Equivalent Circuit Parameters for Large Brushless Doubly Fed Machines (BDFMs)

Abstract

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.

Full text

706 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 3, SEPTEMBER 2014
Equivalent Circuit Parameters for Large Brushless
Doubly Fed Machines (BDFMs)
Salman Abdi, Ehsan Abdi, Senior Member, IEEE, Ashknaz Oraee, and Richard McMahon
Abstract—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 magnetiz-
ing 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 ef-
fects can be modeled by numerical models, the proposed analytical
methods are particularly helpful in optimizing machine design, in-
verter 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 mag-
netic 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 meth-
ods are assessed by experimentally verified finite-element models
for a 250 kW BDFM.
Index Terms—Brushless doubly fed machine (BDFM), carter
factor, coupled-circuit model, finite-element (FE) method, induc-
tance calculation, magnetic wedges, magnetomotive force (MMF).
I. INTRODUCTION
THE BDFM is an alternative to the well-established doubly
fed induction generator (DFIG) for use in wind turbines
as it retains the benefit of utilizing a fractional-size converter,
but it also offers higher reliability and lower maintenance costs
than the DFIG due to absence of brush gear and slip-rings [1]. In
addition, the BDFM is intrinsically a medium-speed machine,
enabling the use of a simplified one or two-stage gearbox, hence
reducing the cost of the overall drivetrain and giving further
reliability improvement.
The modern BDFM as a variable speed drive or generator
comprises two electrically separate stator windings: one con-
nected directly to the grid, called the power winding (PW), and
the other supplied from a variable voltage and frequency con-
verter, called the control winding (CW). 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 [2]. The rotor is especially designed to couple
Manuscript received September 20, 2013; revised January 27, 2014; accepted
February 28, 2014. Date of publication April 17, 2014; current version date Au-
gust 18, 2014. This work was supported by the European Union’s Seventh
Framework Program managed by REA Research Executive Agency (FP7/2007
2013) under Grant N.315485. Paper no. TEC-00556-2013.
S. Abdi, A. Oraee, and R. McMahon are with the Electrical Engineer-
ing Division, Cambridge University, Cambridge, CB3 0FA, U.K. (e-mail:
s.abdi.jalebi@gmail.com; ashknaz.oraee@gmail.com; ram1@eng.cam.ac.uk).
E. Abdi is with Wind Technologies Limited, St. Johns Innovation Park,
Cambridge, CB4 0WS, 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/TEC.2014.2311736
to the two air-gap fields associated with the two stator windings,
the nested-loop design being commonly used [3]. The normal
mode of BDFM operation is as a synchronous machine with
the rotor rotating at a speed determined by the winding pole
numbers and the mains and converter frequencies.
The equivalent circuit is a simple method of representing
the steady-state performance of the BDFM, allowing the rapid
calculation of operating conditions of the BDFM and its asso-
ciated converter and grid connection. The equivalent circuit is
also helpful for the design and optimization of the machine. The
equivalent circuit parameters can be calculated from the ma-
chine geometry using the analytical methods described in [4].
They can also be extracted from BDFM steady-state operation,
provided from numerical models or experimental tests, by ap-
plying curve-fitting methods [2]. Extracted parameters are likely
to be more accurate as it is hard to obtain analytically precise
values of certain quantities such as the leakage reactances of
end-windings.
To date, there have been several attempts to manufacture
large BDFMs, for example, in Brazil with a 75 kW machine [5],
in China with the design of a 200 kW machine [6] and the
250 kW BDFM reported by the authors [7]. The latter was built
in a frame size D400 as a stepping-stone toward a megawatt-
scale BDFM wind turbine and it involved construction and wind-
ing techniques appropriate to large machines including mag-
netic wedges. These are often used in large machines with open
slots [8] to reduce the effective air gap length, thereby result-
ing in lower magnetizing currents, which is advantageous in the
BDFM design. Wedges, typically comprising 75% iron powder,
7% glass mat, and 18% epoxy resin [9], also reduce core losses,
and hence, the machine temperature rises [8], [10], [11]. On the
other hand, magnetic wedges will increase the slot leakage by
providing easier paths for fluxes not crossing the air gap [12].
Though the effects of magnetic wedges on the performance
of induction and permanent magnet machines have been studied
by others, for example, in [11] and [13], the methods of anal-
ysis proposed in those papers cannot be readily applied to the
BDFM due to its complex design and magnetic field distribu-
tion [14]. This paper investigates the effects of magnetic wedges
on the performance of the BDFM by analyzing their impacts on
the equivalent circuit parameters. Analytical methods are pre-
sented to calculate the magnetizing and leakage inductances for
a 250 kW BDFM with magnetic wedges and finite-element (FE)
models are used to verify the accuracy of the proposed methods.
II. BDFM EQUIVALENT CIRCUIT MODEL
A simplified equivalent circuit for the BDFM is shown in
Fig. 1, where all parameters are referred to the PW side and iron
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ABDI et al.: EQUIVALENT CIRCUIT PARAMETERS FOR LARGE BRUSHLESS DOUBLY FED MACHINES (BDFMS) 707
Fig. 1. Simplified equivalent circuit.
losses are neglected [2]. The circuit is valid for all the modes
of operation, including the induction, cascade, and synchronous
modes and can be utilized for the analysis of steady-state per-
formance of the BDFM. s1and s2are the power and CW slips
and are defined as
s1=ω1−p1ωr
ω1
(1)
s2=ω2−p2ωr
ω2
.(2)
