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Abstract-- The brushless doubly-fed machine (BDFM) has two
stator windings with different pole numbers, supplied with
different frequencies. Therefore, the distribution of magnetic
fields in stator and rotor iron is complex. In addition, the stator
flux density distribution is non-sinusoidal and has a DC offset at
the natural speed. This makes the use of conventional hysteresis
models utilised for sinusoidal fields impractical for the BDFM. In
this paper a new hysteresis model is proposed for the BDFM stator
iron based on the scalar Preisach model. The rotational
characteristics of the magnetic fields in the BDFM are also
considered and their effects in generating iron losses are assessed.
2-D time-stepping finite element (FE) models are developed for a
prototype D160 BDFM to estimate iron losses and are validated by
experiments.
Index Terms-- Brushless Doubly Fed Machine (BDFM), Epstein
frame, Finite element analysis, Hysteresis models, Iron loss
calculation, Rotational magnetic fields.
I. INTRODUCTION
HE Brushless Doubly Fed Machine (BDFM), previously
known as self-cascaded machine [1], is an adjustable speed
AC electrical machine and can operate as both a generator and
a motor. The machine could be conceptually considered as two
induction machines of different pole numbers and hence
different synchronous speeds for the same supply frequency,
with their rotors connected together both physically and
electrically.
The most promising applications for the BDFM are those
requiring variable speed operation with limited speed range so
the advantage of a partially-rated convertor can be realised. The
advantages of fractional converter and adjustable power factor
have already promoted the use of doubly-fed induction
generator in wind power generation [2]. The BDFM maintains
these advantages but also achieves brushless operation which,
particularly for offshore installations, would be of considerable
benefit; it reduces the failure rate of generators in wind turbines
[3]. In addition, because the BDFM is a medium-speed
machine, its gearbox system is simplified from three stages to
two or one stage, reducing the cost and weight of drive train
system and increasing the reliability.
The BDFM has two stator windings with different pole
numbers, supplied with different frequencies [4]. Therefore, the
distribution of magnetic fields in stator and rotor iron is
complex. In addition, the motion of magnetic flux is not a
simple rotation as in induction machines [5]. In an ordinary
S. Abdi is with School of Engineering, University of East Anglia (UEA), Norwich, NR4 7TJ, UK (e-mail: s.abdi-jalebi@uea.ac.uk).
E. Abdi is with Wind Technologies Ltd, St. Johns Innovation Park, Cambridge, CB4 0WS, UK (e-mail: ehsan.abdi@windtechnologies.com).
R. McMahon is with Warwick Manufacturing Group (WMG), University of Warwick, Coventry, CV4 7AL, UK (e-mail: r.mcmahon.1@warwick.ac.uk).
squirrel cage induction machine and under normal operating
conditions, the slip is relatively low and therefore the rotor core
loss could be neglected. But the rotor electrical frequency in the
BDFM is relatively high and can be as high as 30 Hz. For this
reason and due to the existence of rotor field spatial harmonics
and rotor current time harmonics, the iron loss in the BDFM is
higher than that of conventional induction machines. Moreover,
iron losses can affect flux and torque dynamic responses.
Therefore, accurate modelling of iron loss for the BDFM is
essential in order to optimise the design and performance of the
machine.
Several works have been reported on the modelling of the
BDFM virtually as the connection of two induction motors with
different pole numbers with their rotors electrically and
mechanically connected known as the Cascaded Doubly Fed
Machine (CDFM) [6]. However, in modern BDFMs where both
stator windings are wound in a single frame, more complexity
arises especially when the stator hysteresis loss is to be
analysed. This is because two simultaneous stator fields exist in
the same air gap in the BDFM while there is only one field in
each air gap of a CDFM. In addition, the nonlinearity of the
machine due to the presence of hysteresis effect does not allow
the principle of superposition to be generally applied. This
subject was studied in [7] using the concepts of dissipation and
restoring functions. It was assumed that all the elements of iron
losses including eddy current and hysteresis losses of both
stator and rotor can be considered separately, but the fact that
the stator hysteresis loss from the two fields cannot be
decoupled, was neglected.
An important contribution to investigating the iron loss in
the BDFM is due to Ferreira [8]. They incorporated the iron loss
model using the conventional three-component equation i.e.
hysteresis, eddy current and excess losses, in finite element
time-stepping analysis, and compared the calculated input
power with measurements at the same operating conditions.
However, applying the conventional iron loss model to the
BDFM did not give accurate results mainly because the
assumption of sinusoidal magnetic field distribution cannot be
made for the BDFM stator iron circuit. Zhang et. al. [9] used a
similar method for iron loss calculation with additional
consideration for the rotational effects of the magnetic fields,
however, no experimental iron loss measurement was reported.
Hashemnia et. al. [10] added parallel iron loss resistances to
the BDFM’s equivalent circuit with an aim to improve stead-
Salman Abdi, Ehsan Abdi, Senior Member, IEEE and Richard McMahon
A New Iron Loss Model for Brushless Doubly-
Fed Machines with Hysteresis and Field
Rotational Losses
T
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state performance predictions. The resistances were computed
by considering the BDFM as a CDFM and taking into account
the slip of the rotor with respect to the PW and CW. This, in
part, improved equivalent circuit’s estimation of the steady-
state performance, but considerable mismatch between
experimental and simulation results remained, because the
effects of the BDFM’s complex magnetic fields in the
calculation of stator hysteresis loss were ignored.
Yu et. al. [11] proposed a vector hysteresis model to
calculate the iron losses in the BDFM, but the effects of eddy
current and excess losses were ignored and no experimental
verification was presented. A number of studies have been
conducted on the modelling of iron losses in Brushless Doubly
Fed Reluctance Machines [12, 13]. In [12], a modified
equivalent circuit was proposed which incorporates iron loss
effects. A 2-D finite element method with each lamination
modelled individually was used, as the prototype machine was
axially laminated. It was shown that the rotor in an axially
laminated machine produces higher iron losses, which will
affect the efficiency and thermal stability of the machine.
