A new stator back iron design for brushless doubly fed machines

Full text

Abstract-- The brushless doubly-fed machine (BDFM) has
been under investigation in its modern forms for 50 years.
The reason for this interest is the machine’s ability to
operate as a brushless variable speed motor or generator
through the use of a partially-rated power electronic
converter connected to its second stator winding. Interest
has increased in the last 20 years because of greater
penetration of wind turbines with Doubly-Fed induction
Generators (DFIGs) where brush gear and slip-ring
maintenance and reliability have been a major issue. The
BDFM appears in two forms, the brushless doubly fed
induction machine (BDFIM), with two stator windings and a
third closed rotor winding or the brushless doubly fed
reluctance machine (BDFRM), with two stator windings and
a reluctance rotor with no winding. This paper sets out to
provide a rotor-centric view of BDFM operation, which
shows the basis of its rotating flux pattern, aligning it with
the known Natural Speed, and clarifying the synchronous
and induction modes of operation of the BDFIM and the
synchronous BDFRM. Based on rotor-centric view of the
BDFM, it is shown that the conventional design methods for
the BDFM stator back iron can be modified, leading to a
lighter and smaller machine. The proposed design concepts
are supported by analytical methods and their practicality is
verified using 2-D Finite Element (FE) modeling and
analysis of three experimental BDFMs.
Index Terms-- brushless doubly-fed machine; brushless
doubly-fed induction machine; brushless doubly-fed
reluctance machine; electromagnetic theory; magneto-
motive force; rotor winding; rotor reluctance; finite element
analysis (FEA)
I. INTRODUCTION
The Brushless Doubly-Fed Machine (BDFM) has
been under development in its modern forms for nearly 50
years particularly in the UK, US, Brazil and China. The
BDFM appears in two forms, both having two stator
windings, the Brushless Doubly Fed Induction Machine
(BDFIM), with two stator windings and a closed rotor
winding, and the Brushless Doubly Fed Reluctance
Machine (BDFRM), with two stator windings and a
reluctance rotor with no winding.
Broadway et al. [1] analysed the BDFM from a stator
winding magneto-motive force (MMF) point of view,
following their pioneering work on the pole-amplitude
modulated (PAM) Induction Motor. Wallace et al. [2]
developed the first stator d-q axis analysis of the BDFM
aimed at studying the machine’s transient behaviour, for
the purpose of controlling the Voltage-Source Converter
(VSC) connected to the BDFM’s Control Winding.
Williamson et al [3, 4] applied the machine air-gap MMF
analysis, pioneered on the Induction Motor, to the BDFM
in order to simplify operational understanding. A BDFM
prototype was also constructed and finite element analysis
of its magnetic field was performed. Roberts et al [5, 6]
developed a mathematical theory for BDFM operation
including an equivalent circuit, based on the Induction
Motor, and showed that the equivalent circuit parameters
can be extracted from cascade tests. McMahon et al [7, 8]
used the equivalent circuit model to analyse BDFM
practical aspects such as machine rating, stator core
design, ride-through and rotor winding designs. A number
of BDFMs ranging from 7-250 kW were constructed in
collaboration with a spin-out company, Wind
Technologies Ltd. A 20 kW BDFM wind turbine was
successfully demonstrated [9] and a 250 kW BDFM in a
D400 frame was built, see Fig. 1, demonstrating that
large-scale BDFM is practical and constructible [10].
Rüncos et al [11] in Brazil worked with WEG Industries
and built prototype BDFMs up to 75 kW. Works in China
have been spread amongst a number of universities
working on both BDFIM and BDFRM design and control
theory [12]. In addition, a number of experimental work
has been performed in Netherland and Spain and reported
for example in [13] and [14], mainly on BDFM design
optimization and control strategies.
Almost all of the studies described above are based on
stator-centric approach resulting in unified equivalent
circuits to produce reliable predictions [6]. The stator-
centric approach is necessary for studying the machine’s
performance because the rotor winding or reluctance
structure is not easily accessible. However, it will be
shown in this paper that previously proposed analytical
calculation of the BDFM stator back iron based on stator-
centric approach leads to overestimation of the back iron
depth, because it was assumed that the BDFM stator back
iron needs to be deep enough to carry two separate
asynchronous magnetic fluxes. However, the resulting
BDFM magnetic field pattern has no clear polar
symmetry and the motion of the field is not a simple
rotation. An alternative analytical method based on a
rotor-centric approach is proposed for the stator back iron
depth calculation and is validated by FE analysis, leading
to an optimal value for the stator back iron depth.
