Calculation of Core and Stray Load Losses in Brushless Doubly Fed Induction Generators

Abstract

The brushless doubly fed induction generator (BDFIG) has substantial benefits, which make it an attractive alternative as a wind generator. However, it suffers from lower efficiency and larger dimensions in comparison with the doubly fed induction generator. A major part of drawbacks arises from undesirable spatial harmonics of air-gap magnetic field. Calculation of core loss is an important issue in optimal design studies to improve the performance characteristics. The iron loss is higher and has a more complex nature in BDFIGs in contrast with conventional machines. Furthermore, additional losses cannot be ignored due to a high level of spatial harmonics distortion. This paper aims to formulate core loss and stray load loss (SLL) in BDFIGs based on the design data and experimental parameters provided by electrical steel manufacturers. In addition, an analytical procedure for calculating spatial harmonic components of the stator and rotor magnetic fields is presented. The acceptable accuracy of the theoretical equations is verified by comparing the calculated core loss and SLL of a D-180 prototype BDFIG with the measured data. The results show that SLL in BDFIGs is higher than the predicted per-unit values given by standards for induction machines, e.g., IEC 60034sim2 and NEMA MG1.

Full text

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014 3167
Calculation of Core and Stray Load Losses in
Brushless Doubly Fed Induction Generators
Hamed Gorginpour, Hashem Oraee, Senior Member, IEEE, and Ehsan Abdi, Senior Member, IEEE
Abstract—The brushless doubly fed induction generator
(BDFIG) has substantial benefits, which make it an attractive
alternative as a wind generator. However, it suffers from lower ef-
ficiency and larger dimensions in comparison with the doubly fed
induction generator. A major part of drawbacks arises from unde-
sirable spatial harmonics of air-gap magnetic field. Calculation of
core loss is an important issue in optimal design studies to improve
the performance characteristics. The iron loss is higher and has
a more complex nature in BDFIGs in contrast with conventional
machines. Furthermore, additional losses cannot be ignored due
to a high level of spatial harmonics distortion. This paper aims to
formulate core loss and stray load loss (SLL) in BDFIGs based
on the design data and experimental parameters provided by
electrical steel manufacturers. In addition, an analytical procedure
for calculating spatial harmonic components of the stator and
rotor magnetic fields is presented. The acceptable accuracy of the
theoretical equations is verified by comparing the calculated core
loss and SLL of a D-180 prototype BDFIG with the measured data.
The results show that SLL in BDFIGs is higher than the predicted
per-unit values given by standards for induction machines, e.g.,
IEC 60034∼2 and NEMA MG1.
Index Terms—Brushless doubly fed induction generators
(BDFIGs), core loss, nested-loop rotor, spatial harmonics, stray
load loss (SLL).
I. INTRODUCTION
NOWADAYS, up to 70% of the installed wind turbines
incorporate doubly fed induction generators (DFIGs) [1].
In recent decades, various generating systems have been pro-
posed for wind turbines [2], [3]. However, most of them still
have technological and economic penalties, such as high cost
of active materials, complexity of the manufacturing process
and control systems, and a large and expensive converter, which
limit their spread in wind power plants [3]. Among these, a
brushless DFIG (BDFIG) has attractive features to be the next
generation of wind generators. Having no brush and slip rings, a
robust structure, and lower operating and maintenance costs as
well as requiring a smaller mechanical gearbox and better low-
voltage ride-through capability are the advantages of BDFIGs
over DFIGs [4]–[8]. The disadvantages of the BDFIG are re-
Manuscript received January 7, 2013; revised April 14, 2013 and June 24,
2013; accepted August 5, 2013. Date of publication August 22, 2013; date of
current version January 31, 2014.
H. Gorginpour and H. Oraee are with the Department of Electrical Engi-
neering, Sharif University of Technology, Tehran 11365-9363, Iran (e-mail:
h_gorgin@ee.sharif.ir; oraee@sharif.edu).
E. Abdi is with Wind Technologies Ltd., CB4 0EY Cambridge, U.K. (e-mail:
ehsan.abdi@windtechnologies.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2013.2279357
Fig. 1. Nested-loop rotor of a D-180 BDFIG with three loops per nest (Inner,
Middle, and Outer loops) [8].
Fig. 2. Wind turbine system based on the BDFIG.
lated to its design, since it has slightly larger dimensions, higher
manufacturing cost, and lower efficiency in comparison with a
DFIG of the same rating [9]. However, its promising features
warrant further investigation on the design possibilities. It is
therefore necessary to optimize the machine structure in order
to enhance its performance, both technical and economic.
The BDFIG has two stator windings with different pole-pair
numbers to avoid direct magnetic coupling, which are known
as power winding (PW) and control winding (CW). PW is
directly connected to the grid, and CW is excited via a partially
rated bidirectional converter, which allows the machine to
synchronously operate in a limited range of rotational speeds.
The rotor is traditionally designed as nested loop (see Fig. 1)
[8], which couples the stator magnetic fields indirectly. The
process, which is called cross coupling, is studied in several
references [4], [10]. The number of rotor nests or poles should
be equal to the summation of PW and CW pole-pair numbers,
i.e., Ppand Pc, respectively, to create indirect coupling between
stator magnetic fields [8]. A wind turbine system based on the
BDFIG is shown in Fig. 2.
In the generating mode of operation, the magnitude and
frequency of CW excitation voltage depends on the rotor speed.
The relation between CW excitation frequency, shaft speed, and
grid frequency is [7]
fc=P
rnm/60 −fp(1)
0278-0046 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