However, no generic iron loss model capable of predicting
different iron losses at different operating conditions was
provided.
This paper proposes a new iron loss model for the BDFM
drawing from the conventional three-termed iron loss model for
electrical machines i.e. eddy current loss, excess loss and
hysteresis loss. The method utilises the magnetic field obtained
for each element of the iron circuit from FE analysis to calculate
the iron losses in that element. It is therefore based on the post-
processing of the FE analysis. The main contribution of this
paper is the use of scalar Preisach model to estimate the
hysteresis losses in the BDFM, as well as taking into account
variable loss coefficients and the rotational characteristics of
the magnetic field in the stator and rotor iron in generating iron
losses.
II. BRUSHLESS DOUBLY FED INDUCTION MACHINES
The BDFM has two sets of balanced three-phase stator
windings which produce two fields of different pole numbers
(2P1 and 2P2). The pole numbers are selected in a way to avoid
direct transformer coupling between the stator windings and the
coupling between the windings is through the rotor. For this
purpose, P1 and P2 must be different from each other. The rules
for choosing pole numbers for the stator windings are discussed
in [14].
The rotor has a short-circuited configuration and couples the
fields of both stator windings by induction. The nested-loop
design, which was first proposed by Broadway and Burbridge
in [1], is the most widely used, although other configurations
are possible [15].
Typically, there are three different operating modes for the
BDFM. Induction mode is obtained by connecting one stator
winding to the supply and leaving the other winding open. The
characteristics of the machine in this mode are the same as those
of a standard induction machine, but with poorer performance.
If the non-connected stator winding is short-circuited, the
behavior of the machine will be similar to an induction machine
with P1+P2 pole pairs, which is called the cascade mode.
The previous two modes are both asynchronous operating
modes in which the shaft speed is dependent on the loading of
the machine as well as the supply frequency. However, the third
and desirable mode of operation for which the design of the
machine is optimised, is the synchronous mode and is used for
controlled variable-speed operation [16]. In this mode, one
winding, the power winding (PW) is connected directly to the
grid and the other winding, the control winding (CW), is
supplied with variable voltage at variable frequency from a
converter also connected to the grid. A schematic of the BDFM
and the way it is connected to the grid is shown in Fig. 1. In the
synchronous mode, the speed of the rotor shaft in rpm is a
function of the supplied frequencies of two stator windings (f1
and f2) given by:
III. THE PROPOSED IRON LOSS MODELLING OF THE BDFM
The iron losses can be conventionally separated into three
categories: eddy current loss, excess loss and hysteresis loss [8].
These losses are functions of the magnetic flux density and
frequency i.e. B and f, and lamination material characteristics,
which are reflected in Ke, Kex and Khyst factors. The prediction
of these loss terms requires knowledge of the field distribution
in the iron as a function of time. This may be computed using a
finite element model or a magnetic equivalent circuit model
[10]. The computation of flux density distribution in the finite
element method is more precise, hence the distribution of core
losses can be computed more accurately.
Fig. 1. Stator PW and CW grid connection.
A. Rotational Magnetic Field in the BDFM
The rotational variations of flux vectors in the core cause
iron losses to increase compared with the situation in which
there is only an alternating field. In the presence of rotational
magnetic fields, not only there is a 180o movement of the
domain wall, but also a 90o shift occurs [17]. A higher rotational
variation of the flux vectors leads to more iron losses. The
nr=60 f1+f2
p1+p2
P
c=P
e+P
ex +P
hyst =
Kef2B2+Kex f1.5B1.5 +Khyst fB2
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conventional equation for calculating the iron losses given in
(2) accounts only for the losses in a lamination with a purely
alternating field and do not take into account the calculation of
the losses produced by rotating fields.
A simple approach to estimate the iron losses is to apply the
conventional equation to the component of flux density along
its major axis only. This approach entirely neglects the
contribution of the minor axis component [18]. A more accurate
approximation is to take into account the losses produced by the
major and minor axes components of the field independently
and summing them up to estimate the total rotational losses
[19]. It is shown in [19] that this approach can lead to the
estimation of iron losses with acceptable accuracy.
In order to investigate the rotational behaviour of the
magnetic fields in a machine’s iron circuit, Kochmann [20]
proposed using an aspect ratio, defined as the ratio of the flux
along the minor axis (Bminor) to that along the major axis (Bmajor)
of the flux density locus:
𝜆 =
|
𝐵!"#$%
|
%
𝐵!&'$%
%
(3)
A value of zero corresponds to a pure alternating field; and
the closer the ratio is to 1, the more the nature of flux density is
rotational. The aspect ratio is used in our study to investigate
the rotational behavior of the magnetic fields in the BDFM.
The FE models have been developed for a prototype BDFM
with specifications shown in Table I. The radial and tangential
components of flux density, i.e. Br and Bt respectively, in
various locations in the stator and rotor iron shown in Figs. 2
and 3, are obtained by post-processing of FE simulation data.
The values of Bminor, Bmajor, and aspect ratio (λ) when the BDFM
is operating in the synchronous mode and at rated conditions
are shown in Table II. The loci of flux density for the stator and
rotor tooth tip and back iron (points P2 and P7 in Figs. 2 and 3)
are shown in Fig. 4.
TABLE I
SPECIFICATIONS OF THE PROTOTYPE BDFM
Frame size
D160
PW rated flux density
0.28 T
Stack length
190 mm
CW rated flux density
0.34 T
Lamination
M530-
65A
Rated speed
700 rpm
PW pole pair number
2
PW rated voltage
415 V at
50 Hz
CW pole pair number
4
PW rated current
12 A
Stator number of slots
36
CW rated voltage
415 V at
50 Hz
Rotor number of slots
24
CW rated current
5.3 A
In the stator core, the region at the bottom of stator tooth (P6
and P7) shows the largest value of λ with the back iron being
next in importance. The flux density along the tooth depth is
nearly alternating as expected. The only tangential components
found in the stator tooth are in P1 and P2 due to the leakage flux.
For the rotor core, the region close to the air gap presents the
largest aspect ratio. The behaviour of the field along the rotor
teeth and in rotor back iron is nearly alternating.