A New Stator Back Iron Design For Brushless
Doubly Fed Machines
Salman Abdi1, Ehsan Abdi 2, Senior Member, IEEE, and Richard McMahon1
1 Warwick Manufacturing Group (WMG), University of Warwick, Coventry, UK
2 Wind Technologies Limited, St Johns Innovation Park, Cambridge, UK
Email: s.abdi.jalebi@gmail.com
978-1-5386-3246-8/17/$31.00 © 2017 IEEE
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II. BACKGROUND THEORY
A. Previous Studies on BDFM
The BDFM originated as a single-frame self-cascaded
induction machine, in which two stator windings of
different pole numbers share the same iron circuit with a
rotor winding of related pole number [15]. The
contemporary BDFM has two stator windings connected
to different frequency supplies, producing different pole
number magneto-motive forces (MMF) with no direct
coupling between them, coupling being through the rotor
only. The separate stator windings facilitate double
feeding, with one winding connected to the grid called the
power winding (PW) and the other via a partially rated
power electronic converter called the control winding
(CW), as shown in Fig. 2. The rotor winding carries an
MMF induced by the stator windings and the rotor, and
stator windings are coupled by the flux rotating in the
common iron circuit.
Fig. 1. 250 kW D400 BDFIM (right front) on test bed.
The equivalent circuit for a BDFM, as shown in Fig.
3, was developed in [6] and improved in [7], the latter
introducing the effects of stator and rotor core losses. The
BDFIM flux is the resultant of three winding MMFs
applied to the machine magnetic circuit. Based on
Broadway et al’s analysis [1] the BDFIM consists of:
• The stator power and control windings, with MMFs
of differing pole pairs, p1 and p2," fed at different
angular frequencies,
ω
1 and
ω
2; PW and CW are
designed for no direct coupling, p1 ≠ p2, but coupled
via the rotor only;
• Rotor Winding (RW), q-poles, fixed by the stator
winding pole pair numbers, q=(p1±p2), carrying an
MMF induced by those stator windings;
• The PW, CW and RW MMFs act upon the machine
air-gap to develop a flux, rotating in the machine’s
common iron magnetic circuit, coupling all those
windings.
Williamson et al. [3] analysed the instantaneous BDFIM
air-gap flux waves, bg1 and bg2 based upon the two stator
windings as follows:
Bidirectional
Converter
1
Power Winding
(p pole pairs)
2
Control Winding
(p pole pairs)
Supply Voltage
2
ControlWinding Frequency(f )
1
Power Winding
Frequency(f )
Rotor structure,
wound
or reluctance
q=(p
1
+p
2
)/2
Fig. 2. Schematic of BDFM grid connection.
b
g
1
(
θ
,
t
)=
B
g
1
cos(
ω
1
t
−
p
1
θ
−
α
1
)
(1)
b
g
2
(
θ
,
t
)=
B
g
2
cos(
ω
2
t
−
p
2
θ
−
α
2
)
(2)
where α1 and α2 are the relative phases of the two flux
waves and p1 and p2 are the PW and CW number of pole
pairs. This reflects the total air-gap flux density but in
terms of the two stator windings rather than the rotor side,
thus perpetuating the concept of two machines in one
frame. Williamson et al. also presented in [4] the first
BDFM FE analysis and found out that the flux pattern has
no clear polar symmetry and the motion of the field is not
a matter of simple rotation and does not have a
distinguishing N-S pattern. As a result, the conventional
analytical method for determining depth of back iron used
for an induction machine can not be utilised for the
BDFM.
Roberts in [5] developed an analytical modeling for
the BDFM rotor, which uses only a single d-q pair for
each set of rotor circuits, while, the machine has two
stator supplies of different pole numbers and as such, it is
expected that two d-q pairs is required for each set of
rotor circuits. This reinforces the fact that the BDFM
performance is dominated with one flux system which is
dedicated by its rotor structure complying with the rules
mentioned in [5].