3168 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
where fp(\c)is PW(\CW) frequency, Pris the number of rotor
nests, and nmis the mechanical speed in revolution per minute.
Since core loss has a significant effect on efficiency and
temperature distribution, it is vital to accurately predict such
loss component in an optimal design procedure. Besides core
loss, stray load loss (SLL) should be included due to its con-
siderable value. Core loss and SLL have more complex natures
in BDFIGs compared with conventional machines. The flow of
two fundamental flux components with different frequencies in
stator core, flux density distributions with unequal magnitudes
in rotor core sections, and operating around natural speed
(60fp/Pr), which does not coincide with the synchronous
speeds of stator fields, make calculation of core loss compli-
cated. In addition, a high level of spatial harmonic distortion
produced by nested-loop rotor results in additional loss, noise,
vibration, and increased temperatures.
The aim of this paper is to present an analytical approach
for calculating core loss and SLL in BDFIGs. Comparison
of calculated results with measured data shows acceptable
accuracy of the presented approach for machine analysis and
design purposes.
II. CORE LOSS
Core loss constitutes a significant portion of total dissipated
power in an electrical machine. Hence, derivation of accurate
analytical expressions for this loss component is necessary in
machine analysis and design studies. The principles and origin
of power loss in soft magnetic materials, rotational core loss,
and variation of core loss factors in electrical machines are
studied in [11]–[13].
The following points should be considered in order to calcu-
late BDFIG core loss.
1) The magnetic field waveform in stator yoke and teeth
has two fundamental frequencies, i.e., fpand fc.Dueto
operation around natural speed, the CW voltage, current,
and frequency are lower than those for the PW [9]. Hence,
the time oscillations of the Pppole-pair component is
superimposed on the lower frequency waveform with Pc
pole pairs. Fig. 3(a) shows the magnetic field waveform
in a stator tooth of the D-180 prototype BDFIG (see
Appendix), where oscillations with two frequencies are
evident.
2) The magnetic field in a stator tooth and a stator yoke
sector has the same variation as the other teeth and yoke
parts, respectively.
3) The magnetic field waveforms of different parts of the ro-
tor yoke and teeth have a single frequency, whereas their
amplitudes are different. Fig. 3(b) shows the magnetic
field variations in three consecutive teeth of a nest of the
rotor in Fig. 1. It should be noted that the field variations
in Fig. 3 are obtained using finite-element (FE) analysis.
A. Eddy Current Loss
Eddy current loss of a sinusoidal flux density wave is propor-
tional to the square of the field magnitude. The time average of
the square of a flux density waveform comprising two compo-
Fig. 3. Time variations of magnetic field in (a) stator tooth, (b) the teeth of a
rotor nest in Fig. 1 (fp=50 Hz, fc=6.2Hz, Pp=2,Pc=4,andnm=
562 r/min).
nents with different frequencies results in the separation of their
related eddy current losses. Hence, eddy current loss in each
part of the stator core is the summation of losses corresponding
to Ppand Pcpole-pair fields. Eddy current loss in stator and
rotor cores can therefore be expressed as
Pec =Ke
Nss
f2
pBst
p,max
2+f2
cBst
c,max
2vst
+Nss
f2
pBsy
p,max
2+f2
cBsy
c,max
2vsy
+Pr2Nrl

m=1
f2
rBrt
m,max
2vrt
m+
Nrl

m=1
f2
rBry
m,max
2vry
m
(2)
where Keis the eddy current loss factor, Nss is the number
of stator slots, Bp(\c),max is the magnitude of the Pp(\Pc)
pole-pair component of flux density, and superscripts st and
sy denote stator tooth and yoke regions, respectively. Nrl is
the number of loops per nest, and the rotor frequency (fr)in
synchronous operation is
fr=fp−Ppnm/60 (3)
vdenotes the volume of the region specified by its subscripts
and superscripts. Investigations using FE analysis reveal the
flux pattern depicted in Fig. 4. The volume of each region is
calculated by multiplying its corresponding hachured area in
Fig. 4 and the stack length (lfe). The volumes of the rotor
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

GORGINPOUR et al.: CALC ULATI ON OF CO RE A ND ST RAY LOAD LOS SES IN BDFIGs 3169
Fig. 4. Stator and rotor quantities (nested-loop rotor with two loops per nest).
teeth and yoke sections depend on the magnetic flux behavior
in the rotor core. The penetration depths in the rotor yoke
(hry,m,m=1...N
rl)are required in the calculation of the
corresponding volumes. The fact that magnetic flux passes
through the route with smallest reluctance is used for evaluating
the penetration depths. The relation of hry,m, as stated in (4),
leads to the lowest value of the net reluctance in the mth path
of the yoke region [14], i.e.,
hry,m =2πwrt(2m+1)
×⎛
⎝0.5Dro −hrs1−hrs2−
m−1

j=1,m=1
hry,j⎞
⎠Nrs
⎤
⎦
0.5
(4)
where Nrs is the number of rotor slots.
B. Hysteresis Loss
Various models have been presented for the calculation of
hysteresis loss, e.g., Preisach, Steinmez, and Jiles–Atherton.
The models require experiments to determine their related
coefficients and parameters. The Steinmetz model is the most
appropriate one due to its acceptable accuracy and simplicity.
Since the rotor magnetic field has a single frequency, the
hysteresis loss arising in the rotor core is simply evaluated using
Steinmetz formula as
Protor
hys =KhPrNrl

