Rotor parameter determination for the Brushless Doubly Fed (Induction) Machine (BDFM)

Abstract

A procedure has been established for obtaining equivalent circuit parameters for the brushless doubly fed machine, also known as the brushless doubly fed induction machine, by transforming a reduced coupled-circuit model into a sequence component form. This approach takes advantage of the model reduction procedure's ability to reduce a complex rotor structure to a single equivalent loop without significant loss of accuracy. An alternative method based on winding factors has also been developed and includes coupling via harmonic fields. Parameters determined in both ways are in close agreement with those determined from finite element analysis and experimentally. The advantages of each method are discussed in the conclusion section.

Full text

Rotor parameter determination for the
brushless doubly fed (induction) machine
ISSN 1751-8660
Received on 15th July 2014
Revised on 7th April 2015
Accepted on 26th May 2015
doi: 10.1049/iet-epa.2015.0023
www.ietdl.org
Richard McMahon1, Paul Roberts2, Mark Tatlow1✉, Ehsan Abdi3, Alex Broekhof1, Salman Abdi1
1
Electrical Engineering Division, University of Cambridge, 9 JJ Thomson Avenue, Cambridge CB3 0FA, UK
2
Cambridge Medical Robotics Ltd., Crome Lea Business Park, Madingley Road, Cambridge CB23 7PH, UK
3
Wind Technologies Ltd., Cambridge Science Park, Cambridge CB4 0EY, UK
✉E-mail: mrt39@cam.ac.uk
Abstract: A procedure has been established for obtaining equivalent circuit parameters for the brushless doubly fed
machine, also known as the brushless doubly fed induction machine, by transforming a reduced coupled-circuit model
into a sequence component form. This approach takes advantage of the model reduction procedure’s ability to reduce
a complex rotor structure to a single equivalent loop without significant loss of accuracy. An alternative method based
on winding factors has also been developed and includes coupling via harmonic fields. Parameters determined in both
ways are in close agreement with those determined from finite element analysis and experimentally. The advantages
of each method are discussed in the conclusion section.
Nomenclature
List of symbols
X
1
,X
2
,X
r
indicating a stators 1, 2 or rotor quantity X
p
1
,p
2
stator winding pole pairs (principal fields)
geffective air-gap
l,dstack length and mean air-gap diameter
w
s
slot mouth opening
ω
r
rotor angular velocity
ℝ,ℂ,ℕ,ℤfields of real and complex numbers, sets of
natural numbers and integers
diag(X
1
,X
2
,…) diagonal matrix with scalars or matrices X
1
,X
2
etc.
k
w
winding factor for a single loop in a nest
N
eff
effective turns
Z
eff
impedance of individual rotor loop (Ω)
n
r
rotor turns ratio
1 Introduction
Contemporary interest in the brushless doubly fed (induction)
machine (BDFM) principally arises as it is a potential alternative
to the slip-ring induction generator widely used in wind turbines
although it can also be used as a drive. In recent years, several
large prototype machines have been reported [1–3]. The BDFM
has two windings on a common stator core, configured for
different, non-coupling pole numbers ( p
1
and p
2
associated with
stator 1 and stator 2, respectively). A special rotor is used to
couple with the two stator windings. One stator winding is
connected directly to the grid; the other is supplied with variable
voltage and frequency through a converter. The machine is
normally run in a synchronous mode, with an appropriate
controller, in which the shaft speed is set by the frequencies
supplied to the stator windings. In this mode, the BDFM operates
in a similar way to the doubly fed induction generator with the
torque related to the load angle and a grid side power factor which
can be varied by adjusting the control winding voltage.
The presence of two stators means that there are more variables to
consider than in a single winding machine and the determination of
the ratings of the second stator and associated inverter is complicated
by the fact that control of torque and power factor are not fully
decoupled. The machine’s operating conditions can be directly
calculated using the method described by Williamson et al. [4],
the coupled-circuit (CC) method [5]orbyfinite element analysis
(FEA) [6]. As an alternative, the equivalent circuit approach
provides an effective way of determining operating conditions but
relies on having accurate machine parameters. In addition, the
equivalent circuit forms a valuable design tool, enabling candidate
machine designs to be rapidly evaluated. In particular, rotor
parameters are important in determining the pull out torque of the
machine as well as low-voltage ride through performance in wind
turbine applications. Some parameters can be found by standard
methods but the calculation of parameters for the usual type of
rotor with nests of multiple loops is not straightforward [7]. In
addition, they are speed dependent.
Two methods of obtaining rotor parameters are given in this paper
and applied to the nested loop type of rotor with p
1
+p
2
equally
spaced sets of Nnested loops, as shown in Fig. 1. One approach is
based on the model reduction procedure starting from a CC model
which leads to a single set of dq parameters for the rotor,
independent of operating speed [5]. Implicit in this is that stator to
rotor couplings are assumed to take place only via the principal
(p
1
+p
2
)fields but couplings between rotor loops via harmonic
fields are included. The dq components are then transformed to
give the rotor parameters.
The alternative method is an extension of the approach based on
winding factors (WFs) given in [7] to include coupling via
harmonic fields. The values obtained from the two models are
compared with those found experimentally and by FEA.
1.1 Equivalent circuit
The operation of the BDFM can be described by a per-phase
equivalent circuit of the form shown in Fig. 2awith the
parameters summarised in Table 1. Parameters are referred to the
power winding and iron losses are neglected. The ‘,’modifier
denotes that the quantity is referred. Slips s
1
and s
2
are given by
(11) and (12). The leakage inductances cannot be directly
measured so the simplified equivalent circuit [8] shown in Fig. 2b
is used. In the simplified equivalent circuit, L
1
and L
2
have been
absorbed into L
m1,2
and as a result the values of the magnetising
IET Electric Power Applications
Research Article
IET Electr. Power Appl., 2015, Vol. 9, Iss. 8, pp. 549–555
549
&The Institution of Engineering and Technology 2015