Fig. 2. Location of elements in stator iron for which flux densities are
calculated.
Fig. 3. Location of elements in rotor iron for which flux densities are calculated.
x
(a)
(b)
(c)
****UNSET****
FILE sync1 TOTAL
1
CURRENT
1
DISPLAY
1
RANGE
1 --- 1
****UNSET****
FILE sync1 TOTAL
1
CURRENT
1
DISPLAY
1
RANGE
1 --- 1
-2 -1 0 1 2
-2
-1
0
1
2
Br (T)
Bt (T)
-1.5 -1 -0.5 0 0.5 1 1.5
-2
-1
0
1
2
Br (T)
Bt (T)
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Br (T)
Bt (T)
P7
P8
P9
P10
P11
P12
P13
P3
P2
P1
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P3
P2
P1
P4
P5
P6
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(d)
Fig. 4. Loci of B for: (a) stator tooth tip (P2), (b) stator back iron (P7), (c) rotor
tooth tip (P2), (d) rotor back iron (P7).
B. Eddy Current Loss
When a conductive material is exposed to time-varying
magnetic fields, loops of eddy currents are induced. Despite the
fact that in electrical machines iron cores are laminated in order
to reduce the flow of eddy currents, there are still losses due to
the Joule’s effect. The time average value of the losses is given
by:
(4)
where T is the period of induction, d is the lamination thickness,
σ is the iron conductivity and ρ is the iron density. To take into
account the effects of rotational magnetic flux in the BDFM,
the eddy current loss can be expressed as [19]:
(5)
TABLE II
FLUX DENSITY ALONG THE MINOR AND MAJOR AXIS, AND THE ASPECT
RATIO IN THE STATOR AND ROTOR IRON CIRCUITS
Points
Bminor (T)
Bmajor (T)
ls
Bminor (T)
Bmajor (T)
lr
Stator iron circuit
Rotor iron circuit
P1
0.56
1.82
0.31
0.37
1.60
0.23
P2
0.50
1.83
0.27
0.32
1.48
0.23
P3
0.41
1.87
0.22
0.19
1.31
0.15
P4
0.14
1.63
0.09
0.06
1.41
0.04
P5
0.11
1.51
0.07
0.04
1.53
0.03
P6
0.88
1.29
0.68
0.18
1.62
0.11
P7
0.66
1.25
0.53
0.18
1.25
0.14
P8
0.34
1.46
0.23
0.10
0.98
0.10
P9
0.09
1.46
0.06
0.05
0.69
0.07
P10
0.10
1.44
0.07
0.07
0.91
0.08
P11
0.31
1.46
0.21
0.06
1.03
0.06
P12
0.45
1.49
0.30
0.22
1.07
0.21
P13
0.06
1.49
0.04
0.04
0.91
0.04
C. Excess Losses
Staumberger et. al. in [21] presented a new physical concept
of Magnetic object. Under this concept, the magnetic domain
wall movements dislocate other domain walls and they are all
related in the same correlation region. Each correlation region
corresponds a magnetic object. A magnetic field is originated
by the currents created by the magnetic object movement.
Therefore, an external field is needed to compensate this field,
causing excess losses, which may exceed the eddy current loss
predicted using the classical model of (4). The mean value of
the excess loss is:
Kex is the excess loss coefficient, which depends on the iron
material characteristics. In the presence of rotational magnetic
field in the iron circuit, the excess loss can be obtained from
[18]:
D. Hysteresis Loss
The hysteresis loss is the energy required to overcome the
impedance of the domain walls motion, which occurs when a
material is magnetised by defects in the magnetic material [22].
For the BDFM rotor iron circuit, where the main field
components have the same frequency, the hysteresis loss is
given by:
where Pxhyst and Pyhyst are the hysteresis losses computed for the
spatially orthogonal components of the flux density, x and y in
this case. This may be combined from different formulations to
give the resulting hysteresis loss. Stumberger et. al. [21]
showed that the formulations that have been used in the
literature give core loss predictions that are not considerably
different. Therefore, a simple summation is adopted. To take
the effects of rotational rotor magnetic field into account, the
hysteresis loss can be expressed as:
𝑃!"#$ = 𝑃!"#$
%+ 𝑃!"#$
$
where Prhyst and Pthyst are given by
(10)
(11)
Khyst and α are constants and determined by the core material
characteristics. K(B(t)) is an empirically determined minor loop
correction factor to account for minor hysteresis loops that are
caused by time harmonics in the flux densities and is given by
[22]:
(12)
where ΔBi is the difference between the local minimum and
local maximum values of the flux density waveform.
The prediction of the stator hysteresis loss component
requires a different approach because the stator flux density
distribution in the BDFM is non-sinusoidal. It also has a DC
offset at the natural speed. Hashemnia et. al. in [10] proposed a
method to calculate the hysteresis loop area without the
knowledge of the hysteresis loop shapes when the stator iron
circuit is subjected to a non-sinusoidal magnetic flux density of
the type in the BDFM. Accurate computation of the hysteresis
loops requires that the hysteresis model be incorporated into the
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Br (T)
Bt (T)
Pd
T
B
t
dt
12
1
e
x
T
22
σ
ρ
=
P
e=
σ
d2
12
ρ
1
T
Br
t
2
+Bt
t
2
T
dt
P
ex =Kex
T
dB
dt
1.5
dt
T
P
ex =Kex
T
dBr
dt
2
+dBt
dt
2
T
0.75
dt
P
hyst =f(P
hyst
x,P
hyst
y)
P
hyst
r=Khyst fBpr
α
K(Br(t))
P
hyst
t=Khyst fBpt
α
K(Bt(t))
K(B(t)) =1+0.65
Bm
ΔBi
i
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FE model when computing the magnetic field distribution [23].
Solving the resulting FE equations at each time-step will
however involve hundreds of iterations, leading to excessive
simulation time.