B. Stator Back Iron Depth Calculation
The pole number and magnetic loading determine the
back iron depth in conventional electrical machines. For
the case of an induction machine, the stator back iron
depth is calculated as:
y
c
=2
Bd
2
p B
c
=
Bπd
4
p B
c
(3)
where
B
c
is the back iron maximum flux density,
B
is
the magnetic loading, p is the pole pair number,
and d is
the air gap diameter. However, the BDFM has a complex
magnetic field pattern with no obvious polar symmetry
and simple rotation. Using the BDFM stator-centric
approach, McMahon et. al proposed an analytical method
in [6] to calculate the stator back iron depth as:
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1
R
11
jL
ω
fe
ps
R
11m
jL
ω
1
1.3
1
||
fe
pr
sR
s
'
1
r
R
s
'
1r
jL
ω
1
1.3
1
||
fe
cr
sR
s
'
12m
jL
ω
1.3
||
fe
cs
sR
s
''
2
R
s
''
12
jL
ω
1
V
1
I
'
r
I
''
2
2
1
sV
s
''
2
I
fe
ps
I
fe
pr
I
fe
cs
I
fe
cr
I
Fig. 3. Two terminal equivalent circuit of the BDFIM [11]
y
c
=2
B
1
d
2
p
1
B
c
+2
B
2
d
2
p
2
B
c
=2
d
2
B
c
B
1
p
1
+
B
2
p
2
!
"
#
#
$
%
&
&
(4)
0 5 10 15 20 25 30 35 40
2
4
6
8
10
12
X: 25
Y: 9.4
Stator back iron depth reduction (%)
Stator winding currents (A)
I
pw
FL
I
cw
FL
I
pw
NL
I
cw
NL
Fig. 4. Full-load (T ≈100 N.m) and No-load (T ≈ 0 N.m) stator PW and
CW currents obtained for D180 BDFIM in synchronous mode of
operation.
0 5 10 15 20 25 30 35 40 45 50
0
100
200
300
400
500
Stator back iron depth reduction (%)
Stator winding currents (A)
I
pw
FL
I
CW
FL
I
PW
NL
I
CW
NL
Fig. 5: Full-load (T ≈3600 N.m) and No-load (T ≈ 0 N.m) stator PW and
CW currents obtained for D400 BDFIM in synchronous mode of
operation.
where, B1 and B2 are the rms values of stator PW and CW
flux densities. The FE analysis of prototype BDFMs
performed in synchronous mode of operation shows that
the stator back iron depth obtained from stator-centric
approach is significantly larger than the required depth to
avoid undesirable saturation. In order to investigate
potential weight reduction in the stator back iron, the
performance of two prototype BDFIMs with frame size
D180 and D400 are analysed in their synchronous mode
of operation when the stator back iron depth is reduced.
The stator PW and CW currents in full-load and no-load
conditions are shown in Figs. 4 and 5 for D180 and D400
BDFIMs, respectively. As it is obvious in Figs. 4 and 5,
there are no significant increases in PW and CW currents
in both full-load and no-load conditions when the stator
back iron depth is reduced up to 30%. At this level of
depth reduction, the maximum increase in a stator current
from its rated value is below 3%. In the next section, an
optimal design value for the BDFM stator back iron depth
will be proposed based on a rotor-centric approach.
III. BDFM ROTOR CENTRIC APPROACH
In this section the fundamental air gap MMF waves
are considered to find the total MMF acting across the air
gap. It will then be shown that the response of the BDFM
rotor winding structure to that MMF determines the field
pattern in the machine’s iron region. Hence, the
fundamental flux wave of the BDFM, can be defined by
the (p1+p2)-pole rotor design, linking stator and rotor
iron, rather than the 2p1 and 2p2 poles stator windings.
The authors propose considering first the fundamental
air-gap MMF waves i.e. MMFgs"due to the stator PW and
CW, and MMFgr due to the rotor winding reaction. Then
considering the combined action of these MMFs summing
to the total air gap MMF wave, MMFg , which develops
the machine magnetic field and therefore the rotor torque.
Additional harmonics, due to the winding configuration
add complexity but do not alter the fundamental result.
