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Published in IET Renewable Power Generation
Received on 28th December 2012
Revised on 3rd July 2013
Accepted on 23rd July 2013
doi: 10.1049/iet-rpg.2012.0383
ISSN 1752-1416
Magnetic equivalent circuit modelling of brushless
doubly-fed induction generator
Hamed Gorginpour1, Behzad Jandaghi1, Hashem Oraee1, Ehsan Abdi2
1
Department of Electrical Engineering, Sharif University of Technology, P.O. Box 11365-9363, Tehran, Iran
2
Wind Technologies Limited, St Johns Innovation Park, P.O. Box CB4 0EY, Cambridge, UK
E-mail: h_gorgin@ee.sharif.edu
Abstract: The brushless doubly-fed induction generator (BDFIG) has substantial benefits, which make it an attractive alternative
as a wind turbine generator. The aim of this work is to present a nodal-based magnetic equivalent circuit (MEC) model of the
BDFIG which provides performance characteristics and flux density distributions. The model takes into account stator
winding distributions, special configuration of rotor bars, slotting effects, teeth saturation, flux fringing and current
displacement effects. The real flux tubes are considered for creating an MEC network and calculating its non-linear elements.
A method for simplifying the rotor magnetic network has been applied and Gauss elimination with partial pivoting approach
is used to solve the equation system with sparse coefficient matrix. The model parameters are based solely on geometrical
data and thus it is an appropriate tool for population-based design studies instead of computationally intense analysis of the
finite element method. The steady-state results of the proposed model are verified experimentally. The comparisons
demonstrate the effectiveness of the proposed model especially when the core is magnetically saturated.
Nomenclature
ℑmagnetomotive force
Amagnetic potential
Bmagnetic flux density
D
ri
\D
ro
inner\outer diameter of rotor
D
si
\D
so
inner\outer diameter of stator
Ffrequency
Hmagnetic field intensity
h
sy
height of stator yoke
i,Icurrent
Jmoment of inertia
Linductance
l
fe
stack length
Npw
c/Ncw
cnumber of conductors per coil of power\control
winding
1n
m
rotational speed
N
rl
number of loops per nest
N
ss
\N
rs
number of stator\rotor slots
P
p
\P
c
number of power\control winding pole pairs
P
r
number of rotor nests
Rresistance
T
el
electromagnetic torque
Vvoltage
w
sso
\w
rso
stator\rotor slot opening width
α
rs
rotor slot pitch angle
Θspatial angle
λflux linkage
μmagnetic permeability
ρpermeance
σ
cu
copper conductivity
t
ss
\
t
rs
stator\rotor slot pitch length
Φmagnetic flux
Subscripts and superscripts
_ denotes a vector/matrix variable
p(pw)\c(cw) power winding\control winding
st\rt stator\rotor tooth
stt\(rtt) stator\rotor tooth tip
sy\ry stator\rotor yoke
1 Introduction
Nowadays, up to 70% of the installed wind turbines
incorporate doubly-fed induction generator (DFIG) [1].
In recent decades, various generating systems have been
proposed for wind turbines [2,3]. However, most of them
still have technological and economic penalties such as high
cost of active materials, complexity of machine
manufacturing and control systems and large and expensive
converter which limit their spread in wind power plants [3].
Among these, brushless DFIG (BDFIG) has attractive
features to be the next generation of wind generators.
Having no brush and slip rings, a robust structure, lower
operating and maintenance costs as well as requiring a
smaller mechanical gearbox besides its advantages in grid
connection issues such as better low-voltage ride through
capability are the benefits of BDFIG over DFIG [4–6]. The
disadvantages of BDFIG are related to its relatively poor
design, since it has slightly larger dimensions and
manufacturing cost as well as lower efficiency in comparison
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doi: 10.1049/iet-rpg.2012.0383
with a DFIG with the same rating [7]. However, its promising
feature warrants further investigations on the design
possibilities. It is therefore necessary to optimise the machine
structure in order to enhance its performance, both technical
and economic, against other options.
1.1 General description of the BDFIG
The BDFIG has two stator windings with different pole pair
numbers to avoid direct magnetic coupling, known as power
winding (PW) and control winding (CW). The PW is
connected to the grid and the CW is excited via a partially
rated bidirectional converter, which allows the machine to
operate synchronously in a limited range of rotational speeds.