Belahcen et. al. [24] showed that the core loss can also be
predicted accurately using B and H vectors that are computed
using vector hysteresis model posteriori. It essentially means
that B, computed using non-linear single-valued B-H curve, is
used as the input into a scalar hysteresis model in order to
compute H. A scalar Preisach model is used to determine the
static hysteresis magnetic field strength, Hhyst, corresponding to
a given magnetic flux density obtained from the FE simulation
based on the Maxwell’s equations. In the scalar Preisach
hysteresis model, a ferromagnetic material is represented as a
superposition of shifted rectangular elementary hysteresis
operators with h1 switching down and h2 switching up fields.
The magnetic flux density in this model is expressed as [25]:
(13)
where
g
(h1, h2, H(t)) represents the elementary operator and is
equal to:
(14)
The
µ
(h1, h2) is the Preisach distribution function that can be
considered as a weight for the elementary operator. Several
expressions are given for the Preisach distribution in the
literature, one is proposed by [26] as:
(15)
where
j
is a one-dimensional function. For a ferromagnetic iron
material used in electrical machines’ lamination,
j
can be
represented by:
(16)
where a and b are dependent on the material characteristics and
are obtained from the lamination B-H curves. It was shown in
[26] that the gradient of flux density over magnetic field
strength can be expressed as:
(17)
By numerical integration of (17), the hysteresis field strength
Hhyst for a given flux density value can be obtained. Finally, the
hysteresis loss component is obtained by numerical
computation of the following equation [25]:
(18)
The proposed iron loss computation procedure is summarised
in Fig. 5.
Fig. 5: Proposed iron loss calculation procedure (i.e. Model 3).
IV. CALCULATION OF BDFM IRON LOSS COEFFICIENTS
The conventional three-term iron loss model of (2) uses
material-dependent coefficients for the iron loss terms.
However, the use of constant coefficients may not be practical
in some operating conditions, especially when a non-sinusoidal
magnetic field is present [27]. Chen et. al. proposed a model
with constant coefficients for the eddy-current and excess
losses and variable coefficients for the hysteresis loss [28]. In
[27] a mathematical procedure was proposed to determine the
iron loss coefficients which are varied with frequency and flux
density. The method was shown to give good estimation of iron
losses in the laminations when compared to experiments.
The specific core loss data for the BDFM lamination (type
M530-65A as given in Table I) have been obtained from
Epstein frame loss measurements, where the sample under test
is subjected to sinusoidal excitation on the primary winding,
while the open-circuit voltage on the secondary is measured
[29]. The measured specific core losses for a frequency range
of 10-50 Hz and flux densities from 0.5 to 2 T are shown in Fig.
6. It is then possible to obtain the variable loss coefficients by
fitting loss data as described below.
Under a sinusoidal alternating excitation, the specific core
losses wFe in watts per kilogram can be expressed as:
(19)
B(t)=
µ
(h
1,h2)
γ
(h
1,h2,H(t))dh
1dh2
T
γ
(h
1,h2,H(tk)) =
1if H (tk)h
1
+1if H (tk)h2
γ
(h
1,h2,H(tk1)) if h1H(tk)h2
µ
(h
1,h2)=
ϕ
(h
1)
ϕ
(h2)
ϕ
(x)=ae
γ
x
(1+be
γ
x)2
dB
dH =2
ϕ
(H)
ϕ
(h
1)dh
1
H
H
P
hyst =1
TH.B
t
T
.dt
wFe =Kef2B2+Kex f1.5B1.5 +Khyst fB
α
The flux density, B(t), is obtained for every element of the
stator and rotor iron using post processing of nonlinear FE
analysis
The rotational components of the flux density, Br(t) and
Bt(t), are computed
The eddy current loss is calculated for each element of the
stator and rotor iron using (5) and summed to obtain the
total eddy current loss
The excess loss is calculated for each element of the stator
and rotor iron using (7) and summed to obtain the total
excess loss
The rotor hysteresis loss is calculated for each element of
the rotor iron using (9) - (12) and summed to obtain the
total rotor hysteresis loss
The stator hysteresis loss is calculated for each element of
the stator iron using (13) - (18) and summed to obtain the
total stator hysteresis loss
The above losses are summed up to determine the total iron
loss in the machine
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6
6
In the first step of identifying the coefficients, (19) is divided
by frequency resulting in:
(20)
where
(21)
Fig. 6. Specific core loss for ranges of frequency and flux density in the
lamination sheet obtained from experiments.
The coefficients a, b, and c are determined by quadratic
fitting based on a minimum of three points. Fig. 7 shows ratio
of core loss to frequency
&!"
'
as a function of $
𝑓
according to
(20) for different flux densities. Each curve is obtained from
five measurements carried out at the same flux density but
different frequencies to improve the accuracy of the numerical
procedure.
Fig. 7. Specific core loss per frequency versus square root of frequency.
The eddy current coefficient
𝐾(
and the excess loss
coefficient
𝐾()
are derived from specific core loss
measurements for a single lamination using (20) at different
values of flux density. These coefficients are independent of
frequency, but unlike those for conventional models, they show
a significant variation with flux density as illustrated in Figs. 8
and 9. Hence, the following third-order polynomials were
employed for curve fitting to obtain
𝐾(
and
𝐾()
:
(22)
(23)
In order to identify the power
a
for the hysteresis loss, a
third-order polynomial is used:
(24)
Substituting (24) in (21) and applying a logarithmic
operator, it can be shown that:
(25)
Coefficient a represents the ratio of hysteresis loss to
frequency and is extracted from (20) after substituting b, c,
𝐾(
and
𝐾()
with (21) - (23). The logarithm of a versus flux density
is shown in Fig. 10. As can be seen, there are two regions with
distinct variation patterns and therefore two flux density regions
may be defined which approximately include the ranges 0.5-1.2
T and 1.2 - 2 T. It is worth noting that in [30] and [27], a two-
region and a three-region approximation of
𝐾!"#$
and
a
was
used, respectively. For a given frequency and flux density
range, (25) is solved by linear regression using at least five
values for log B. The hysteresis loss parameters for different
frequencies and flux density ranges are obtained as shown in
Table III.