m=1 fα
rBry
m,maxβvry
m
+
2Nrl

m=1 fα
rBrt
m,maxβvrt
m(5)
where Khis the hysteresis loss factor, and αand βare material-
dependent constant exponents.
The situation is different in the case of stator due to the
presence of double frequency magnetic field variations. Under
normal operation of the BDFIG, the hysteresis loop of each
stator core sector can be modeled as a dc-biased hysteresis
loop of the Pppole-pair field with fposcillations per second.
The dc bias field is the average value of the Pcpole-pair
field in that core sector, which varies with time. Hence, the
hysteresis loop center is displaced from the coordinate center
with the frequency of fc. Although the loss associated with
a hysteresis loop of nonzero average is equal to that of zero
average, the hysteresis loop is subjected to changes because
of the dc bias. This phenomenon complicates the hysteresis
loss calculation. While none of the steel manufacturers provide
information on core loss factors of a premagnetized core, it
is experimentally shown that hysteresis loss increases with
increasing the amplitude of the dc magnetic field in a given ac
field [12].
Many researchers have investigated the calculation of hys-
teresis loss under nonsinusoidal conditions arising from time
harmonics, which create minor hysteresis loops in the main
frequency hysteresis loop. In [15], the effect of minor loops
on hysteresis loss under pulsewidth modulation and dc-biased
sources is investigated, and it is shown that the locations of
the minor loops have a significant effect on hysteresis loss.
In addition, an equation for calculating hysteresis loss under
these conditions is presented, whereas its parameters are ex-
perimentally determined. A correction factor is proposed in
[16] to modify the hysteresis loss equation in the presence of
minor loops, which is linearly proportional to unweighted sum
of harmonic reversals. Furthermore, the findings reported in this
paper illustrate that harmonic loss depends on the amplitude,
phase, and order of the harmonics. Other research efforts on the
calculation of hysteresis loss under nonsinusoidal conditions
have been carried out using mathematical expressions of the
hysteresis loop [12].
Core loss sources are discussed in [12], and it is concluded
that they arise from the movement of magnetic regions due
to eddy currents and spinal rotations. In the following, the
modified Steinmetz equation (MSE) is presented based on the
physical interpretation of core loss, which is in direct relation
to the magnetizing rate. The magnetizing rate is different from
the frequency of applied external magnetic field. The modified
method utilizes the parameters provided by electrical steel
manufacturers, and only the modified frequency should be sub-
stituted. The equivalent frequency relation is as (6) [12]. Note
that the equivalent frequency of a pure sinusoidal waveform
equals its frequency, i.e.,
feq (B(t),f)= 2
ΔB2π2
1/f

0∂B
∂t 2
dt (6)
where ΔBis the peak-to-peak amplitude of the magnetic field
(Bmax −Bmin).
In [9], it is shown that Beq, as stated in (7), is an appropriate
approximation of BDFIG magnetic loading. Hence, in order to
calculate stator hysteresis loss, the magnetic field in the stator
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

3170 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
Fig. 5. Normalized hysteresis loss of two alternating fields of different
frequencies.
core may be assumed as a pure sinusoidal waveform with the
frequency of fpand the amplitude of Beq, i.e.,
Beq =B2
p,max +B2
c,max.(7)
The normalized hysteresis loss curves predicted by two latter
approaches versus different amplitudes of CW field are depicted
in Fig. 5. It can be seen that considering an equivalent field
yields underestimated results in all cases. Hence, the MSE
approach is used for the calculation of stator core hysteresis
loss. The relation of stator hysteresis loss is stated as follows:
Pstator
hys =KhNssfpfeq Bst(t),f
pα−1Bst
maxβvst
+(feq (Bsy (t),f
p))α−1(Bsy
max)βvsy (8)
where Bst(\sy)(t)is the magnetic field waveform in the stator
tooth (\yoke) region, which comprises two main components
of Ppand Pcpole pairs.
C. Core Loss Calculation Algorithm
The algorithm of the core loss calculation procedure is shown
in Fig. 6.
It should be noted that the PW field (Bpw
p(t, θ)) and the
Pppole-pair component of the rotor field (Brw
p(t, θ)) have the
same speed from the stator point of view and create the Pppole-
pair component of the air-gap field (Bag
p). A similar situation
exists for CW field (Bcw
c(t, θ)) and the Pcpole-pair component
of the rotor field (Brw
c(t, θ)) at a different speed and form the
Pcpole-pair component of the air-gap field (Bag
c).
As discussed before, the air-gap field waveform varies with
time because of unequal rotational speeds of its fundamental
components, i.e., Bag
pand Bag
c. Hence, the average value of the
calculated core losses over one period of Bag, i.e., reciprocal of
the greatest common divisor of fpand fc, should be stated as
the dissipated power in laminated core.
III. ADDITIONAL LOSSES
Additional losses, which are broadly termed as SLL, consist
of several components, i.e., stator surface loss, rotor surface
loss, stator teeth pulsation loss, rotor teeth pulsation loss, rotor
bar loss, and loss due to uninsulated bars of rotor winding [17].
Fig. 6. Flowchart of core loss calculation.
SLL is defined as the difference between the total loss and
the sum of conventional losses, i.e., resistive loss, iron loss, and
mechanical loss. Some sources of the additional losses can be
analytically interpreted and formulated. A portion of the SLL
is due to mechanical imperfections. Hence, SLL has different
values from manufacturer to manufacturer and from machine
to machine. Consideration of these issues is beyond the scope
of this paper.
Most literatures on SLL have performed experiments using
an input–output method and/or calorimetric methods. Compar-
ison between measured values of SLL and predicted values
in standards for IMs is reported in [18]–[21]. A modified
equivalent circuit of IM for considering SLL is developed in
[22]. A resistor is added in series with the stator impedance,
whereas its value is determined using test results on several
machines of different ratings. Considering the amplitude of
each loss components in squirrel-cage IMs, measurements on
small- and medium-size machines up to 100 kW show addi-
tional fundamental frequency losses of 10%, high-frequency
stray losses arising from a slotting effect of 12%, and losses
due to magnetomotive force (MMF) harmonics of 78% of total
full-load additional loss [17]. In uninsulated skewed bar cage
rotors, the rotor stray loss constitutes 30%–40% of the total
high-frequency full-load loss [17].
Here, analytical equations for each component of SLL are
presented. Although time and spatial distributions of flux
density in different parts of an electrical machine can be
obtained using FE analysis, it is important to analytically in-
vestigate the distributions since it gives a clear vision of the
electromagnetic phenomena as well as order and magnitude
of the MMF harmonics. Hence, the relations of stator and
rotor flux density distributions considering MMF and slotting
harmonics are presented at first.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