and rotor leakage inductances will change. There will also be a
modification to the turns ratio, n
r
, and hence all referred parameters.
2 CC method
2.1 Steady-state analysis by complex decomposition
The CC model relating the stator and rotor voltages and currents has
been presented in [5]. It was shown that the model can be reduced so
that the rotor–rotor and stator–rotor couplings are dependent on a
single dq pair, each. This reduction is equivalent to modelling the
rotor as having a single-loop per nest. The model can then be
written as
vs
0

=MsMsr
Msr
TMr

d
dt
is
ir

+Rs0
0Rr1

+
v
r
QsQsr
00

is
ir
 (1)
A transformation into a complex form, as first undertaken by Li et al.
[9] for the case of a BDFM with a nested loop rotor, ultimately
allows parameters to be found for an equivalent circuit. The dq0
current components may be transformed into two counter-rotating
vectors (plus a zero component) using the following relationship:
i+
i−
i0
⎡
⎣⎤
⎦=T+
c
T−
c

0
01
⎡
⎣⎤
⎦
id
iq
i0
⎡
⎣⎤
⎦(2)
where T+
c=1
2
√1j

,T−
c=1
2
√1−j
(3)
where superscripts + and −represent the forward and backward
rotating vectors. This transform assigns the dand qaxes to the
real and imaginary axes, respectively. Along with ensuring a
unitary transform, the counter-rotating vector is used for the stator
2, whose rotating magnetic field appears anti-phase in the rotor
reference frame. The transformation matrix for the reduced-order
model will consist of three copies of T+
cand T−
cand two 1 s
along the diagonal. Although, from a mathematical point of view,
these elements comprising the transformation matrix may be put in
any order, the whole transformation matrix for the reduced-order
model can be chosen to be
Tcmplx =
T+
c0000
00T−
c00
0000T+
c
T−
c0000
00T+
c00
0000T−
c
01000
00010
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(4)
The matrix has been organised to group forward rotating, backward
rotating and zero current components together. The two final 1 s are
for the stator zero components. As Tcmplx is unitary, the currents can
be transformed as follows:
i+
cs1
i−
cs2
i+
cr
i−
cs1
i+
cs2
i−
cr
ic∅
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=Tcmplx
idqs1
idqs2
idqr
⎡
⎢
⎣⎤
⎥
⎦⇒
idqs1
idqs2
idqr
⎡
⎢
⎣⎤
⎥
⎦=T∗
cmplx
i+
cs1
i−
cs2
i+
cr
i−
cs1
i+
cs2
i−
cr
ic∅
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(5)
where subscript cindicates that the quantity has been transformed to
the complex form. The voltages can be transformed similarly. These
vectors can then be condensed by defining
i+
cs1
i−
cs2
i+
cr
⎡
⎢
⎣⎤
⎥
⎦
i−
cs1
i+
cs2
i−
cr
⎡
⎢
⎣⎤
⎥
⎦
ic∅
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
:=
i+
c
i−
c
ic∅
⎡
⎣⎤
⎦,
v+
cs1
v−
cs2
v+
cr
⎡
⎢
⎣⎤
⎥
⎦
v−
cs1
v+
cs2
v−
cr
⎡
⎢
⎣⎤
⎥
⎦
vc∅
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
:=
v+
c
v−
c
vc∅
⎡
⎣⎤
⎦(6)
This transformation is applied to the reduced-order BDFM dq model
given in (1) multiplying by Tcmplx from the left and by T∗
cmplx from
the right. Substituting for the complex voltages and currents using
Fig. 1 Nested loop rotor design, with three loops per nest
Fig. 2 Per-phase BDFM equivalent circuit referred to the power winding
aFull per-phase equivalent circuit
bReduced per-phase equivalent circuit
Table 1 Equivalent circuit parameters
Parameter Description
R
1
,R
2
stators 1 and 2 resistances
R
r
rotor resistance
Lm1,Lm2stators 1 and 2 magnetising inductances
L
1
,L
2
stators 1 and 2 leakage inductances
L
r
rotor inductance
IET Electr. Power Appl., 2015, Vol. 9, Iss. 8, pp. 549–555
550 &The Institution of Engineering and Technology 2015