Fig. 8. Variation of the eddy current coefficient with flux density.
𝐾!
is
independent of frequency.
Fig. 9. Variation of the excess loss coefficient with flux density.
𝐾!"
is
independent of frequency.
The specific core loss at different flux density and frequency
is calculated from (19) using the parameters extracted from
experimental test, and results are compared in Fig. 11 with
measured losses. The relative error between the estimated and
measured specific core losses is less than 9% across all
measurements.
( )
2
Fe
wabf c f
f=+ +
a=Khyst B
α
,b=Kex B1.5,c=KeB2
0.5 1 1.5 2
Flux Density (T)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Specific Core Loss (W/kg)
f = 10Hz
f = 20Hz
f = 30Hz
f = 40Hz
f = 50Hz
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5
Square root of frequency (Hz
0.5)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Specific core loss per frequency (W/kg/Hz)
B = 0.5 T
B = 1 T
B = 1.5 T
B = 2 T
23
01 2 3ee e e e
KK KBKBKB=+ + +
Kex =Kex0+Kex1B+Kex 2B2+Kex 3B3
23
01 2 3
BB B
aa a a a
=+ + +
log a=log Khyst +
α
0+
α
1B+
α
2B2+
α
3B3
( )
log B
0.5 1 1.5 2
Flux Density (T)
0
0.00005
0.0001
0.00015
0.0002
0.00025
0.0003
0.00035
0.0004
Coefficient Ke (W/kg/Hz 2/T 2)
extracted from experiments
fitted curve
0.5 1 1.5 2
Flux Density (T)
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
Coefficient Kex (W/kg/Hz 1.5/B 1.5)
extracted from experiments
fitted curve
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Fig. 10. Logarithm of the ratio of hysteresis loss to frequency. Curves for
different frequencies are overlapping.
Fig. 11. Relative error between calculated and measured specific core losses at
various flux and frequency conditions.
V. BDFM IRON LOSS COMPUTATION
The specifications of the prototype BDFM considered in this
study are given in Table I. The FE analysis of the machine is
performed using a commercial software application EFFE [14].
A voltage-fed time-stepping analysis is used to compute the
flux density in the stator and rotor iron circuits when the BDFM
is operating in the synchronous mode. A 2-D FE model is
developed to reduce the computational time by assuming that
the effects of axial flux are negligible. The end region leakage
effects are incorporated into the analysis using lumped
parameters [3]. The modelling is performed using the time-
stepping method for accurate analysis and took into account the
nonlinear properties of the iron.
In order to validate the proposed iron loss computation
method, an open-circuited rotor shown in Fig. 12 is used. Thus,
the total iron losses can be obtained from no-load locked-rotor
tests using stator winding measurements since both mechanical
and rotor copper losses are eliminated.
(26)
Pin is the total input power to stator PW and CW and the Pcu-
PW and Pcu-CW are the copper losses dissipated in the stator PW
and CW, respectively.
Once the FE model is solved, the local flux density
waveforms for both the stator and rotor regions are extracted
from the FE solution. The flux density data for every element
in the mesh at each time-step is logged in a file for further
analysis. The process is incremented to the next time-step and
repeated until the required data for a complete period of the flux
density is obtained. Each data set includes the element number,
the x and y components of the flux density, are the coordinates
of the centroid of the elements. Initially, the data is processed
to decompose the flux density into radial and tangential
components, enabling the losses resulting from rotational flux
patterns in the stator and rotor laminations to be calculated.
TABLE III
HYSTERESIS LOSS PARAMETERS FOR THE LAMINATION STEEL
Flux Density (T)
Frequency
(Hz)
𝐾#$%&(𝑊 𝐾𝑔/𝐻𝑧/𝑇'
)
a
0.5 < B < 1.2
10
0.0132
2.5787
20
0.0132
2.5748
30
0.0132
2.5716
40
0.0131
2.5685
50
0.0131
2.5651
1.2 < B < 2
10
0.0147
1.2311
20
0.0148
1.2219
30
0.0149
1.2150
40
0.0149
1.2092
50
0.0150
1.2042
Next, the local loss densities are calculated for the radial and
tangential components of the flux density, then summed to give
the elemental iron loss density. The local eddy current and
excess loss densities for every element are computed using (5)
and (7), respectively. The local hysteresis loss density for the
rotor elements are computed using (9) - (12). For the stator iron
circuit, however, the scalar Preisach model presented in Section
III-D is employed to compute the hysteresis loss for each
element using (13) - (18). These iron loss densities are
multiplied by the mass of the iron calculated using the element
areas and length of the iron core. Finally, these localised iron
losses are summed to give the total iron loss dissipated in the
machine.
Fig. 12. The BDFM rotor used in this study with open-circuited winding.
VI. EXPERIMENTAL VALIDATION OF THE BDFM IRON LOSS
MODEL
The test bench for the BDFM is shown in Fig. 13. The
machine is operated at the no-load locked-rotor condition with
an open-circuited rotor winding. The delta-connected PW is
connected to the grid through a variac and is supplied at 230
Vrms, 50 Hz. The CW is also connected in delta and supplied
by a unidirectional converter at a constant V/f ratio. A sinusoidal
filter is connected between the converter output and the CW to
0.5 1 1.5 2
Flux density (T)
-2.8
-2.6
-2.4
-2.2
-2
-1.8
-1.6
-1.4
Log a (W/kg/Hz)
f = 10Hz
f = 20Hz
f = 30Hz
f = 40Hz
f = 50Hz
0.5 1 1.5 2
Flux density (T)
-10
-8
-6
-4
-2
0
2
4
6
8
10
Relative error (%)
10Hz 20Hz 30Hz 40Hz 50Hz
P
iron =P
in P
cuPW P
cuCW
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filter out the harmonics produced by the converter’s switching
frequency. At each CW voltage level, the iron loss is measured
using (26) by subtracting the PW and CW copper losses from
the input power. The per-phase resistance values of PW and
CW were obtained by DC measurements and are 6.3 Ω and 3.9
Ω, respectively. The voltages and currents of each stator phase
are measured by LEM LV 25-p and LEM LTA 100-p
transducers, respectively. A V/f control algorithm is
implemented in MATLAB, which generates PWM signals for
the converter.