The rotor is traditionally designed as a ‘nested loop’[8]
(Fig. 1a), which couples the stator magnetic fields indirectly.
The coupling process, which is called ‘cross coupling’,is
studied in several references such as [4,9]. The number of
rotor nests or poles should be equal to the summation
of stator windings pole pair numbers to provide indirect
coupling between PW and CW magnetic fields [8]. A wind
turbine system based on the BDFIG is shown in Fig. 1b.
In the generating mode, the magnitude and frequency of
CW excitation voltage depends on the rotor speed. The
relation between CW excitation frequency with shaft speed
and grid frequency is [7]
fc=Prnm/60 −fp(1)
1.2 Magnetic equivalent circuit (MEC)
literature review
Up to now, the presented models for the BDFIG include: (i)
finite element (FE) model [10,11], (ii) coupled circuit model
[12], (iii) d−qand reduced d−qmodels [13] and (iv) EEC
model [13]. Furthermore, a simplified version of the EEC is
used in design of the BDFIG which is insufficiently accurate
[7]. Finite element analysis can directly calculate the flux
patterns. However, the entire process is computationally
expensive and changing the design parameters (dimensions,
slot numbers and winding arrangements) often requires the
model to be reconstructed. The dq model is generally
acceptable for control and fundamental-component dynamic
and steady-state analysis. The magnetic saturation is ignored
in coupled-circuit, EEC and dq models. Hence, the results of
these models cannot be trusted when the core is magnetically
saturated and it is impossible to determine the flux density
distributions. Correction factors are proposed for the EEC
parameters of induction machines (IMs) in order to include
saturation. However, the presence of two rotating magnetic
fields with different speeds makes the definitions of
correction factors complicated in the case of the BDFIG.
MEC is widely used for electric machine analysis and design.
For example, MEC has been applied to induction machine
analysis under faulty conditions [14]. In addition, Sudhoff
et al. [15] provided full details of a nodal-based MEC model
to analyse IMs under no load, full load and unbalanced
operating conditions as well as broken rotor end ring. In [16],
the MEC model of the synchronous machine is presented.
Also, the MEC approach has been used in surface
permanent-magnet machine design [17]. Prediction of flux
density distribution and core loss in linear permanent magnet
synchronous motors using MEC is reported in [18].
Three-dimensional (3D) MEC is presented based on nodal
equations for IMs in [19,20], which improves the model
accuracy whereas the complexity increases. The MEC
equations can also be developed based on mesh equations
instead of node equations, where magnetic fluxes flowing in
magnetic circuit branches are the unknowns [21]. It is shown
that the two formulations exhibit similar performance
under linear magnetic operating conditions. However, the
convergence properties of the mesh-based model are
significantly better than those of the nodal-based model under
non-linear operation [22]. In the case of using mesh
equations, the variable number of air-gap loops which
depends on the rotor teeth positions, increases the model
complexity [21]. Hence, the number of reported mesh-based
MEC models is less than the number based on node
equations. To solve non-linear equations of the MEC arising
from consideration of non-linear properties of the magnetic
materials, iterative methods such as Newton–Raphson,
Gauss–Seidel and others have been used [15].
1.3 Aims and scopes
This paper aims to present a nodal-based MEC model of the
BDFIG, which has some advantages including reasonable
accuracy as well as acceptable computational time,
considering stator winding distributions, special rotor
configuration, slotting effects and magnetic saturation. The
real flux tubes are considered in developing the MEC network
and calculating its non-linear elements. The original
contributions of this paper are the following: (i) presenting a
MEC network for BDFIGs, (ii) calculation of air-gap
reluctances considering fringing flux, (iii) presenting accurate
relations for resistance and leakage inductance terms taking
into account current displacement in deep conductors, (iv)
presenting a method for simplifying the rotor network and (v)
application of a numerically stable approach to solve the
matrix equation system with high condition number. Having
determined the flux density distributions, the iron loss, which
has a complex nature in the BDFIGs, can be formulated
considering spatial harmonic effects. The steady-state results
of the proposed model are verified experimentally. The
comparisons demonstrate the effectiveness of the proposed
model especially when the core is magnetically saturated.