(5) gives a model of the form
v+
c
v−
c
vc∅
⎡
⎣⎤
⎦=Rc+
v
rQc

i+
c
i−
c
ic∅
⎡
⎣⎤
⎦+Mc
d
dt
i+
c
i−
c
ic∅
⎡
⎣⎤
⎦(7)
Details for R
c
,Q
c
and M
c
are given in Appendix.
The structure of (7) shows that the equations relating i+
cto v+
c,i−
c
to v−
cand ic∅to vc∅are all independent of one another. Furthermore, it
is easy to see that due to the structure of (7) if v−
c=
v+
cthen i−
c=
i+
c,
where an overbar denotes the complex conjugate, as shown in
Table 2.
Therefore, in balanced three-phase conditions, the BDFM may be
fully described using the forward-rotating components only
v+
c=(R+
c+
v
rQ+
c)i+
c+M+
c
di+
c
dt(8)
2.2 Equivalent circuit representation
In the steady state, an equivalent circuit representation of (8) can be
derived. With the machine operating in synchronous mode, that is
v
r=
v
1+
v
2
p1+p2
(9)
the frequency of the currents in the rotor reference frame are given by
ω
sync
=p
1
ω
r
−ω
1
=−p
2
ω
r
+ω
2
[5]. Therefore, (8) becomes
3
2
Vc=(R+
c+
v
rQ+
c+j
v
syncM+
c)3
2
Ic(10)
where 
3/2
√Vcand 
3/2
√Ic(V
c
,I
c
∈ℂ
3
) are vectors of appropriate
steady-state components of current and voltage;
v
rQ+
c+j
v
syncM+
cis
given in Appendix by (29).
Following [10], slips between stators 1 and 2 and the rotor are
defined as follows:
s1W
v
1/p1−
v
r
v
1/p1=
v
1−p1
v
r
v
1=−
v
sync
v
1
(11)
s2W
v
2/p2−
v
r
v
2/p2=
v
2−p2
v
r
v
2=
v
sync
v
2
(12)
Substituting (11), (12) and (29) into (10) gives (subject to ω
1
≠0 and
ω
sync
≠0) (see (13))
where Lcs1=L1−M1and Lcs2=L2−M2. Furthermore, the
scaling- and frequency-dependent terms of (10) cancel out,
yielding complex phasor sets
V1
V2
0
⎡
⎣⎤
⎦and
I1
I2
Ir
⎡
⎣⎤
⎦with physical
quantities.
Equation (13) is in a form which admits an equivalent circuit
representation. If the rotor has (or is modelled to have) only one
set of coils (i.e. N= 1), then any phase shift will be eliminated by
an appropriate rotation matrix, Trot, and hence Mcr,Mcs1rand Mcs2r
become purely real.
In this case, a useful equivalent circuit representation may be
derived based on (13), as shown in Fig. 3. This form of the
equivalent circuit is the same as that found in [9,11].
To obtain the equivalent circuit depicted in Fig. 2a, it is necessary
to split the inductance terms into their constituent parts
Lcs1=L1f+L1h+L1l
Lcs2=L2f+L2h+L2l
Mcr=Lr1+Lr2+Lrh+Lrl
(14)
L1fand L2frepresent the self-inductance of the stator windings
arising from the principal fields ( p
1
and p
2
) and the Lr1and Lr2
terms represent the rotor self-inductances linking the rotor to p
1
and p
2
, respectively. The inductances due to all remaining
harmonics are grouped under the subscript hterms. Inductance
due to leakage is self-inductance arising from flux which does not
cross the air-gap, and is represented by subscript lterms.
The individual harmonic terms can be obtained by calculating the
machine parameters of (13) from a Fourier series representation [5].
For each harmonic a separate inductance matrix will be given. As the
dq0 transformation and model reduction procedure given in [5], and
complex form transformation described above, are linear, they may
be applied directly to each harmonic component individually (note
that the matrix used in the model reduction for each harmonic will
always be the one derived for the full system).
Table 2 Complex stator voltages. |V
1
| and |V
2
| are root-mean-square
magnitudes and α
1
and α
2
are phase offsets
Complex form voltages
stator 1
voltage: v+
cs1
v−
cs1
vc∅1
⎡
⎢
⎣⎤
⎥
⎦=3
2
|V1|exp (j( p1
u
r−
v
1t−
a
1))
3
2
|V1|exp ( −j( p1
u
r−
v
1t−
a
1))
0
⎡
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎦
stator 2
voltage:
v+
cs2
v−
cs2
vc∅2
⎡
⎢
⎣⎤
⎥
⎦=3
2
|V2|exp (j( p2
u
r−
v
2dt−
a
2))
3
2
|V2|exp ( −j( p2
u
r−
v
2dt−
a
2))
0
⎡
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Fig. 3 Coupled coils per-phase equivalent circuit for a single rotor circuit
BDFM
V1
s2
s1
V2
0
⎡
⎢
⎢
⎣⎤
⎥
⎥
⎦=−j
v
1
Lcs1
0Mcs1r
0Lcs2
Mcs2r
M∗
cs1rM∗
cs2rMcr
⎡
⎢
⎣⎤
⎥
⎦+
R100
0s2
s1
R20
001
s1
Rr
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
⎛
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎠
I1
I2
Ir
⎡
⎣⎤
⎦(13)
IET Electr. Power Appl., 2015, Vol. 9, Iss. 8, pp. 549–555
551
&The Institution of Engineering and Technology 2015