Three different modelling approaches are used to compute
the BDFM iron losses as described below. A summary of the
iron loss models is also shown in Table IV.
Model 1 uses iron loss equations of (4), (6) and (8) for
computing eddy current, excess and hysteresis losses,
respectively. It ignores the presence of rotational fields in the
BDFM.
Model 2 takes into account the rotational characteristics of
the magnetic field by using (5) and (7) for computing eddy
current and excess losses, respectively, and (9) - (12) for
computing hysteresis losses in the stator and rotor iron.
Model 3, proposed in this paper, uses the same method as
Model 2 to calculate eddy current, excess and rotor hysteresis
losses. However, it employs (13) - (18) for the calculation of
stator hysteresis loss.
Fig. 14 compares the iron losses computed from the above
three models with experimental results at different CW
voltages. Close agreement can be seen between the
computational results from Model 3 and experimental
measurements which validates the practicality of the proposed
iron loss modelling approach for the BDFM. The rise in the iron
losses as the CW voltage is increased is due to the increase in
the CW supply frequency set by the v/f controller.
Fig. 13. Prototype D160 BDFM on the test rig.
Fig. 14. Iron loss values at different CW voltages, obtained from experiments
and iron loss models. Vpw = 230 V and fpw = 50 Hz.
TABLE IV
THE EQUATIONS USED IN CONSIDERED IRON LOSS MODELS. MODELS 1 AND 2
ARE CONVENTIONAL IRON LOSS MODELS AND MODEL 3 IS THE ONE PROPOSED
IN THIS PAPER.
Eddy
current loss
Excess
loss
Rotor hysteresis
loss
Stator
hysteresis loss
Model 1
(4)
(6)
(8)
(8)
Model 2
(5)
(7)
(9) - (12)
(9) - (12)
Model 3
(5)
(7)
(9) - (12)
(13) - (18)
Fig. 15 shows the relative error between the iron losses
measured by experiment and computed by the models. The
highest error is attributed to Model 1, as expected. The error
associated to Model 2 is noticeably lower than Model 1, which
shows that the rotational characteristics of the magnetic field
have important effects on the machine iron losses. The least
error, by far, is due to Model 3. Larger error is generally seen
as the CW voltage is increased, which is mainly because of
excessive saturation of the iron circuit, especially in the stator
teeth where the highest levels of flux density exist, leading to
additional losses that are not modelled by the analytical
methods.
Fig. 15. Relative error between the measured and modelled iron losses.
Fig. 16 shows the breakdown of the iron loss components
computed using Models 1, 2 and 3. As it is evident from Figs.
16a and 16b, the loss curves follow a similar rising trend,
however, accounting for the rotational characteristics of the
magnetic fields in Models 2 and 3 has led the eddy current,
excess and rotor hysteresis losses to be notably larger
throughout the CW voltage range compared to when those
effects are ignored in Model 1.
The most noticeable difference can be observed in the stator
hysteresis loss computed by Model 3 compared to Models 1 and
2. The difference becomes significant above the CW voltage of
200 V where the hysteresis loss increases sharply in Model 3,
while only a slight increase can be observed in Model 1 and 2.
This explains the significant difference between the predictions
of Models 2 and 3 in Fig. 14 above the CW voltage of 200.
Thus, the proposed stator hysteresis model given by (13) - (18)
has enabled the iron loss predictions by Model 3 to closely trace
the experimental results, maintaining the error within an
acceptable range of 6 to 11%.
Consequently, it can be concluded that the conventional
method used in Model 1 for iron loss calculation, which was
originally developed for electrical machines with single
frequency and alternating magnetic fields is not suitable to the
BDFM with a complex magnetic field pattern resulted from the
presence of two magnetic fields with different frequencies and
pole numbers. In addition, while incorporating the rotational
0 50 100 150 200 250 300 350 400
CW Voltage (V)
450
500
550
600
650
700
750
800
850
Iron Loss (W)
Experiment
Model 3
Model 2
Model 1
0 50 100 150 200 250 300 350 400
CW Voltage (V)
5
10
15
20
25
30
Error (%)
Model 3
Model 2
Model 1
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magnetic field effects into the conventional model i.e. Model 2
has improved the accuracy of iron loss prediction, there is still
significant disagreement between the predicted and measured
iron loss values.
The proposed iron loss model in this study i.e. Model 3, has
led to significant improvement in the accuracy of iron loss
prediction in the BDFM. This is mainly because, in comparison
with Model 2, the scalar Preisach model has been utilised for
the calculation of stator hysteresis loss which is known to give
more accurate estimate when nonlinear and non-stationary
magnetic fields are present, as in the BDFM.
(a)
(b)
(c)
Fig. 16. Iron loss components computed by Models 1 to 3; (a) sum of eddy
current and excess losses, (b) rotor hysteresis loss, (c) stator hysteresis loss.
VII. CONCLUSIONS
In this paper, a new approach for modelling the iron losses
in the BDFM has been proposed. The iron loss calculation is
particularly challenging for the BDFM since two magnetic
fields with different frequencies and pole numbers are present
in the iron circuit. This causes a nonlinear magnetic field to be
induced in the BDFM stator iron with a DC offset and hence the
conventional hysteresis models may not be suitable for
computing the stator hysteresis losses. A new method based on
the scalar Preisach model has been developed to estimate the
hysteresis loss in the stator. It essentially uses the flux density
computed from the FE model that incorporates a nonlinear
single-valued B-H curve to compute H by utilising the scalar
hysteresis model. Then the computed B and H values are used
to calculate the stator hysteresis losses. The effects of rotational
magnetic field have also been considered in the iron loss model.
The main limitation of the proposed method is that iron losses
are calculated offline from post-processing of FE results, which
compromises the accuracy of iron loss calculations.
Nevertheless, experimental tests, conducted on a laboratory
BDFM, has validated the proposed model, with predictions
being <11% lower than measurements.