GORGINPOUR et al.: CALC ULATI ON OF CO RE A ND ST RAY LOAD LOS SES IN BDFIGs 3171
Fig. 7. Three different arrangements of the nested-loop rotor.
A. Air-Gap Flux Density Distribution
1) Stator and Rotor MMFs: The MMF distribution of a
three-phase Ppole-pair winding can be expressed as
s(t, θ)= 3Imax Nsw
π⎧
⎨
⎩
∞

h=P(6n+1)
1
hkwh sin(ωt −hθ)
+∞

h=P(6n−1)
1
hkwh sin(ωt +hθ)⎫
⎬
⎭
,n=0,1,2,... (9)
where Imax is the magnetizing current amplitude, Nsw is the
number of winding turns per phase, and kwh is the winding
factor of the hth spatial harmonic order.
Three different arrangements of the nested-loop rotor are
shown in Fig. 7. The arrangements of the rotor loops signifi-
cantly affect the magnitude of spatial harmonics.
The MMF distribution of the nested-loop arrangements
shown in Fig. 7(a) and (b) is as
r(t, θr)
=LPr
π×
Nrl

m=1
Im,max
×∞

h=nPr+Pc
1
hsinhαm
2sin(ωrt+hθr−nπ(Pr−1))
+∞

h=nPr+Pp
1
hsinhαm
2sin(ωrt−hθr+nπ(Pr−1))
⎫
⎬
⎭
(10)
where Im,max is the magnitude of current flowing in the mth
loop with αmspan angle.
The MMF distribution of the nested loops in configuration
in Fig. 7(c) is obtained using (10). In this configuration, the
relation of the MMF produced by the squirrel cage is as
r(t, θr)
=I1,maxPr
2π
×⎧
⎨
⎩
∞

h=nPr+Pp
1
hcos (ωrt−hθr+nπ(Pr−1))
−∞

h=nPr−Pp
1
hcos (ωrt+hθr−nπ(Pr−1))⎫
⎬
⎭
(11)
where I1,max is the magnitude of current flowing in the squirrel-
cage bars.
Furthermore, a novel rotor configuration is proposed in [23],
which has the benefits of reducing spatial harmonic distortion
of air-gap magnetic field as well as improving some drawbacks
of the conventional structure, including unequal magnitudes of
current in rotor bars, rotor teeth saturation at low average
air-gap magnetic fields, high core loss, and inefficient mag-
netic material utilization. The rotor loops are proposed to be
connected in series in the new scheme rather than in nested
arrangement of the conventional design.
The pole number changing action is performed using a single
rotor at the cost of producing many undesirable spatial harmon-
ics of nPr+Ppand nPr+Pcorders, where nis an integer
number [10]. This is different from the phenomenon, which
takes place in a conventional squirrel-cage rotor. For example,
the air-gap magnetic field distribution produced by the nested-
loop rotor in Fig. 1 under specified operating conditions is
plotted in Fig. 8(a), and its spatial harmonic spectrum is shown
in Fig. 8(b). The spatial distribution of air-gap field in this figure
is obtained using the FE method. As can be seen, in addition to
Ppand Pcpole-pair components, low-order spatial harmonics
with considerable magnitudes contribute in the rotor magnetic
field. The amplitudes of undesired harmonics are subjected to
increase when the rotor teeth are magnetically saturated. It
is shown that saturation causes harmonics of nPr±3Ppand
nPr±3Pcorders to be superimposed on the air-gap magnetic
field [24].
The rotor performance is a function of the number of nests
and thus PW and CW pole-pair numbers, number of loops per
nest, loop spans, and their arrangement in the nest. In [25],
the appropriate selection of the number of stator winding pole
pairs for increasing cross-coupling capability by minimizing
the undesirable harmonics content is investigated. Analytical
and experimental studies reveal that 2/4 pole-pair combination
is the most appropriate one for the operational speed range of
600–900 r/min. Similar studies should be conducted for each
desired speed range in order to choose the optimal combination
of winding pole pairs between the possible sets. In addition,
the number of loops per nest and loop spans can be considered
as optimization variables in a population-based optimal design
procedure.
2) Slotting Effects: The effect of the variable air-gap length
due to slots on the magnetic field was first taken into account
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