Using the relationships shown in (14) Fig. 3can be converted to
Fig. 4.
Fig. 4may be further transformed into a ‘T’equivalent form with
transformers [12]. It is convenient to choose the turns ratio in each
case so that it is the effective turns ratio for each coupling. The
effective turns ratio is the square root of the ratio of the
fundamental space harmonic self-inductance terms for the rotor
and the stator [13, p. 381]. Therefore, the effective turns ratios for
stator 1–rotor and stator 2–rotor are
N1=
L1f
Lr1
,N2=
L2f
Lr2
(15)
Since L1f,Lr1and L2f,Lr2represent couplings between same pole
number fields with no parasitic effects, the coupling in each case
is perfect, so
|Mcs1r|= 
L1fLr1
,|Mcs2r|= 
L2fLr2
(16)
Then, converting the coupled coils arrangements of Fig. 4gives the
equivalent circuit as shown in Fig. 5.
Fig. 5can be easily converted to Fig. 2aby noting that, from (11)
and (12), ω
sync
=−s
1
ω
1
. The circuit of Fig. 5may be recognised as
two standard induction machine equivalent circuits with rotors
connected together. Furthermore, Fig. 5may be referred to stator 1
or 2 as in [8]. This equivalent circuit expresses the parameters in
Table 1and is therefore the endpoint of the CC
parameter-determination method.
3 WF method
3.1 Rotor model
The WF method of analysis centres on the loops of the nested loop
rotor which are effectively in parallel and have mutual couplings via
the principal fields and their space harmonics. Each loop has WFs for
the principal fields, magnetising inductances for each space
harmonic, emfs produced by mutual inductances for the various
space harmonics in response to the flow of currents in other rotor
loops and via coupling to the stator. In addition, each loop has a
leakage reactance which can be estimated by conventional means
and a resistance which can be calculated or measured. These are
shown in Fig. 6. In addition to the rotor loops, fictitious stator
coils, with one effective turn, for the two principal fields are
shown for convenience of analysis.
Space harmonics of the principal fields manifest themselves as
magnetising and mutual inductances in each loop. The space
harmonics, pthat exist (harmonic order n) are described in [7]as
p[{{ p1+m(p1+p2)} <{p2+m(p1+p2)}}, m[N(17)
The WF, k
w
, for a single loop in a nest, as shown in Fig. 6is
kw=sin
b
p
2

sin
v
srp/2

v
srp/2(18)
where βis the individual coil span of the loop, pis the rotor harmonic
pole pair given in the set (17) and ω
sr
is the angular slot mouth span.
The self-inductance evaluated for all rotor loops of a specific span,
taking into account each harmonic pole-pair is calculated as
Lm1, 1
... Lm1, n
.
.
...
..
.
.
Lmi,1
... Lmi,n
⎡
⎢
⎢
⎣⎤
⎥
⎥
⎦=
m
0ldq
p
g