It is worth noting that although the proposed iron loss model
is developed and verified on a 10 kW laboratory BDFM, the
loss calculation procedure can be generalised and applied to
larger machines and with different designs and configurations.
Future work may include more accurate measurement of iron
losses using calorimetric measurements and the thermal
modelling of the machine for further optimisation of the thermal
design. In addition, by utilising a wireless rotor current
measurement technique, as shown in [31], iron losses may be
measured at more practical operating conditions, especially
when the machine is loaded.
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0 50 100 150 200 250 300 350 400
CW Voltage (V)
240
260
280
300
320
Pe + Pex (W)
Model 1
Model 2 & 3
0 50 100 150 200 250 300 350 400
CW Voltage (V)
100
105
110
115
120
Physt-rotor (W)
Model 1
Model 2 & 3
0 50 100 150 200 250 300 350 400
CW Voltage (V)
100
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200
250
300
Physt-stator (W)
Model 1
Model 2
Model 3
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Transactions on Energy Conversion
IEEE TRANSACTIONS ON ENERGY CONVERSION
10
10
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[26] Z. Szabo, I. Tugyi, G. Kadar, J. Fuzi, “Identification procedures for scalar
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IX. REFERENCES
Salman Abdi received the B.Sc. degree from
Ferdowsi University, Mashhad, Iran, in 2009, and
the M.Sc. degree from the Sharif University of
Technology, Tehran, Iran, in 2011, both in electrical
engineering. He then completed the Ph.D. degree in
electrical machines design and optimisation from
Cambridge University, Cambridge, U.K, in 2015.
He is currently an Assistant Professor in Electrical
Engineering at the University of East Anglia (UEA),
Norwich, UK. His main research interests include
electrical machines and drives for renewable power
generation and automotive applications.
Ehsan Abdi (SM’12) received the B.Sc. degree from the Sharif University of
Technology, Tehran, Iran, in 2002, and the M.Phil.
and Ph.D. degrees, from Cambridge University,
Cambridge, U.K., in 2003 and 2006, respectively, all
in electrical engineering. He is currently the
Managing Director of Wind Technologies Ltd.,
Cambridge, where he has been involved with
commercial exploitation of the brushless doubly fed
induction generator technology for wind power
applications. He became a Senior Member of the
IEEE in 2012. His main research interests include
electrical machines and drives, renewable power
generation, and electrical measurements and
instrumentation.
Richard McMahon received the B.A. degree in
electrical sciences and the Ph.D. degree from the
University of Cambridge, Cambridge, U.K., in 1976
and 1980, respectively. Following post-doctoral work
on semiconductor device processing, he became a
University Lecturer in electrical engineering with the
Engineering Department, University of Cambridge, in
1989, where he was a Senior Lecturer in 2000. In 2016,
he joined the Warwick Manufacturing Group (WMG),
University of Warwick, Coventry, U.K., as a Professor
of power electronics. His current research includes
electrical machines, power electronics and the
electrification of transport.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on April 29,2021 at 13:08:54 UTC from IEEE Xplore. Restrictions apply.
... The differences are dependent on the frequency, the magnetic flux density and the axis ratio of the flux density loci. To consider such rotational magnetizations, Zhu et al. (2019) and Abdi et al.(2021) suggested to divide the field trajectories for the calculation of the losses into two components, the rolling and transverse direction. As shown by Ragusa et al. (2007) and Appino et al. (2009), the hysteresis losses and excess losses for low values of flux densities (B < 1.5 T) increase by increasing axis ratio between the maximum flux density in the rolling and transverse direction. ...
... For flux density above a certain high value (B >1.5 T), the hysteresis and excess losses for rotating magnetizations begin to decrease, whereas the hysteresis losses and excess losses for alternating will increase further. This effect is not considered in the models by Zhu et al. (2019) and Abdi et al.(2021). To consider these effects in the loss modeling, different approaches have been developed. ...
... The detailed theory of the adapted IEM loss model for rotational magnetization and detailed measurement results are presented in a study by Schauerte et al. (2021) as a preceding publication. In this paper, the adapted IEM loss model for rotational COMPEL magnetization is used for the iron loss calculation in a finite element method (FEM) simulation of an exemplary induction machine (IM) as it is done in studies by Abdi et al. (2021), Belahcen et al. (2014) and Hernandez-Aramburo et al. (2003). In addition, two frequently used frequency-based and one time-based model are used for the iron loss calculation in the FEM. ...
Article
Purpose Various iron loss models can be used for the simulation of electrical machines. In particular, the effect of rotating magnetic flux density at certain geometric locations in a machine is often neglected by conventional iron loss models. The purpose of this paper is to compare the adapted IEM loss model for rotational magnetization that is developed within the context of this work with other existing models in the framework of a finite element simulation of an exemplary induction machine. Design/methodology/approach In this paper, an adapted IEM loss model for rotational magnetization, developed within the context of the paper, is implemented in a finite element method simulation and used to calculate the iron losses of an exemplary induction machine. The resulting iron losses are compared with the iron losses simulated using three other already existing iron loss models that do not consider the effects of rotational flux densities. The used iron loss models are the modified Bertotti model, the IEM-5 parameter model and a dynamic core loss model. For the analysis, different operating points and different locations within the machine are examined, leading to the analysis of different shapes and amplitudes of the flux density curves. Findings The modified Bertotti model, the IEM-5 parameter model and the dynamic core loss model underestimate the hysteresis and excess losses in locations of rotational magnetizations and low-flux densities, while they overestimate the losses for rotational magnetization and high-flux densities. The error is reduced by the adapted IEM loss model for rotational magnetization. Furthermore, it is shown that the dynamic core loss model results in significant higher hysteresis losses for magnetizations with a high amount of harmonics. Originality/value The simulation results show that the adapted IEM loss model for rotational magnetization provides very similar results to existing iron loss models in the case of unidirectional magnetization. Furthermore, it is able to reproduce the effects of rotational flux densities on iron losses within a machine simulation.