On these bases, the stator MMF acting on the air gap can
be given as:
MMF
gs
=k
w1
N
1
I
1
cos
ω
1
t−p
1
θ
( )
+K
w2
N
2
I
2
cos
ω
2
t−p
2
θ
−
λ
( )
(5)
Equation (5) can be resolved as:
MMF
gs
=2k
w1
N
1
I
1
cos (
ω
1
+
ω
2
)t/ 2 +(p
1
+p
2
)
θ
/ 2 −
γ
( )
×cos (
ω
1
−
ω
2
)t/ 2 +(p
1
−p
2
)
θ
/ 2 +
γ
( )
−(k
w2
N
2
I
2
−k
w1
N
1
I
1
)cos
ω
2
t−p
2
θ
−2
γ
( )
(6)
The BDFM rotor winding must have q = p1±p2 poles
to meet the BDFM rules [5]. The rotor structure with q =
p1+p2 poles is popular, since it is difficult to achieve
acceptable performance in a BDFM when a q = p1-p2 pole
rotor is utilised. MMFgr in this case is taken the form:
12
12
{cos(( ) / 2
()/2}
gr wr r r
MMF k N I t
ppt
ωω
=+
++
(7)
It can be shown that for a BDFM to have reasonable
PW and CW air gap magnetic fields, the stator windings
are designed in a way to maintain the ratio of
kw2N2I2/kw1N1I1 close to unity. In this case, in (6) the
residual term (with the magnitude of kw2N2I2 – kw1N1I1) is
negligible compared to the main term and hence can be
omitted from further analysis. Hence the stator MMF can
be considered as two waves, one with p1+p2 pole pairs,
and (ω1+ω2)/(p1+p2) rad/sec rotational speed, and another
with p1-p2 pole pairs and (ω1-ω2)/(p1-p2) rad/sec rotational
speed. For a BDFM with q = p1+p2 pole rotor design, the
MMF reflected from the rotor is given by (7). With this
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design therefore, the resulting MMF across the air gap
will be:
MMF
g
=MMF
gs
−MMF
gr
≈2k
w1
N
1
I
1
cos (
ω
1
+
ω
2
)t/ 2 +(p
1
+p
2
)
θ
/ 2 −
γ
( )
×cos (
ω
1
−
ω
2
)t/ 2 +(p
1
−p
2
)
θ
/ 2 +
γ
( )
− k
wr
N
r
I
r
{cos((
ω
1
+
ω
2
)t/ 2
+p
1
+p
2
( )
θ
/ 2}
(8)
The result is that the rotor structure can only respond
to one of the two fundamental components of the stator
MMF wave (with (p1 + p2)/2 pole pairs) and the rotor
suppresses the other stator MMF wave (with (p1–p2)/2
pole pairs), resulting in end winding and air gap flux
leakage. This ensures that the (p1–p2)/2 wave effectively
appears in the leakage path only. Therefore, the
fundamental flux wave of the BDFM, can be defined by
the q-pole rotor design, linking stator and rotor, and
rotating at (ω1+ω2)/(p1+p2). This finding can be used to
obtain an optimum design value for stator back iron depth
if the BDFM is treated as a single-field induction machine
with (p1 + p2) poles and the magnetic loading of
2 2
π
B
1
2
+
B
2
2
. This is the BDFM magnetic loading,
which is shown in [7] to be independent of the pole
number combinations. The optimal value of the BDFM
back iron depth can therefore be calculated using (3).
TABLE I
A SUMMARY OF BDFM STATOR BACK IRON DEPTH CALCULATION
METHODS STUDIED IN THIS PAPER
Old Design
New Design
yc Term
2
d
2
B
c
B
1
p
1
+
B
2
p
2
!
"
#
#
$
%
&
&
Bπd
4
p B
c
,
p
=
p
1
+
p
2
!
,
B
=2 2
π
B
1
2
+
B
2
2
yc Value
D180
21.4 (mm)
14.5 (mm)
D400
54.2 (mm)
37.4 (mm)
Reductio
n in yc
D180
–
32%
D400
–
31%
This is summarised in Table I. The stator core back
length in the new design method is about 32% and 31%
smaller than in original design method in D180, and D400
BDFIMs, respectively. This level of reduction has been
shown in previous sections to be an acceptable limit
before the machine is prone to saturation effects.
IV. EXPERIMENTAL VERIFICATION
Experimental verification for the proposed rotor-
centric approach can be obtained from finite element
analysis of magnetic flux for prototype BDFMs at
different speeds and loads. In this paper, three prototype
BDFMs with frame size D180 and D400 are being
considered, their specifications are described in Table II.
The air gap scalar magnetic potentials, MMFg(θ), have
been extracted from FE analysis for porototype BDFMs
and shown in Figs. 6 to 8 alongside their magnetic flux
plots. In addition, the air gap flux density of D400
BDFIM in synchronous mode of operation obtained from
FE analysis and shown in Fig. 9.