×
Neff 1, 1
p1

2
...
Neff 1, n
p1

2
.
.
...
..
.
.
Neff i,1
pi

2
...
Neff i,n
pi

2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(19)
where qis the summation of the pole-pairs ( p
1
+p
2
), nis the loop
number (A, B, C etc), N
eff
is the effective rotor turns found for the
ith harmonic being analysed, the first two components being the
principal fields (i.e. i= 1 corresponds to the first harmonic pole
pair in the set p).
Fig. 6 Nested loop electrical circuit, with three loops per nest
Fig. 4 Coupled coils equivalent circuit separating Mcr,L
cs1
and Lcs2
Fig. 5 BDFM equivalent circuit with separate inductance components
IET Electr. Power Appl., 2015, Vol. 9, Iss. 8, pp. 549–555
552 &The Institution of Engineering and Technology 2015

The total magnetising inductance for each loop from the rotor
space harmonics (excluding the two principal fields) is calculated as
Lmloop, 1
Lmloop, 2
.
.
.
Lmloop, n
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦=

1
i=3
Lmi,1

1
i=3
Lmi,2
.
.
.

1
i=3
Lmi,n
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(20)
By considering the rotor space harmonics, an emf is induced in one
loop by the mmf in another loop. This is accounted for by
considering the mutual inductance, M
x,y
, for each loop xin relation
to another loop y. Noting that the self-inductance of the principal
fields (e.g. M
x,x
) are excluded. In calculating these mutual
inductances, and indeed the magnetising inductances, the effect of
all the nests is included
[M]=
m
0ldq
p
g

×
0
1
i=3
kw1, ikw2, i
(pi)2... 
1
i=3
kw1, ikwn,i
(pi)2

1
i=3
kw2, ikw1, i
(pi)20... 
1
i=3
kw2, ikwn,i
(pi)2
.
.
..
.
...
..
.
.

1
i=3
kwn,ikw1, i
(pi)2
1
i=3
kwn,ikw2, i
(pi)2... 0
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(21)
3.2 Rotor–stator coupling via space harmonics
Space harmonics produced by the rotor will also couple to the stator
and effect rotor parameters. The stator resistance and reactance
referred to the rotor (can be considered a short circuit on stator 1)
is calculated as
R′
s1=
Rs1Neff r/Nef f s1

2
s1
(22)
X′
s1=j
v
Ls1
Neff r
Neff s1

2
(23)
where R
s1
and L
s1
are the stator resistance and inductance,
respectively. Neff rand Nef f s1
are the rotor and stator effective turns
for the harmonic being analysed and s
1
is the slip calculated in
(11) extended to include all space harmonics. Similar equations
can be applied to calculate R′
s2and X′
s2assuming a short circuit on
stator 2 and using s
2
calculated in (12).
The total impedance, not including the effect of mutual couplings
via space harmonics, is given as the sum of rotor coils resistance, R
n
,
the conventional leakage inductance components, overhang, slot and
zig-zag, (Lrn) found using the methods described in [14] and the
magnetising inductance (Lmcoil ) in parallel with the referred stator
impedance (Z
s
′
=R
s
′
+X
s
′
)
Z

=diag Rn+j
v
Lrn+(j
v
Lmcoil )(Z′
s)
j
v
Lmcoil +Z′
s

(24)
The current flowing through the individual rotor loops, for unit
voltage applied to the one-turn coil is given by
I1
I2
.
.
.
In
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦=([M]+[Z])−1
kw1, 1
kw1, 2
.
.
.
kw1, n
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
(25)
Using the current calculated in (25), the impedance, Z
eff
, of the
individual loops, n, in the nested loop rotor taking into account
rotor–rotor and rotor–stator coupling via space harmonics is
Zeff 1
Zeff 2
.
.
.
Zeff n
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦=
kw1, 1
I1

kw1, 2
I2

.
.
.
kw1, n
In

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(26)
3.3 Parameter calculation
The impedance presented by one loop, referred to the fictitious p
1
stator winding, can be determined by considering a short circuit on
stator 2. The loop impedances are taken from (26). Multiplying
this by k2
w1irefers these to the one-turn stator winding and the
impedance presented by one nest is given by the parallel
combination of the loop impedances as shown in (27)
Zeq =
n
i=1
k2
w1i
Zeff i