Fig. 1 Rotor is traditionally designed as ‘nested loop’
aNested-loop rotor of D-180 prototype BDFIG
bWind turbine system based on BDFIG
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IET Renew. Power Gener., 2014, Vol. 8, Iss. 3, pp. 334–346
doi: 10.1049/iet-rpg.2012.0383
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&The Institution of Engineering and Technology 2014
2 MEC network
In this section, the MEC model of a BDFIG is developed. The
MEC approach provides a compromise between the accuracy
of the FE method and the computational time of analytical
models, for example, EEC. The accuracy of the MEC
model is related to the accuracy in calculating network
elements and creating appropriate connections between
them, so that the network represents the real flux routes.
In this way, the graphical patterns of flux lines obtained
from FE simulations can be used as a good representative
of the magnetic phenomena.
In the conventional squirrel cage winding, a loop consists
of a rotor bar on one side and an identical bar under
adjacent pole on the other side. Hence, the rotor winding
acts as a three-phase full-pitch winding with the same pole
number as the stator winding. In the case of nested-loop
winding, special connection of bars on one side leads to the
formation of an magneto-motive force (MMF) distribution
with two fundamental pole pairs components. The magnetic
flux flows between two nodes with different magnetic
potentials. The symmetry of the nested-loop configuration
yields the odd symmetry of the magnetic potential
distribution with respect to the centre line of the nest.
Hence, it can be concluded that each flux line entering the
rotor core from a tooth surrounded by bars of two loops of
a nest, leaves it from the tooth between the other bars of
those loops. This phenomenon arises because of the nests
symmetry independent of the rotor position which is
different from the pattern occurring in a conventional cage
rotor. Fig. 2shows the magnetic flux lines in a radial sector
of D180 prototype BDFIG.
The MEC network of a part of the stator and one nest of
rotor with two loops are depicted in Fig. 3. It should be
noted that the structure can be meshed finer at the cost of
more system complexity and higher computational time. For
example, extra nodes can be placed in the tooth region in
order to calculate saturation level at different points of the
tooth height.
Since the flux routes in the rotor yoke region have different
lengths and widths, the rotor teeth and yoke reluctances are
not identical. Furthermore, the MMF source in a rotor loop
branch is a function of its related loop current and the other
loop currents [9]. Hence, the rotor network is complex and
numerical implementation of the resulting MEC is difficult.
A simplified network can be developed by considering an
equivalent flux tube in the rotor yoke region with rotor bar
currents as MMF sources. Hence, the simplified rotor
network consists of the same sub-circuits as the stator
sub-circuits. The bar currents are defined according to the
bar connections in nested-loop configuration. Therefore the
resulting network of a nest is symmetrical around the centre
line of the nest and the desired MMF pattern is produced.
The specifications of the equivalent flux tubes in the
simplified network should be calculated with the goal of
minimising the inserted error.
The general form of the equations in the stator yoke nodes
and their matrix form, according to Fig. 4a, are as (2) and (3),
respectively. Throughout this work, scalar quantities are
designated with italics while matrix/vector quantities appear
in italic-boldface
Asy
i−Asy
i−1−ℑ
s
i
r
sy
i+Asy
i−Ast
i
r
st
i
+Asy
i−Asy
i+1+ℑ
s
i+1
r
sy
i+1=0(2)
M1Asy −M2Ast =M3TT
sIs,
Is=iA
pw,iB
pw,iC
pw
Ip
,iA
cw,iB
cw,iC
cw
Ic
T
(3)
The elements of the Mmatrices are the network
permeances and each column of the turn function matrix,
T
s
, with a size of 6 ×N
ss
, are determined from the MMF
expression of the corresponding slot considering both CW
and PW currents
ℑs
i=+Npw,A
c,iiA
pw +Npw,B
c,iiB
pw +Npw,C
c,iiC
pw +Ncw,A
c,iiA
cw
+Ncw,B
c,iiB
cw +Ncw,C
c,iiC
cw =Ip,Ic
×Ts(:, i)
(4)
where N
c,i
(i=1:N
ss
) is the number of conductors of the
specified winding in ith slot and the sign corresponds to its
current flow direction.