3172 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
Fig. 8. (a) Air-gap magnetic field distribution of the nested-loop rotor in Fig. 1
and (b) its spatial harmonics spectrum under specified operating conditions.
by Carter’s factor. However, in derivation of this factor, the slot
depth and pitch are considered infinite, which causes significant
errors in some cases. Modifications on Carter’s factor were
experimentally proposed by Ossana, Oberretl, Weber, and Foit
[17]. Examinations show that an appropriate description of
magnetic field distribution on a stator slot pitch is [17]
B(θ)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
1−βs−βs
×cos πDag
1.6wsso θBmax |θ|≤0.8wsso
Dag
Bmax 0.8wsso
Dag ≤|θ|≤ π
Nss
(12)
where
βs=(1 −u)2
2(1 + u2),u=wsso
2g+1+wsso
2g2
.(13)
The distribution of air-gap conductance, i.e., 1/g(θ), is obtained
as [17]
Gag(θ, θr)
≈1
gkcskcr
+a1
kcr
cos(Nssθ)+ a2
kcr
cos(2Nssθ)
+b1
kcs
cos (Nrs(θ−θr)) + b2
kcs
cos (2Nrs(θ−θr))
+a1b1
2g[cos ((Nss +Nrs)θ−Nrsθr)
+cos((Nss −Nrs)θ+Nrs θr)] (14)
Fig. 9. Plot of 1/tan αfunction of M47 electrical steel versus tooth flux
density.
where kcs and kcr are stator and rotor Carter’s factors, respec-
tively. These factors are calculated using (15) by substituting
appropriate parameters of stator and rotor quantities. anand
bnare related to stator and rotor conductance distributions,
respectively, which can be obtained using Fourier series cal-
culations of (12). The general form of anand bncoefficients
is stated in (16). Hence, the air-gap conductance consists of
spatial harmonics of c1Nrs,c2Nss, and c3Nss ±c4Nsr orders
superimposed on the large average value of 1/(gkcskcr ), where
c1−4=1,2,..., i.e.,
kcs(\r)=τss(\rs)
τss(\rs)−1.6βs(\r)wsso(\rso)
(15)
an=βs
gFn(wsso/τss ),b
n=βr
gFn(wrso/τrs)
Fn(x)= 4
nπ sin(1.6nπx) 0.5+ (nx)2
0.78 −2(nx)2!.(16)
The air-gap distributions of stator and rotor magnetic fields are
evaluated by multiplying their related MMF distributions by
μ0Gag.
3) Saturation Effects: The magnetic saturation reduces the
permeability in tooth regions and affects the SLL significantly.
In the saturated area, reversible permeability (μrev)should
be inserted instead of linear permeability. Measurements show
that μrev can be estimated with differential permeability [17].
Furthermore, effective air gap increases due to teeth satura-
tion, which decreases the spatial harmonic amplitudes. Hence,
Carter’s factor should be modified. The correction factor for the
hth harmonic is as [17]
kgh=1+ sin(hπ/Nss)
hπ/Nss
τss
τ
ss −0.5(Wss1+Wss2)hss
1
tan α"""Bst
max
+sin(hπ/Nrs)
hπ/Nrs
τrs
Wrt
hrs
1
tan α"""Brt
max !μ0
0.92g(17)
where τ
ss is the stator slot pitch length in 1/3 of the slot height.
tan αis the ratio of the flux density to field intensity in the tooth
region. The curve of the 1/tan αfunction for M47 electrical
steel (M800-65A according to IEC 60404∼8) is depicted in
Fig. 9 using the PowerMagnetic Material Toolbox [26]. The
value of this function for tooth flux densities lower than 1.4T
is negligible, and thus, the saturation effect can be ignored.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

GORGINPOUR et al.: CALC ULATI ON OF CO RE A ND ST RAY LOAD LOS SES IN BDFIGs 3173
Fig. 10. Flux routes of a Ppole-pair harmonic field and induced surface
currents.
B. No-Load Surface Losses
The difference between the speed of a harmonic field and
the mechanical speed causes eddy currents to be induced in the
rotor surface. A similar phenomenon occurs in the stator sur-
face. A magnetic field intensity Hmax cos(ωt −Pθ)circulating
around a very long cylindrical rotor is considered in order to
calculate the surface loss. The magnetic field creates intensity
components of Hxand Hyin the rotor, while the induced eddy
currents flow in the z-direction (see Fig. 10). The Maxwell
equation of the resulting system and the expression of the eddy
current density in Cartesian coordinate are as (18) and (19),
respectively, where the rotor core has surface permeability of
μsurf and the resistivity of ρfe [17], i.e.,
ρfe∇×∇×
H−∇
2
H
=−μsurf
∂
H
∂t ,
H=Hxˆax+Hyˆay(18)

J=∂Hy
∂x −∂Hy
∂y2ˆaz.(19)
The current density magnitude exponentially decreases by pen-
etrating the surface. The surface resistive loss due to harmonic
field taking into account the laminated structure is as [17], [27]
Pr
surf =Kexp
1
√ρfeμsurf ""2πfr
surf ""
1.5
×τ2
p
4(Bmax)2πDrol(τrs −wrso)
τrs
(20)
where Bmax is the magnitude of Ppole-pair spatial harmonic,
and fr
surf is the frequency of induced surface currents. Kexp (≈
0.08 ...0.13) is an experimental factor indicating the ratio of
measured surface loss of laminated stack and calculated loss
value using thick sheet equation [27].
The relation of stator surface loss due to rotor spatial fields
can be written from (19) by suitable substitutions.
The dominant spatial harmonic orders of stator and rotor
fields and the frequencies of their related surface currents are
listed in Table I.
C. No-Load Pulsation Losses
Due to slotting of stator and rotor surfaces, the magnitudes of
teeth magnetic fields depend on the relative rotor position. For
TAB L E I
FREQUENCIES OF STATOR AND ROTO R SURFACE CURRENTS DUE TO
DOMINANT SPATI AL HARMONIC FIELDS
example, the magnetic field of a stator tooth varies between a
maximum value (when the stator tooth is located completely
opposite a rotor tooth) and a minimum value (when a rotor
slot mouth is located under the stator tooth). Variations of tooth
magnetic flux lead to the creation of eddy currents and, thus, ad-
ditional loss in the iron core, which is called pulsation loss. The
pulsation loss is calculated similar to conventional eddy current
loss, whereas the frequency and field amplitudes are substituted
with the frequency of tooth magnetic field oscillations and the
amplitude of tooth flux density pulsations, respectively. The
expressions of stator and rotor pulsation losses, respectively,
are as
Ps
pls =4.6×10−4KeNss Nrsnm
60
γrg
2τss 2
×Bst
p,max2+Bst
c,max2Mst (21)
Pr
pls =4.6×10−4Ke(2LPr)
×Nssnpw
s/60 γsg
2τrs 22L

m=1 Brt
p,max2
+Nssncw
s/60 γsg
2τrs 22L

m=1 Brt
c,max2×Mrt
(22)
where npw(\cw)
sis the synchronous speed of the PW (\CW)
magnetic field, Mst(\rt)is the weight of a stator tooth (\rotor
tooth), and γs(\r), which is a function of the slot opening width
to air-gap length ratio, is defined as
γs(\r)=2
πwrso(\sso)
gtan−1wrso(\sso)
2g
−ln #1+wrso(\sso)
2g2$.(23)
D. No-Load Losses in Rotor Loops
The stator spatial harmonics induce voltages and currents
in rotor loops and cause resistive loss. For example, the PW
magnetic field consists of dominant spatial harmonics of Nss ±
Pporder rotating at angular speeds of ±ωp/(Nss ±Pp)with
the amplitude of Bp,maxa1/(2Kcr ).
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