−1
(27)
A further step is the transformation from the p
1
+p
2
phase system of
the rotor to the three-phase system of the stator multiplying by ( p
1
+
p
2
)/3; this point is discussed in [7]. It is important to note that these
parameters are rotor frequency, that is, speed, dependent.
The turns ratio for the nested loop rotor can be found by
considering an mmf balance with one stator open circuit [7]
nr=
n
i=1
kw1ikw2i/Zi

k2
w2i/Zi

!!!!!!!
!!!!!!!
(28)
These rotor parameters are for the full equivalent circuit shown in
Fig. 2a.
4 Comparison of parameter values
A comparison between the CC and WF methods was carried out
using two BDFMs, a frame size D180 machine [8] and a D400,
250 kW machine [3]. Stator parameters calculated from machine
geometry are given in Table 3. To account for the magnetic
wedges fitted between the stator teeth of the D400 machine, a
modified stator Carter factor was calculated using the procedure
Table 3 Stator parameters calculated from machine geometry
Parameter D180 D400
R
1
,Ω2.3 0.0907
R
2
,Ω4 0.667
L
1
, mH 5.34 2.73
Lm1, mH 410 95.6
L
2
, mH 14.8 45.9
Lm2, mH 417 306
IET Electr. Power Appl., 2015, Vol. 9, Iss. 8, pp. 549–555
553
&The Institution of Engineering and Technology 2015

described in [6]. Rotor parameters found using the CC and WF
methods, evaluated at natural speed, are given in Table 4and are
in close agreement. As discussed in Section 3, rotor parameter
values obtained from the WF method are speed dependent, as
shown in Table 5. However, the variation with speed is negligible.
Table 6shows the effect on rotor parameters if:
(i) All space harmonic couplings: rotor–stator and rotor–rotor,
generated by the rotor are included.
(ii) Only rotor–rotor coupling via space harmonics is included.
(iii) No coupling via space harmonics included.
It can be seen that rotor–stator coupling has a very small effect on
rotor parameters and in most cases does not need to be considered in
contrast to rotor–rotor space harmonic coupling.
5 Experimental parameter extraction
Machine parameters were extracted from the D180 BDFM and 250 kW
machine from cascade tests using a modification of the procedure
described by Roberts et al. [8], in which parameters were obtained
from experimentally determined torque-speed characteristics in the
cascade mode. Finding the best fit to experimental data is done
using non-linear least-squares optimisation. To avoid being trapped
in local minima, the Levenberg–Marquardt algorithm (LMA) [15]
has been employed instead of the simple random search used in
[8]. The LMA shows better resilience against local minima, and
converges faster and more reliably, especially by using multiple
start points. In this case, a speed increase of about 100 times was
achieved using 100 starting points.
Accuracy was improved in comparison with [8]byfixing
parameters that can be accurately measured or to which the fitting
shows little sensitivity. The stator winding resistances, R
1
and R
2
,
were obtained from DC measurements at working temperature (the
skin effect can be ignored). The sum of L
1
and Lm1and L
2
and
Lm2was determined from a no-load test applied, in turn, to each
stator winding, with the other open circuit. The conditions under
which the parameters were extracted are described in [16].
The D180 experimental and FEA results were obtained from
cascade and induction mode tests carried out at a power winding
(PW) and control winding (CW) voltages of 120 V, 50 Hz. The
D400 experimental and FEA results were carried out with the PW
and CW supplied at 70 V, 20 Hz and 240 V, 20 Hz, respectively,
due to restrictions on stator currents; hence, the extracted
parameters do not represent the effect of iron saturation.
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 [8]enablesa
comparison to be made with the experimentally determined values
and those found using the CC and WF methods. The results are
shown in Table 7as well as the values found from FEA in [6]. The
two analytical methods and FEA all lead to values of rotor
resistance and turns ratio which are within 10% of the experimental
values but the inductance parameters can show larger discrepancies.
Inclusion of saturation effects (non-linear) in the FEA shows that
the machines parameters are not greatly altered, showing that the
iron circuits of these machines are conservatively designed.
With the D400 machine, there is about 15% difference in the rotor
inductance, L
r
′
. Abdi et al. [6] attributed this error to the difficulty in
modelling leakage effects (e.g. winding overhang). The agreement
between FEA (linear) and finite element (non-linear), both using
identical geometrical data, is close, which shows that the
non-linearity of the iron circuit can be neglected due to reduced
voltage levels in the tests. Another reason for the discrepancy
between analytical and experimental results is due to the variation
of the air-gap along the length of the rotor as a consequence of
aiming for a small air-gap which turned out to be a challenge to
achieve. The air-gap variation will have an effect on the equivalent
circuit parameters which assume a uniform air-gap.
Table 5 Rotor parameters found by the WF method as a function of
speed
Speed, rev/min D180 D400
L
r
′
,mH R
r
′
,ΩL
r
′
,mH R
r
′
,Ω
350 42.37 1.621 6.275 0.10899
500 42.30 1.623 6.275 0.10903
650 42.26 1.627 6.275 0.10906
Table 6 Comparison of different WF methods for calculating rotor
parameters at natural speed
Machine Parameter All coupling No R–S
a
No R–R
b
D180 R
r
′
,Ω1.62 1.65 1.67
L
r
′
, mH 42.3 42.9 65.4
n
r
0.732 0.732 0.721
D400 R
r
′
,Ω0.109 0.109 0.111
L
r
′
, mH 6.28 6.30 10.48
n
r
0.409 0.409 0.408
a
No rotor–stator coupling via space harmonics.
b
No rotor–rotor or rotor–stator coupling via space harmonics.
Table 4 Comparison of CC and WF methods for determining rotor
parameters
Machine Parameter CC WF
D180 R
r
′
,Ω1.64 1.62
L
r
′
, mH 42.4 42.3
n
r
0.718 0.732
D400 R
r
′
,Ω0.109 0.109
L
r
′
, mH 6.09 6.28
n
r
0.401 0.409
Table 7 Reduced form of rotor and stator parameters extracted from CC, WF, FEA and experimental (Exp) methods
Machine Parameter CC WF FEA (linear) FEA (non-linear) Exp
D180 R
r
′
,Ω1.68 1.67 1.53 1.53 1.6
L
r
′
, mH 56.5 56.7 56 56 54
n
r
0.703 0.716 0.678 0.705 0.72
R
1
,Ω2.3 2.42
R
2
,Ω4 4.04
Lm1, mH 415 416 425 431 457
Lm2, mH 431 435 451 463 493
D400 R
r
′
,Ω0.115 0.115 0.12 0.122 0.114
L
r
′
, mH 16 16.5 15.2 14.3 12.5
n
r
0.359 0.366 0.355 0.365 0.38
R
1
,Ω0.0907 0.0971
R
2
,Ω0.667 0.706
Lm1, mH 98.3 98.3 96.8 97.1 104
Lm2, mH 352 383 349 341 368
IET Electr. Power Appl., 2015, Vol. 9, Iss. 8, pp. 549–555
554 &The Institution of Engineering and Technology 2015