The general and systematic forms of the equations in the
stator teeth nodes can be written as the following equations,
respectively
Ast
i−Ast
i−1
r
stt +Ast
i−Ast
i+1
r
stt
i+1+
Nrs
j=1
Ast
i−Art
j
r
ag
i,j
+Ast
i−Asy
i
r
st
i=0 (5)
−M2Asy +M4Ast +M5Art =0 (6)
The node equations of the rotor MEC network are stated in
Fig. 2 2D pattern of flux lines in the cross section of a D-180 frame
size prototype BDFIG
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doi: 10.1049/iet-rpg.2012.0383
the following equations
Art
j−Ary
j
r
rt
j+Art
j−Art
j−1
r
rtt +Art
j−Art
j+1
r
rtt
+
Nss
i=1
Art
j−Ast
i
r
ag
i,j=0
(7)
Hence
−M6Ary +M7Art +MT
5Ast =0 (8)
and
Ary
j−Ary
j−1+ir
j
r
ry
j+Ary
j−Ary
j+1−ir
j+1
r
ry
j+1
+Ary
j−Art
j
r
rt
j+Ary
j
r
sh
j=0
(9)
M8Ary −M6Art −M9M10Inest =0 (10)
ir=M10Inest ,ir=ir
1ir
2··· ir
j··· ir
Nrs
T(11)
where M
10
represents the dependency of the rotor bar currents
on the loop currents and thus on the configuration of loops in
the nest. In the nested loop arrangement, the dimension of M
10
is N
rl
P
r
×2N
rl
P
r
and each loop current is a variable. The
elements of this matrix are determined by using the
following equation (see (12))
The stator voltage equations are expressed as
vpvc
T=Rp0
0Rc
Ip
Ic
+d
dt
l
s,
l
s=
l
p
l
c
(13)
where the stator flux linkage vector, λ
s
,isdefined in the
following equation
l
s=WsM2Asy −Ast
+Ls
leak.Is(14)
The dimensions of the stator leakage inductance matrix
Ls
leak.
and the winding function matrix (W
s
) are 6 × 6 and
6×N
ss
, respectively. The elements of the winding function
matrix are calculated by using (15) which is valid for both
Fig. 3 MEC network of a BDFIG with two loops per nest
M10(k,j)=
1(k−1) ×2Nrl +1≤j≤k×2Nrl
j≤(k−1)Pr+Nrl
−1(k−1) ×2Nrl +1≤j≤k×2Nrl
j.(k−1)Pr+Nrl
0o.w.
⎧
⎨
⎩
(12)
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&The Institution of Engineering and Technology 2014
single and double layer windings
W∗
s(i,j)=
j
k=1
Ts(i,k)
Ws(i,j)=W∗
s(i,j)−
Nss
k=1
W∗
s(i,k)/Nss
(15)
The matrix form of the rotor voltage equations is
d
dt
l
r+RrIr=0(16)
where
l
r=
l
r
1
l
r
2···
l
r
j···
l
r
Nrs
T
=WT
rM6Art −Ary
+Lr
leak.Inest
(17)
The rotor winding function matrix, W
r
, is created from M
10
in
the same manner as W
s
. The rotor leakage inductance matrix,
Ls
leak., is a square matrix of N
rl
P
r
×N
rl
P
r
dimension.
By combining all the rotor and stator equations, the MEC
equation system can be written as the following equation
(see (18))
Since the network permeances depend on the flux crossing
them, the equation system is non-linear. Gauss–Seidel is a
relatively simple method to solve such non-linear equations.
However, the high number of iterations in this method is
not desired. Another method, which is faster but requires
more computational effort, is the Newton–Raphson method.
It is shown that utilisation of this method for solving
nodal-based MEC equations yields an ill-conditioned
Jacobian matrix and leads to divergent results [22]. As an
alternative approximation, the Jacobian matrix can be
estimated as a coefficient matrix. This improves the
response convergence rate [22]. In addition, instead of
direct inversing of the coefficient matrix, Gauss elimination
with partial pivoting method is used in the numerical
process, which has a better stability, accuracy and
convergence properties [23].