3174 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
The resistance and leakage inductance of rotor loops is
required in order to calculate the induced currents. It is
reasonable to ignore the effect of induced rotor currents on
the air-gap magnetic field. The loop leakage inductance is the
sum of slot leakage inductance, overhang leakage inductance,
and differential leakage inductance. A detailed formulation for
calculating the leakage inductance of slot, end-ring, and joint
bars are presented in [8]. The differential leakage inductance of
the mth loop with αmspan angle is as
Drilμ0P2
r
πg 
h
(sin(hαm/2)/h)2.(24)
E. Full-Load Stray Losses
The vector summation of MMF harmonics and slotting har-
monics due to load current is used in SLL equations under a
loading condition. For example, PW produces field harmonics
of cNss ±Pporders with amplitudes of Ppa1/(2kcr).The
same harmonic orders are produced due to slotting effects from
winding current with the amplitude of cNss±Pp/(gkcskcr ).
Hence, the magnitudes of resultant magnetic fields are
B(cNss∓Pp)=μ0cNss ∓Pp
1
gkcskcr
cos φ2
+cNss∓Pp
1
gkcskcr
sin φ±
Pp
a1
2kcr 20.5
(25)
where his the magnitude of the hth spatial harmonic com-
ponent of MMF distribution, and φis the phase shift an-
gle between winding voltage and current vectors under load
conditions.
IV. R ESULTS AND DISCUSSION
Two experimental studies have been conducted in order to
evaluate the accuracy of the extracted equations. The experi-
mental setup and schematic diagrams of scenarios are shown in
Fig. 11. In the first case, PW is supplied via a 240-Vrms 50-Hz
voltage source, and CW is connected to a three-phase 11.7-Ω
resistor. A dc machine equipped with an ABB DCS800 drive
system is mechanically coupled to the D-180 BDFIG as the
prime mover. At subnatural speeds (nm<500 r/min), the dc
motor regulates the rotational speed by providing appropriate
load torque (motoring mode), whereas at supernatural speeds
(nm>500 r/min), the machine works in generating mode. The
specifications of the test machine are listed in the Appendix.
A Magtrol TMB312 torque transducer and an incremental
encoder with 10 000 pulses per revolution are used to measure
torque and shaft rotational speed, respectively. The voltage and
current of each stator phase are measured using LEM LV 25-p
and LEM LTA 100-p transducers, respectively.
Loss segregation using the input–output method (according
to IEEE 112) is widely used in order to measure total loss and,
particularly, SLL in small IMs with low efficiencies. Since the
Fig. 11. (a) Test rig. (b) First experimental case. (c) Second experimental case.
TAB L E I I
ACCURACIES OF MEASURING INSTRUMENTS
efficiency of the test machine is low (0.6–0.8), the input–output
method can be utilized with satisfactory accuracy. The max-
imum uncertainty in the determination of total loss based on
sensitivity analysis carried out in [28] and the accuracies of
measuring devices (see Table II) is lower than 5%.
Using the input–output method, the difference between total
loss and resistive and mechanical losses is equal to the sum-
mation of core loss and SLL. Because of a small slip under
nominal operating conditions and negligible rotor core loss, the
core loss can be simply measured in an IM using the no-load
test. However, the rotor frequency is high throughout the speed
range in the case of the BDFIG. Hence, the rotor core loss
cannot be ignored. Furthermore, in order to have a constant
terminal voltage, the magnitude of the CW excitation voltage
is a function of loading. It means that the stator and rotor core
losses vary with load current, which makes direct measurement
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