The experimental results, with all parameters referred to stator 1,
are expressed in per-unit and shown in Table 8to provide more
generality to the results and help with comparisons between the
two machines. As expected the per-unit resistance is lower for the
larger machines.
6 Conclusions
Two analytical procedures for obtaining rotor parameters for BDFMs
with nested loop rotors have been described. One transforms a
reduced CC model into a complex form to give speed independent
values. An alternative procedure uses WFs to give parameters
valid for one operating condition but in fact there is little
dependence on speed. Parameters determined by the two methods
are close to those determined experimentally and by FEA. The
ability to calculate parameters accurately enables the exploration of
rotor designs, and the overall machine design. Although this paper
has looked at the nested loop type of rotor, the methods can be
readily extended to other arrangements, including loops of
arbitrary pitch and placement. Using both analytical methods gives
a valuable cross-check, and the WF method gives more obvious
insight into the effects of space harmonics. FEA, of course, is
essential in assessing the effects of saturation.
7 References
1 Carlson, R., Voltolini, H., Runcos, F., Kuo-Peng, P., Batistela, N.: ‘Performance
analysis with power factor compensation of a 75 kw brushless doubly fed
induction generator prototype’. IEEE Int. Electric Machines Drives Conf., 2007.
IEMDC ‘07, May 2007, vol. 2, pp. 1502–1507
2 Liu, H., Xu, L.: ‘Design and performance analysis of a doubly excited brushless
machine for wind power generator application’. 2010 Second IEEE Int. Symp.
on Power Electronics for Distributed Generation Systems (PEDG), June 2010,
pp. 597–601
3 Abdi, E., McMahon, R., Malliband, P., et al.: ‘Performance analysis and testing of
a 250 kW medium-speed brushless doubly-fed induction generator’,IET Renew.
Power Gener., 2013, 7, pp. 631–638
4 Williamson, S., Ferreira, A.C., Wallace, A.K.: ‘Generalised theory of the brushless
doubly-fed machine. Part 1: analysis’,IEE Proc., Electr. Power Appl., 1997, 144,
(2), pp. 111–122
5 Roberts, P., Long, T., McMahon, R., Shao, S., Abdi, E., Maciejowski, J.: ‘Dynamic
modelling of the brushless doubly fed machine’,IET Electr. Power Appl., 2013, 7,
(7), pp. 544–556
6 Abdi, S., Abdi, E., Oraee, A., McMahon, R.: ‘Equivalent circuit parameters for
large brushless doubly fed machines (BDFMs)’,IEEE Trans. Energy Convers.,
2014, 29, (3), pp. 706–715
7 McMahon, R., Tavner, P., Abdi, E., Malliband, P., Barker, D.: ‘Characterising
brushless doubly fed machine rotors’,IET Electr. Power Appl., 2013, 7, (7),
pp. 535–543
8 Roberts, P.C., McMahon, R.A., Tavner, P.J., Maciejowski, J.M., Flack, T.J.:
‘Equivalent circuit for the brushless doubly-fed machine (BDFM) including
parameter estimation’,Proc. IEE B, Electr. Power Appl., 2005, 152, (4),
pp. 933–942
9 Li, R., Spée, R., Wallace, A.K., Alexander, G.C.: ‘Synchronous drive performance
of brushless doubly-fed motors’,IEEE Trans. Ind. Appl., 1994, 30, (4),
pp. 963–970
10 McMahon, R.A., Roberts, P.C., Wang, X., Tavner, P.J.: ‘Performance of the
BDFM as a generator and a motor’. Proc. IEE B –Electric Power Applications,
March 2006
11 Gorti, B.V., Alexander, G.C., Spée, R.: ‘Power balance considerations for
brushless doubly-fed machines’,IEEE Trans. Energy Convers., 1996, 11, (4),
pp. 687–692
12 Carlson, A.B.: ‘Circuits: engineering concepts and analysis of linear electric
circuits’(John Wiley & Sons Inc., New York, 1996)
13 Slemon, G., Straughe, A.: ‘Electrical machines’(Addison-Wesley, London, 1980)
14 Draper, A.: ‘Electrical machines’(Longmans, Reading, MA, 1967, 2nd edn.)
15 Nash, J.C.: ‘Compact numerical methods for computers: linear algebra and
function minimisation’(Adam Hilger, Bristol, 1990, 2nd edn.)
16 Wang, X., Roberts, P., McMahon, R.: ‘Optimisation of BDFM stator design using
an equivalent circuit model and a search method’. The Third IET Int. Conf. on
Power Electronics, Machines and Drives, 2006, April 2006, pp. 606–610
8 Appendix
8.1 Expansion of complex sequence terms
(see (29))
The terms described in the complex sequence component derivation
are expanded here
Qc=TcmplxQdq T∗
cmplx =
Q+
c00
0
Q+
c0
000
⎡
⎣⎤
⎦
Q+
c=−jp1(L1−M1)0
0jp2(L2−M2)