The instantaneous rotor position is required in calculation
of air-gap permeances. Hence, the dynamic equations
should be solved in conjunction with the MEC equations
Tel −Tload =Jd
dt
v
m,
v
m=d
dt
u
m(19)
where T
load
is the load torque and the electromagnetic torque
(T
el
) is obtained from the rate of change of air-gap magnetic
energy using the following equations
Wmag =
Nss
i=1
Nrs
j=1
Ast
i−Art
j
2d
d
ur
ag
i,j(20)
Tel =d
d
u
Wmag =Ast −Art
Td
d
ur
ag Ast −Art
(21)
The dynamic MEC model of a BDFIG is depicted in Fig. 5.
3 Permeance calculations
In this section, calculation of the network permeances is
presented.
3.1 Stator magnetic circuit permeances
The calculation of stator network permeances is established in
(22)–(24) based on the geometry shown in Fig. 6a. The slots
are semi-closed in low powers and open in higher ratings.
Also, semi-closed slots can be designed as parallel teeth or
Fig. 4 MEC network of
aith stator slot pitch and
bjth rotor slot pitch
M1−M200−M3TT
S0
−M2M40M500
0MT
5−M6M700
00M8−M60−M9M10
WsM2−WsM200Ls
leak.0
00−WT
rM6WT
rM60Lr
leak.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Asy
Ast
Ary
Art
Is
Inest
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
0
0
0
0
l
s
l
r
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(18)
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IET Renew. Power Gener., 2014, Vol. 8, Iss. 3, pp. 334–346
doi: 10.1049/iet-rpg.2012.0383
Fig. 5 Structure of dynamic MEC model
Fig. 6 Geometry notations for calculation of the network permeances
aStator geometry
bRotor geometry
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&The Institution of Engineering and Technology 2014
parallel slots shapes
r
sy
i=
m
sy
fe,ileqhsy Nss
p
(Dso −hsy),i=1:Nss (22)
r
stt
m
0hss1leq
wsso
(23)
r
st
i=
m
st
fe,ileq wst 2 −wst 1
(hss 2 +0.5hsy )ln(wst 2 /wst 1)(24)
where μ
fe
denotes the local iron core permeability. The
magnetic field intensity is computed from the magnetic
potential across the section and the section length (H=ΔA/
l). The magnetic permeability is obtained from the µ–H
curve of the core. To do so, the power magnetic material
toolbox (PMMT) is utilised [24]. The permeability curve fit
equation used in this toolbox is [25]
m
fe(H)=
m
0+sign(H)
k
mk
hk
1
1+H/hk
nk(25)
where m,hand nare constant vectors determined
experimentally.
l
eq
in (22)–(24) is the equivalent length of the magnetic
core considering radial cooling ducts. It is greater than the
iron length because of flux fringing in the ducts and can be
calculated by using the definition of Carter factor [26].
3.2 Rotor magnetic circuit permeances
In the following, a systematic method for calculation of rotor
permeances with arbitrary number of loops per nest is
presented. The relations of slot opening, tooth and yoke
permeances are stated in (26)–(28), respectively, (according
to Fig. 6b). The rotor slots shape, as shown in Fig. 6b,is
assumed to be rectangular for simplicity
r
rtt
m
0hrs 1leq Nrs
p
Dro −hrs 1
−Nrswrst
(26)
r
rt
m,k=
m
rt
fe,mleqwrt
hrs 2 +m−1
j=1,m=1hry,j+0.5hry,m
,
m=1:Nrl,k=1:Pr
(27)
r
ry
m,k=
m
ry
fe,mhry,mleqNrs
2
p
(2m+1) 0.5Dro −hrs1 −hrs2 −m−1
j=1,m=1hry,j−0.5hry,m
(28)
The reluctance of the base of one tooth to the base of the tooth
in the similar position on the other side of the nest is
<m,k=2
r
rt
m,k
+1
r
ry
m,k
(29)
The fact that the magnetic flux seeks the path of least
reluctance (dℜ
m,k
/(dh
ry,m
) = 0)) is used to calculate h
ry,m
,
hry,m=
2
p
wrt(2m+1) 0.5Dro −hrs 1 −hrs2 −
m−1
j=1,m=1
hry,j
/Nrs
0.5
(30)
In the simplified rotor network, the real flux tubes in the teeth
and yoke sections are substituted with equivalent tubes which
have the same dimensions as the other teeth and yoke
sections, respectively. The relations of the tooth and yoke
reluctances in this network are as per the following
equations, respectively.