GORGINPOUR et al.: CALC ULATI ON OF CO RE A ND ST RAY LOAD LOS SES IN BDFIGs 3175
Fig. 12. Measured and calculated results of the first study case [see
Fig. 11(b)].
of core loss impossible. In addition, the dependence of CW and
rotor voltages and frequencies on mechanical speed yields the
variation of core loss with rotational speed.
It should be noted that the rotor resistive loss is not ac-
cessible practically. Hence, a 2-D magnetodynamic FE model
is developed for accurate prediction of this loss component.
Current displacement is automatically considered in the slot
area through the software, whereas this phenomenon is not im-
portant in the overhang region. In FE simulations, the resistance
and leakage inductance of the overhang sections are inserted in
the external circuit of nested-loop arrangement. The analytical
equations required for calculation of these parameters are stated
in [27]. The calculated resistance is thermally modified using
the difference between stator winding resistances measured at
cold and hot conditions. In addition, the friction and windage
losses at each speed are obtained by measuring the mechanical
output power of the dc drive when both stator windings are
opened and speed is constant at the specific value. The calcu-
lated and measured results are shown in Fig. 12. The core loss
and SLL variations, which are evaluated using the presented
analytical equations, are depicted in this figure. Stator and rotor
currents increase by deviating from natural speed (500 r/min).
The same manner occurs for SLL, which shows the dependence
of stray losses on load. It can be seen that there is a good
agreement between measured and calculated remaining losses,
i.e., the sum of core loss and SLL, which proves acceptable
accuracy of the proposed analytical procedure. For the speed
range under consideration, the maximum additional loss is
4.3% of the output power, which is much higher than the
estimated values for IMs, e.g., 0.5% stated in IEC 60034∼2
standard and the 1.2% predicted in NEMA MG1 for ratings
lower than 2500 hp.
The second case is devoted to the actual generating operation.
The CW excitation is controlled at each speed in order to have
a constant PW terminal voltage. In this manner, constant active
Fig. 13. Measured and calculated results of the second study case [see
Fig. 11(c)].
power is delivered to the grid throughout the speed range. The
measured graphs of CW voltage magnitude, mechanical loss,
active powers and torque, and the calculated graphs of remain-
ing loss, core loss, and SLL components are shown in Fig. 13.
Again, the results prove acceptable accuracy of the analytical
relations. It should be noted that the rotor resistive loss due
to harmonic currents is negligible. This is mainly due to the
special rotor configuration that yields high differential leakage
inductance for higher harmonic orders. Finally, it is stressed
that the SLL components are functions of slot dimensions and
winding arrangements. Hence, their patterns may be different
in another machine.
V. C ONCLUSION
This paper has presented analytical equations for calculating
core loss and SLL in brushless doubly fed induction machines.
The equations are based solely on the geometry dimensions,
winding arrangements, and core loss parameters provided by
electrical steel manufacturers. Hence, the presented relations
can be effectively used in machine analysis and optimal design
procedures. It is concluded that due to a high level of spatial
harmonic distortion of the air-gap field, rotor and stator teeth
saturation due to the fundamental components of flux densities,
operating far from synchronous speeds of both stator fields and
nonuniform distribution of the magnetic field in stator and rotor
cores, the core and stray losses are higher in BDFIMs than
IMs with the same rating. It is also shown that the studied loss
components depend on the loading conditions and rotational
speed and increase with deviating from natural speed.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