TABLE II
SUMMARY OF PROTOTYPE BDFMS USED IN FE ANALYSIS
BDFM Type
BDFIM
BDFRM
BDFIM
Reference
[7]
[10]
[11]
Year
2006
2011
2013
Frame Size
D180
D400
D400
Rating (kW)
22
250
250
P1 PW Pole Pairs
2
3
2
P2 CW Pole Pairs
4
2
4
q poles
6
5
6
Ns rev/min
500
600
500
Load Condition
Full load
Full load
No load
FE Plot Speed rev/min
834
1200
500
(a)
(b)
Fig. 6. 4/8 poles D180 BDFIM with 6-nest rotor (a) Finite element
magnetic flux plot. (b) air gap scalar magnetic potential.
The normalised plots of scalar magnetic potential in
all BDFMs show that the flux pulsates as it rotates. It can
be also seen an obvious polar symmetry in these plots.
The polarity of the plots is 6-pole for the 4/8 poles
BDFIMs and 5-pole for 6/4 poles BDFRM, confirming
the validity of q-pole rotor centric approach proposed in
section III. The flux density plot for a 4/8 poles BDFIM
shown in Fig. 9 also confirms the q-pole air gap flux
density pattern of the BDFM demonstrating that the air
gap flux density of a BDFM with p1 and p2 pole-pair
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stator windings is similar to that of an induction machine
with p1+p2 pole pair, except that the flux pulsates as it
rotates due to the effects of the two stator windings with
different pole numbers and frequencies. The pulsation
depends on the relative angular positions of the rotor
elements in relation to the two stator windings.
It should be noted that although the flux plots are not
purely sinusoidal, they show a near-sinusoidal air gap
MMF pattern. The air gap flux form of a BDFM can be
improved by better design of the rotor winding in the case
of the BDFIM and reluctance structure in the case of the
BDFRM [8].
(a)
(b)
Fig. 7. 6/4 poles D400 BDFRM with 5-pole reluctance rotor (a) Finite
element magnetic flux plot. (b) air gap scalar magnetic potential.
V. CONCLUSION
This paper has demonstrated the following:
• The BDFM magnetic field, with stator windings of
p1 and p2 pole pairs and a rotor winding structure of
q=(p1±p2) elements, consists of a flux pattern
coupling stator and rotor rotating at (
ω
1+
ω
2)/(p1±p2)
rad/s. To date (p1+p2) rather than the (p1 –p2) rotor
elements have generally been selected for prototype
BDFMs studied in published work, because of
difficulties in achieving acceptable performance
with q=(p1–p2);
• Since the BDFM rotor has by design a winding or
reluctance structure of q=(p1+p2) elements or poles,
the rotor has a flux pattern of q-poles rotating at
ω
r =
(
ω
1+
ω
2)/q rad/s, with a Natural or Synchronous
Speed of
ω
n =
ω
1/q rad/s. The rotor and stator
magnetic structures therefore need to be designed,
for example in the terms of the back iron depth, to
support a q-pole flux pattern rotating at
ω
r =
(
ω
1+
ω
2)/q rad/s;
• The performance of the BDFM depends upon this
single q-pole flux pattern. That performance also
depends, in the case of a BDFIM, upon whether the
rotor body is rotating synchronously or
asynchronously with the
ω
r=(
ω
1+
ω
2)/q rad/s flux
rotation, performing in synchronous or induction
mode respectively. In the case of the BDFRM the
rotor body will operate synchronously with that flux
pattern and perform in synchronous mode;
• The air-gap flux density produced by the BDFM has
been demonstrated from three prototype machines to
be a q-pole wave, which is not a pure sinusoid just
as the air-gap flux density of an IM, however, in the
BDFM it pulsates as it rotates. This will be common
to all BDFM machines, the non-sinusoidal air-gap
flux wave and pulsation being the result of leakage
flux harmonics due to combinations of the two stator
winding and rotor element harmonics. The
waveform of the air-gap flux density can be
improved by the design of the rotor elements;
• The magnetic field in the BDFM can be
characterized by a p1 + p2 pole field rotating at (ω1
+ω2)/(p1 + p2) rad/sec. In order to determine the
stator core back length, the BDFM can be
considered as an induction machine with p1+p2 poles
and the magnetic loading of
2 2
π
B
1
2
+
B
2
2
. The FE
modeling has then been used to verify the proposed
design methodology.
(a)
(b)
Fig. 8. 4/8 poles D400 BDFIM with 6-nest rotor (a) Finite element
magnetic flux plot. (b) air gap scalar magnetic potential.
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Fig. 9. D400 BDFIM air gap flux density obtained from FE analysis in
synchronous mode of operation.
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