Q+
csr

Mc=TcmplxMdq T∗
cmplx =
M+
c00
0
M+
c0
00Mdq∅
⎡
⎢
⎣⎤
⎥
⎦
Rc=TcmplxRdq T∗
cmplx =
R+
c00
0
R+
c0
00Rdq∅
⎡
⎢
⎣⎤
⎥
⎦
where Q
c
,M
c
,R
c
∈ℂ
5×5
,Q+
c,M+
c,R+
c[C3×3and Q+
csr
[C2×1
and
Q+
csr =−jp1M1srdq ej
f
1
jp2M2srdq ej
f
2
"# (30)
R+
c=diag(R1,R2,Rr) (31)
M+
c=
L1−M10
0L2−M2

M+
csr
M+
csr ∗M+
cr
⎡
⎢
⎣⎤
⎥
⎦(32)
M+
csr =
M1srdq ej
f
1
M2srdq ej
f
2
"# (33)
M+
cr=Ldqr(34)
Table 8 Experiential results expressed in per-unit
Parameter D180 D400
R
r
′
0.19 0.06
L
r
′
2.06 2.06
R
1
0.29 0.05
R′′
20.25 0.05
Lm117.45 17.16
L′′
m29.76 8.77
v
rQ+
c+j
v
syncM+
c=
j(L1−M1)(
v
sync −
v
rp1)0j(
v
sync −
v
rp1)Mcs1r
0j(L2−M2)(
v
sync +
v
rp2)j(
v
sync +
v
rp2)Mcs2r
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
j
v
syncM∗
cs1rj
v
syncM∗
cs2rj
v
syncMcr
⎡
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎦
---------------------------------------
(29)
IET Electr. Power Appl., 2015, Vol. 9, Iss. 8, pp. 549–555
555
&The Institution of Engineering and Technology 2015