for the machine.
V. MODEL IMPLEMENTATION IN MATLAB
In order to employ the vector model developed for the
BDFM in section IV, the followings will have to be
performed:
1) Determine the machine inputs. These include the
mechanical input, Tl, and in general, two independent
electrical inputs. The electrical inputs will depend on whether
it is supplied from voltage source or current source supplies.
2) Arrange the machine state equations. In general the
machine states are: rrss
θλλλ
,,, 21 and r
ω
. Note that if a
stator winding is fed by a current source supply, the
corresponding flux vector quantity will be removed from the
machine states.
3) Assign appropriate values to
γ
η
,and r
ϕ
. This is
equivalent to choosing 21 ,ss
θ
θ
and r
ϕ
. In general, these
parameters can be any arbitrary functions of time in the
machine vector model. However, the choice of these
parameters will depend on the application for which the
vector model is utilised. This is the same argument as for
standard AC machines vector models [15]. Usually, for the
purpose of simulation, the free parameters of the vector
model are chosen such that the simulation time becomes less.
For steady-state analysis of the machine operation, it is
convenient to choose the free parameters such that the
machine quantities become DC quantities in steady-state. In
order to implement a torque, speed or position control system
for the machine, it is desirable to choose the free parameters
to simplify the machine control problem, e.g. decoupled
conditions, less complexity in the control algorithm
implementation, etc.
The vector model of the BDFM was implemented in
Matlab and some simulation results are provided here. The
main operating mode of the BDFM is the so-called
synchronous mode of operation in which both windings are
supplied. The machine has two other modes of operation
which are particularly important for the machine analysis [1].
The BDFM can operate in a cascade mode where one
winding is supplied and the other is shorted. This mode of
operation is useful for the machine parameter estimation [8].
Further, it enables the use of a unidirectional inverter when
starting the machine as a motor [17].
The machine transient behaviour in the cascade mode is
shown in simulation using the developed vector model in
Simulink. The 8 pole stator winding is supplied at 220 Vrms,
50 Hz and the 4 pole winding is shorted. The load torque is
set to 1 Nm.
The simulation of the vector model is performed by setting
0
1
=
s
θ
, 0
2
=
s
θ
and 0
=
r
ϕ
. The results are shown in figure
4.
(4-a)
(4-b)
Fig.4: Simulation results for the cascade mode at start-up. The 8 pole stator
winding is supplied and the 4-pole winding is shorted. a) rotor speed b)
machine torque
610
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(4-c)
Fig.4-cont: Simulation results for the cascade mode at start-up. The 8 pole
stator winding is supplied and the 4-pole winding is shorted. c) 8-pole
winding current
VI. CONCLUSION
A vector model for the BDFM is derived. The effect of
multiple loop in each of rotor nest is taken into account. The
model is able to investigate transient and steady-state
behaviour of the machine. The equations are straight forward
to be implemented in Matlab/Simulink. It can be used to
predict the machine behaviour in several modes of operation
including synchronous, cascade and simple induction modes.
The model can be utilised to develop a vector control
system for the machine. The approach presented in this paper
can be generalised to model BDFMs with any stator pole pair
combination and nested-loop rotors with any number of
loops per nest.
REFERENCES
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