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The brushless doubly fed generator (BDFG) has been a research focus of wind power generation due to its inherent promising features. Although it is important to consider the effect of losses on the design and analysis of the BDFG, there are a few related comprehensive studies. Moreover, a hybrid rotor is studied to improve the performance of BDFG, which brings greater challenges to the loss calculation. The aim of this paper is to propose the accuracy calculation methods of core loss and copper loss for the studied BDFG with hybrid rotor. The core loss calculation model which taken the influence of harmonics, alternating magnetization, rotating magnetization and two-dimensional magnetic property of material into account is proposed based on the analysis of the magnetic field characteristics. In order to calculate the copper loss considering the skin effect, the inductive current of the assisted cage bars which is difficult to obtain directly is calculated. The accuracy and correctness of the proposed approaches and models are validated by experimental results from a 25kW prototype BDFG with hybrid rotor. In addition, the proposed loss calculation method, especially the core loss calculation model, is also applicable to BDFG with other rotor structure which is operated by the magnetic field modulation mechanism.
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The brushless doubly fed induction generator (BDFIG) is an alternative to the doubly fed induction generator (DFIG), widely used in wind turbines which avoids the need for brush gear and slip rings. The choice of pole numbers for the two stator windings present in the BDFIG sets the operating speed range but also affects the torque capability, magnetizing currents and back iron depth. Analytical expressions are developed for these effects and a comparison is made between the BDFIG and the conventional DFIG. The torque capabilities and magnetizing currents are not strongly dependent on the choice of pole numbers but the back iron depth is significantly affected. The torque density of the BDFIG is somewhat reduced compared to a similarly sized DFIG but magnetizing currents per unit torque are the same. However, the required back iron depths are greater. The work also shows that multi-megawatt machines are expected to work within the desired range of power factors at acceptable efficiencies.
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It is desirable to measure rotor quantities such as currents and temperatures in an electrical machine for design verification and condition monitoring purposes. A Bluetooth module which sends data from the rotor was previously reported in literature, but this module was battery powered, and therefore the duration of the tests was limited. This study presents a solution to this problem by developing a rotor-mounted power supply system which can harvest energy from the magnetic field inside the machine, by fixing an external loop to the rotor and making use of the induced voltage in the loop. A full-bridge rectifier, boost converter and battery charging module were developed to supply sufficient power to a bespoke Bluetooth transmission system and associated sensor circuitry.
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The brushless doubly fed machine (BDFM) is an alternative to the doubly fed induction generator, widely used in wind turbines without use of brush gear and slip ring. Rotor design is important for achieving an optimal multi-MW BDFM. To date, nested-loop rotors have been exclusively used in various BDFMs, although they may not be ideal for larger machines. This paper studies, both analytically and experimentally, different rotors for the D160 and D400 frame size BDFMs. An optimization on the rotor equivalent circuit parameters is carried out and verified with different parameter extraction methods. A comparison of the performance of the suggested rotors is also presented.
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This paper presents analytical methods to calculate the equivalent circuit parameters for large-scale brushless doubly fed machines (BDFMs) with magnetic wedges utilized for closing stator open slots. The use of magnetic wedges reduces the magnetizing currents in the machine, reflected in the values of magnetizing inductances, but also increases leakage fluxes affecting the value of series inductances in the equivalent circuit. Though such effects can be modeled by numerical models, the proposed analytical methods are particularly helpful in optimizing machine design, inverter rating, reactive power management, and grid low-voltage ride-through performance. The conventional analytical methods cannot be readily applied to the BDFM due to its complex magnetic field distribution; this paper presents analytical methods to calculate the magnetizing and leakage inductances for the BDFM with magnetic wedges used in the stator slots. The proposed methods are assessed by experimentally verified finite-element models for a 250 kW BDFM.
Conference Paper
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In this paper some fundamental design issues concerning the brushless doubly-fed reluctance machine are addressed. The literature has several examples of electrical machines with different pole number. It has become apparent that a radially-laminated ducted rotor is a better option. In this paper it is illustrated that the pole number combination should not have pole pair number varying by one, otherwise unbalanced magnetic pull occurs. In addition, it is shown that to reduce voltage ripple due to the interaction of the slotting between the stator slots and rotor ducts, two axial rotor sections can be used with a small degree of skew between them. The example machine design is a 2 MW design as used previously, this has 48 stator slots. The machine results are couched in terms of a 4/8 pole and a 4/6 pole combination (for UMP calculation). Finite element analysis and analytical algorithms are used in the paper.
Article
The Brushless Doubly-Fed Generator (BDFG) shows the great potential for use in large variable speed wind turbines due to its high reliability and cost benefits of a partially-rated power electronics converter. However, it suffers from the compromised efficiency and power factor in comparison with conventional doubly fed induction or synchronous generators. Therefore, optimizing the BDFG, especially the rotor, is necessary for enhancing its torque density and market competitiveness. In this paper, a novel cage-assisted magnetic barrier rotor, called the hybrid rotor, is proposed and analyzed. The detailed analytical design approaches based on the magnetic field modulation theory are investigated. In addition, the machine losses and mutual inductance values using the proposed rotor designs are calculated and their performance implications evaluated. Finally, the comparative experimental results for two BDFG prototypes are presented to verify the accuracy and effectiveness of the theoretical studies.
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A multi-harmonic magnetic flux density waveform controller for the Epstein frame is developed. The controller has the ability to produce high quality, single frequency, flux waveforms with low harmonic content and non-sinusoidal waveforms consisting of a superposition of many harmonics. The controller employs proportional control on the real and imaginary parts of the secondary voltage harmonics, where the real axis is determined by the fundamental harmonic, and can be used to either add or cancel higher order harmonics. Specific loss measurements are reported for 29 gauges M19 from AK Steel. Single frequency data are used to determine the coefficients of the frequency-domain Bertotti equation; the coefficients are then applied to the time-domain Bertotti loss equation. Magnetic loss testing is performed on non-sinusoidal flux-density waveforms, and the ability of both the frequency and time-domain Bertotti equations to predict the loss is discussed and compared with the measured loss.