r
rt
j=
m
rt
fe,jleqwrt
hrs 2 +0.5hry,eq
(31)
r
ry
j=
m
ry
fe,jleqhry,eq Nrs
2
p
0.5Dro −hrs 1 −hrs 2 −0.5hry,eq
(32)
where h
ry,eq
is the width of the equivalent flux tube in the yoke
region which should be selected in such a way as to minimise
the simplification error. In this regard, the least square error
approach is effectively used. h
ry,eq
is obtained from the
numerical solution of the following equation
min
Nrl
m=1
2<rt
j+(m+2)<ry
j
−<m,k
2
!"
(33)
Finally, the rotor yoke to shaft permeance can be estimated as
r
sh
j2
pm
sh
fe,jleq
Nrs ln (Dro −2(hrs 1 +hrs 2 +hry,eq ))/Dri
(34)
Fig. 7 Relations of fringing permeance between stator and rotor
teeth
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doi: 10.1049/iet-rpg.2012.0383
3.3 Air-gap magnetic circuit permeances
The air-gap permeance
r
ag
i,j
depends on the relative position
of the rotor and stator teeth. This permeance is computed
using the following equation
r
ag
i,j(
u
)=
m
0
leq
u
rs
i,jDag
2g+
r
f
i,j(35)
where
u
rs
i,jis the overlap angle between ith stator tooth and jth
rotor tooth and the relations of fringing permeance
r
f
i,jcan be
determined by using Fig. 7.
The derivation of air-gap permeance matrix with respect to
the rotor angular position, which is required for calculation of
electromagnetic torque, is obtained as
d
d
ur
ag =
r
ag
u
m+d
u
−
r
ag
u
m−d
u
2d
u
(36)
Stator and rotor teeth angle matrices can be used for calculating
air-gap permeances in software implementation. The elements
of the teeth angle matrix are 0 or 1. The ith row of a teeth
angle matrix is related to the position of the ith slot. The
number of columns is around (2π/Δθ), where Δθis the
differential angle. The columns of the rotor teeth angle matrix
are shifted in accordance with the rotor displacement in each
time step. Element by element multiplication of a row of
stator teeth angle matrix and a row of rotor teeth angle matrix
is proportional to the overlap surface of their related teeth.
3.4 Resistances and leakage inductances
The leakage inductance of a winding can be divided into slot
leakage inductance, differential leakage inductance and
overhang leakage inductance. The differential leakage
inductance of stator winding arises from the spatial harmonic
flux as well as the flux flowing from stator to rotor while its
circular path does not include rotor currents. This component
Fig. 8 Relations of stator slot and end-winding leakage inductance terms
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doi: 10.1049/iet-rpg.2012.0383
341
&The Institution of Engineering and Technology 2014
of the leakage inductance is considered in the MEC. However,
the flux lines, which only embrace the slots and the flux of the
end coils, have to be considered. The relations of these terms are
stated in Fig. 8for semi-closed and open slots and single and
double layers winding.
Circulating current because of slot leakage flux causes
additional resistive loss in the case of stator conductors
having more than one strand in parallel. This loss
component is modelled by increasing the stator winding
resistance. The correction factors for single and double
layer windings are stated in [26].