3176 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
TABLE III
SPECIFICATIONS OF PROTOTYP E D-180 FRAME SIZE BDFIG
TAB L E I V
SPECIFICATIONS OF IRON CORE (M47)
APPENDIX
The specifications of the prototype BDFIG are listed in
Table III [9].
The parameters required for iron loss calculation are given in
Table IV.
REFERENCES
[1] W. Wang, M. Kiani, and B. Fahimi, “Optimal design of doubly fed in-
duction generators using field reconstruction method,” IEE Trans. Magn.,
vol. 46, no. 8, pp. 3453–3456, Aug. 2010.
[2] M. Liserre, R. Cardenas, M. Molinas, and J. Rodrigoez, “Overview of
multi-MW wind turbines and wind parks,” IEEE Trans. Ind. Electron.,
vol. 58, no. 4, pp. 1081–1095, Apr. 2011.
[3] H. Li and Z. Chen, “Overview of different wind generator systems and
their comparisons,” IETRenew.PowerGener., vol. 2, no. 2, pp. 123–138,
Jun. 2008.
[4] F. Barati, S. Shao, E. Abdi, H. Oraee, and R. A. McMahon, “General-
ized vector model for the brushless doubly-fed machine with a nested-
loop rotor,” IEEE Trans. Ind. Electron., vol. 58, no. 6, pp. 2313–2321,
Jun. 2011.
[5] S. Shao, E. Abdi, and R. A. McMahon, “Low-cost variable speed drive
based on a brushless doubly-fed motor and a fractional unidirectional
converter,” IEEE Trans. Ind. Electron., vol. 59, no. 1, pp. 317–325,
Jan. 2012.
[6] S. Shao, T. Long, E. Abdi, and R. A. McMahon, “Dynamic control of the
brushless doubly fed induction generator under unbalanced operation,”
IEEE Trans. Ind. Electron., vol. 60, no. 6, pp. 2465–2476, Jun. 2013.
[7] S. Tohidi, H. Oraee, M. Zolghadri, S. Shao, and P. Tavner, “Analysis and
enhancement of low voltage ride-through capability of brushless doubly
fed induction generator,” IEEE Trans. Ind. Electron., vol. 60, no. 3,
pp. 1146–1155, Mar. 2013.
[8] H. Gorginpour, H. Oraee, and R. A. McMahon, “Electromagnetic-thermal
design optimization of the brushless doubly-fed induction generator,”
IEEE Trans. Ind. Electron., vol. 61, no. 4, pp. 1710–1721, Apr. 2014.
[9] R. A. McMahon, P. C. Roberts, X. Wang, and P. J. Tavner, “Perfor-
mance of BDFM as generator and motor,” IEE Proc.-Electr. Power Appl.,
vol. 153, no. 2, pp. 289–299, Mar. 2006.
[10] F. Blazquez, C. Veganzones, D. Ramirez, and C. Platero, “Character-
ization of the rotor magnetic field in a brushless doubly-fed induc-
tion machine,” IEEE Trans. Energy Conv., vol. 24, no. 3, pp. 599–607,
Sep. 2009.
[11] D. M. Ionel, M. Popescu, S. J. Dellinger, T. J. E. Miller, R. J. Heideman,
and I. McGlip, “On the variation with flux and frequency of the core loss
coefficients in electrical machines,” IEEE Trans. Ind. Appl., vol. 42, no. 3,
pp. 658–667, May/Jun. 2006.
[12] J. Reinert, A. Brockmeyer, and R. W. A. A. De Doncker, “Calculation
of losses in ferro- and ferrimagnetic materials based on the modified
Steinmetz equation,” IEEE Trans. Ind. Appl., vol. 37, no. 4, pp. 1055–
1061, Jul./Aug. 2001.
[13] L. Ma, M. Sanada, S. Morimoto, and Y. Takeda, “Prediction of iron loss in
rotating machines rotational loss included,” IEEE Trans. Magn., vol. 39,
no. 4, pp. 2036–2041, Jul. 2003.
[14] H. Gorginpour, H. Oraee, and R. A. McMahon, “A novel modeling ap-
proach for design studies of brushless doubly-fed induction generator
based on magnetic equivalent circuit,” IEEE Trans. Energy Conv., vol. 28,
no. 4, pp. 902–912, Dec. 2013.
[15] E. Barbisio, F. Fiorillo, and C. Ragusa, “Predicting loss in magnetic steels
under arbitrary induction waveform and with minor hysteresis loops,”
IEEE Trans. Magn., vol. 40, no. 4, pp. 1810–1819, Jul. 2004.
[16] J. D. Lavers, P. P. Biringer, and H. Hollitscher, “A simple method of
estimating the minor loop hysteresis loss in thin laminations,” IEEE Trans.
Magn., vol. 14, no. 5, pp. 386–388, Sep. 1978.
[17] B. Heller and V. Hamata, Harmonics Field Effects in Induction Machines.
New York, NY, USA: Elsevier Scientific Pub., 1977.
[18] K. J. Bradley, W. Cao, and J. Arellano-Padilla, “Evaluation of stray load
loss in induction motors with a comparison of input–output and calori-
metric methods,” IEEE Trans. Energy Conv., vol. 21, no. 3, pp. 682–689,
Sep. 2006.
[19] W. Cao, K. J. Bradley, J. C. Clare, and P. W. Wheeler, “Comparison of
stray load and inverter-induced harmonic losses in induction motors using
calorimetric and harmonic injection methods,” IEEE Trans. Ind. Appl.,
vol. 46, no. 1, pp. 249–255, Jan./Feb. 2010.
[20] E. B. Agamloh, “An evaluation of induction machine stray load loss from
collated test results,” IEEE Trans. Ind. Appl., vol. 46, no. 6, pp. 2311–
2318, Nov./Dec. 2010.
[21] A. Bogliett, A. Cavagnino, M. Lazzari, and M. Pastorelli, “International
standards for the induction motor efficiency evaluation: A critical analysis
of the stray-load loss determination,” IEEE Trans. Ind. Appl., vol. 40,
no. 5, pp. 1294–1301, Sep./Oct. 2004.
[22] A. Boglietti, A. Cavagnino, L. Ferraris, and M. Lazzari, “Induction
motor equivalent circuit including the stray load losses in the machine
power balance,” IEEE Trans. Energy Conv., vol. 23, no. 3, pp. 796–803,
Sep. 2008.
[23] H. Gorginpour, B. Jandaghi, and H. Oraee, “A novel rotor configuration
for brushless doubly-fed induction generators,” IET Electr. Power Appl.,
vol. 7, no. 2, pp. 106–115, Feb. 2013.
[24] H. Gorginpour, H. Oraee, and R. A. McMahon, “Performance description
of brushless doubly-fed induction machine in its asynchronous and vari-
able speed synchronous modes,” J. Electromag. Anal. Appl., vol. 3, no. 12,
pp. 490–511, Dec. 2011.
[25] M. S. Boger, A. K. Wallace, and R. Spee, “Investigation of appropriate
pole number combinations for brushless doubly fed machines applied
to pump drives,” IEEE Trans. Ind. Appl., vol. 32, no. 1, pp. 189–194,
Jan./Feb. 1996.
[26] S. D. Sudhoff and G. Shane, PMMToolbox: PowerMagnetic Material
Toolbox (Version 1.0) [MATLAB Toolbox], West Lafayette, IN, USA:
School Elect. Comput. Eng., Purdue Univ. 2010. [Online].
Available: https://engineering.purdue.edu/ECE/Research/Areas/PEDS/
power_magnetic_material_toolbox
[27] I. Boldea and S. A. Nasar, The Induction Machine Handbook, 2nd ed.
New York, NY, USA: CRC Press LLC, 2010, pp. 119–141.
[28] M. Aoulkadi and A. Binder, “When loads stray, evaluation of differ-
ent measurement methods to determine stray load losses in induction
machines,” IEEE Ind. Electron. Magazine, vol. 2, no. 1, pp. 21–30,
Mar. 2008.
Hamed Gorginpour was born in Bushehr, Iran, in
1985. He received the B.Sc. degree in electrical
engineering from Shiraz University, Tehran, Iran, in
2007 and the M.Sc. degree in electrical engineering
from Sharif University of Technology, Tehran, in
2009, where he is currently working toward the Ph.D.
degree.
His research interests include electrical machine
design and modeling, finite-element analysis, and
power electronics and drives.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.

GORGINPOUR et al.: CALC ULATI ON OF CO RE A ND ST RAY LOAD LOS SES IN BDFIGs 3177
Hashem Oraee (SM’98) received the B.Eng. de-
gree (first-class honors) in electrical and electronic
engineering from the University of Wales, Cardiff,
U.K., in 1980 and the Ph.D. degree in electri-
cal machines from the University of Cambridge,
Cambridge, U.K., in 1984.
He is currently a Professor of electrical engineer-
ing with Sharif University of Technology, Tehran,
Iran. His research interests include electrical energy
conversion and power quality. He is also active in
commercialization of brushless doubly fed induction
generators in wind generation industry.
Ehsan Abdi (M’06–SM’13) received the B.Sc. de-
gree in electrical engineering from Sharif University
of Technology, Tehran, Iran, in 2002, and the M.Phil.
and Ph.D. degrees in electrical engineering from the
University of Cambridge, Cambridge, U.K., in 2003
and 2006, respectively.
He is currently with Wind Technologies Ltd.,
Cambridge, aiming at exploiting the brushless dou-
bly fed machine for commercial applications. He is
also an Embedded Researcher with the Electrical
Engineering Division, University of Cambridge. His
main research interests include electrical machines and drives, wind power
generation, and electrical measurements and instrumentation.
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on March 24,2021 at 14:14:54 UTC from IEEE Xplore. Restrictions apply.