The bars of a rotor coil are connected to a common
end-ring from one side. The connection in the other side of
each loop is created via a joint bar. The rotor slots often
have deep bars which results in low-rotor resistance. The
bar resistance is increased because of slot leakage flux and
its resulting current displacement. This phenomenon
decreases the slot leakage inductance [26]. The machine
performance is greatly dependent on the rotor parameters
especially the rotor leakage inductance [27]. Hence,
appropriate correction factors should be used for
considering the current displacement effects. The factors
(K
R
and K
L
) are functions of slot dimensions and rotor
frequency [26]
kR=
j
sinh (2
j
)+sin (2
j
)
cosh (2
j
)−cos (2
j
)(37)
kL=2
j
sinh (
j
)−sin (
j
)
cosh (
j
)+cos (
j
)(38)
j
=hrs 2 ##########
p
fr
m
0
s
cu
$(39)
The leakage inductance of a rotor loop can be calculated by
using the following equations
Lleak.,m=
m
02leq
l
leak.,rs +(2m+1)
a
rs Dro −hrs 1 −hrs 2
l
leak.,er +Lleak.,oh,m=1:Nrl
(40)
l
leak.,er 1
p
1+4wrs
p
hrs 2
ln 4wrs
4wrs +
p
hrs 2
%&
(41)
l
leak.,rs =2hrs 2
3wrs 1 +wrs 2
KL+hrs 1
wrso
+g
t
rs
(42)
Lleak.,oh 0.3
×0.08leq +(2m+1)
a
rs ODr−hrs 1 −hrs 2
(43)
The rotor loop resistance is simply expressed as a function of
the dimensional parameters. The resistance of the mth loop of
the nest is
Rm=2Rbar,mKR+Rer,m+Roh,m,m=1:Nrl (44)
where R
bar,m
,R
er,m
and R
oh,m
are the resistances of rotor bar,
end-ring segment and overhang connection between the
bars of the mth loop.
4 Results and discussion
The accuracy and validity of the proposed model are verified
using experimental studies. The parameters of the test
machine are given in Table 1. The experimental setup is
shown in Fig. 9. A DC machine equipped with an ABB
DCS800 drive system is mechanically coupled with the
D-180 BDFIG as the prime mover. A Magtrol TMB312
torque transducer and an incremental encoder with 10 000
pulses per revolution are used to measure the 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. The
specifications of electric steel used in the stator and rotor
core are given in Table 2.
In the first case, the PW is excited via a three-phase, 90
V
rms
, 50 Hz voltage source, whereas the control winding is
shorted. The magnetic characteristics of the stator and rotor
cores lie on the linear region of the magnetising curve
under this operating condition. The produced torque and the
effective value of the PW phase current against rotor speed
are measured. The steady-state measured and calculated
curves of torque and PW current are compared in Figs. 10a
and b, respectively.
The measured and calculated curves of the PW current at
different rotational speeds are shown in Fig. 10cwhen the
PW voltage is increased to the nominal value, 240 V
rms
.
The average value of the torque given by the MEC model
is slightly lower than its measured value and has a higher
Table 2 Specifications of Laminated Core (M800-65A) [24]
Parameter Value Description
k
h
273.2 hysteresis loss factor
k
e
0.4786 Eddy current loss factor
d
t
0.65 mm lamination thickness
ρ
fe
390 nΩm resistivity
m
k
[1.86 0.834 −0.517 1.18] m
k
values in (25)
h
k
[290 160 160 297] h
k
values in (25)
n
k
[1.00 3.30 1.26 2.02] n
k
values in (25)
Fig. 9 Test rig
Table 1 Prototype D-180 frame size BDFIG specifications
N
ss
48 w
ss1
8.1 mm w
rs1
9.4 mm
P
p
2w
ss2
5.3 mm w
rs2
5.7 mm
P
c
4h
ss1
1.7 mm h
rs1
1.8 mm
D
so
270 mm h
ss2
19.9 mm h
rs2
19.5 mm
D
si
175 mm N
rs
36 α
rs
10
o
N
p
10 D
ro
174.5 mm L3
N
c
20 D
ri
30 mm l199.5 mm
w
sso
3.2 mm w
rso
2mm J0.143 Nm kg
2
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342
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IET Renew. Power Gener., 2014, Vol. 8, Iss. 3, pp. 334–346
doi: 10.1049/iet-rpg.2012.0383
Fig. 10 Produced torque and the effective value of the PW phase current against rotor speed
aTorque–speed curves
bPW current–speed curves when PW is excited via a three-phase, 90 V
rms
, 50 Hz voltage source and CW is shorted
cPW current–speed curves when PW voltage is increased to 240 V
rms
and CW is shorted through an external resistance of 11.7 Ω
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doi: 10.1049/iet-rpg.2012.0383
343
&The Institution of Engineering and Technology 2014
distortion level, which is due to the fact that a limited
number of nodes are considered in the air-gap region. The
magnitude and frequency of the CW induced voltage is
subject to increase by deviation of rotational speed from
natural speed (n
n
= 500 rpm) [7]. Hence, core and stray
load losses are increased under these operating conditions.
These loss components are not modelled in the MEC
which result in computational error and the imposed error
is increased by deviating from natural speed. However, the
calculation error is acceptable for the speed range of
interest (n
n
± 25%).
The computational errors of the EEC and MEC methods in
calculation of PW current are shown in Fig. 11 for both
operating conditions. The EEC parameters of the test
machine are obtained analytically and experimentally as
reported in previous publications, for example [12].
Although the MEC model has satisfactory accuracy under
linear and saturated conditions, the accuracy of the EEC
model decreases with increasing saturation level.
In design studies, the computing time is also important in
addition to the accuracy of the model. The extracted MEC
consists of 192 unknowns. It has been programmed in
MATLAB software and runs for 14 min per steady-state
value on a PC with a 2.53 GHz Core2Duo processor.
It should be noted that the elapsed time in simulation of a
2D FE model of the machine using a purpose-built
commercial software (Ansoft Maxwell 12.0) is about 2.5 h
per steady-state value. An efficient implementation of the
MEC model further decreases the simulation time.
In contrast to the steady state, where the flux lines have a
uniform distribution, the transient response of the MEC
model cannot be trusted because of its coarse meshing. The
stator windings and rotor loops currents as well as the
magnetic field of the stator and rotor teeth in simulation of
the MEC model under specified operating conditions are
shown in Fig. 12. The flow of two fundamental flux
components with different frequencies in stator core and
flux density distributions with unequal magnitudes in rotor
core sections are evident in this figure.
The steady-state results of the MEC and 2D
magneto-dynamic FE models under operating conditions of
Fig. 12 are given in Table 3. The comparisons show good
agreement between the results and confirm the validity of
the modelling approach.
Fig. 11 Computational errors of EEC and MEC models in calculation of PW current under operating conditions of Fig. 8with PW voltage of
a90 V
rms
b240 V
rms
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344
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IET Renew. Power Gener., 2014, Vol. 8, Iss. 3, pp. 334–346
doi: 10.1049/iet-rpg.2012.0383
Fig. 12 Steady-state results of the MEC and 2D magneto-dynamic FE models under operating conditions
aPW and CW phase currents
bRotor loop currents
cTime variations and spatial distributions of stator and rotor teeth flux density, (PW is excited via a three-phase, 240 V
rms
, 50 Hz voltage source. CW is shorted
and the speed is 562 rpm)
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doi: 10.1049/iet-rpg.2012.0383
345
&The Institution of Engineering and Technology 2014
5 Conclusions
In this paper, an MEC model is proposed and utilised to
predict different performance aspects of a BDFIG. The
model takes into account actual details, including stator
winding distributions, special configuration of rotor
conductors, slotting effects, fringing flux and magnetic
material saturation. The proposed model provides
reasonable accuracy as well as acceptable computation time
which make it an appropriate tool for using in
population-based optimal design studies. The validation of
the model is performed experimentally and through FEA.
The comparisons show considerable benefits of the MEC
model over electric equivalent circuit model, especially
when the core is magnetically saturated.
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Table 3 Calculation and FEM results
Parameter Description MEC 2-D FEM
I
pw
effective value of
PW current
9.39 A
rms
9.74 A
rms
I
cw
effective value of
CW current
4.9 A
rms
5.12 A
rms
T
el
electromagnetic
torque
−87.12 Nm −94.8 Nm
B
st,max
max. value of
stator tooth flux
density
1.75 T 1.83 T
B
sy,max
max. value of
stator yoke flux
density
1.24 T 1.33 T
B
rt,max
max. value of
rotor tooth flux
density
1.88 T 1.93 T
B
ry,max
max. value of
rotor yoke flux
density
1.09 T 1.16 T
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IET Renew. Power Gener., 2014, Vol. 8, Iss. 3, pp. 334–346
doi: 10.1049/iet-rpg.2012.0383