The referred rotor inductance, L
r, in the simplified circuit
shown in Fig. 1 represents the series inductances in the full
equivalent circuit [2], including the stator PW and CW and rotor
leakage inductances. Hence, the effects of stator slot magnetic
wedges on the flux leakage will be reflected in the value of the
L
rin the simplified equivalent circuit.
As described, the equivalent circuit parameters can be calcu-
lated from the machine geometry, during the design stage, using
the method described in [15]. The procedure involves deriving
the machine’s coupled-circuit model, followed by performing a
series of transformations to obtain the dq sequence components
and equivalent circuit parameters, respectively.
The equivalent circuit parameters can also be extracted from
steady-state measures, such as torque, speed, voltages, and cur-
rents, obtained from BDFM’s operation in the induction and
cascade modes [2], as shown in the block diagram of Fig. 3.
The steady-state data can be obtained from numerical models or
experimental tests. The stator winding resistances, R1and R2,
are either calculated from the machine geometry at a certain
operating temperature or obtained from the DC measurements.
The magnetizing inductances Lm1and Lm2are obtained from
the magnetizing tests where a single stator winding is supplied
in turn while the other winding is left open and the rotor is driven
at the synchronous speed to eliminate rotor currents. Finally, the
rotor parameters Lrand Rrare obtained from applying a curve
fitting method to the data from cascade tests, assuming the stator
resistance and magnetizing parameters are fixed [2].
III. MODELS CONSIDERED IN THIS STUDY
Five models have been utilized in this study to obtain the
equivalent circuit parameters, which are described in Table I.
Two of the models utilized are based on the coupled-circuit ap-
proach: one neglecting the effects of magnetic wedges (CC-LN-
NW) and the other incorporating the authors’ proposed approach
to model the effects of wedges (CC-LN-W). The coupled-circuit
model comprises the machine inductance and resistance matri-
ces and the effects of iron saturation are neglected. It has been
TAB LE I
DIFFERENT APPROACHES UTILIZED FOR OBTAINING THE EQUIVALENT
CIRCUIT PARAMETERS
TAB LE I I
SPECIFICATIONS OF THE 250 KW D400 BDFM
shown to give a close agreement with the experimental tests on
a D180 size machine, for example, in [16]. The inductance and
resistance matrices are calculated from the machine geometry
and can be transformed into the equivalent circuit parameters
by performing a series of transformations [17].
The other three models are based on the 2-D FE approach. One
assumes a linear iron circuit and no magnetic wedges (FE-LN-
NW), the second one assumes a linear iron circuit and takes into
account the magnetic wedges with a constant relative perme-
ability (FE-LN-W), and the third model includes the nonlinear
properties of the iron circuit and magnetic wedges(FE-NLN-
W). Clearly, the latter has a significantly more computational
time than the linear models, but will be shown to give the closest
predictions to the experimental results.
IV. PROTOTYPE 250 KW D400 BDFM SPECIFICATIONS
A. Specifications
The specifications of the 250 kW D400 BDFM are shown in
Table II. The D400 BDFM was constructed as a frame size 400
machine with the stack length of 820 mm. The stator windings
were form wound from copper strips. The PW was rated at
690 V, 178 A, at 50 Hz and the CW was designed for 620 V at
18 Hz and rated at 73 A. Both stator windings were connected
in delta. The rotor comprises six sets of nests, each consisting
of a number of concentric loops [18], the conductors being solid
bars with one common end ring [7]. The magnetic properties for
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708 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 3, SEPTEMBER 2014
Fig. 2. 250 kW D400 BDFM (right front) on test bed.
Fig. 3. Extraction of equivalent circuit parameters from numerical modeling
or experimental measurements.
TABLE III
MEASUREMENT ACCURACY
the iron and the magnetic wedges were provided by the machine
manufacturer.
B. Test Rig and Instrumentation Details
Fig. 2 shows the BDFM on the test bed, which was set up at
the manufacturer’s facility. The BDFM was driven via a torque
transducer by a 355 kW eight-pole induction motor fed from
a 400 kW ACS800 inverter. The PW was connected to the
mains and supplied at 690 V, 50 Hz and the CW was supplied
from the converter. Speed and position signals are obtained
from an incremental encoder with a resolution of 10,000 pulses
per revolution. The voltages and currents of each stator phase
are measured by LEM AV100-750 and LEM LA 205/305-s
transducers, respectively. The accuracy of voltage, current,
torque, and speed measurements is shown in Table III, which are
calculated from the specifications provided by the manufacturers
and the accuracies of the conditioning circuitries.
V. MAGNETIZING AND CASCADE TESTS
As described, the BDFM equivalent circuit parameters can be
extracted from the magnetizing and cascade tests performed by
either numerical models or experiments.
A. Magnetizing Tests
When only the PW or CW of the BDFM stator is supplied
and the unsupplied winding is left open, the BDFM operates as
ap1or p2pole pair induction machine, respectively. If the rotor
speed is set to the synchronous speed using an external load
machine, only the field due to the excited winding will exist
in the machine, and hence, the equivalent circuit of Fig. 1 will
reduce to only include the supplied winding’s resistance and the
magnetizing inductance. The magnetizing inductances of PW
and CW, Lm1and Lm2, can be derived from these tests.
The magnetizing characteristic of the D400 BDFM is shown
in Fig. 4 for the two stator windings. Results from the exper-
iments and FE-LN-NW and FE-NLN-W models are shown.
The supply frequency was 20 and 10 Hz for the PW and CW,
respectively.
There are two horizontal lines shown in Fig. 4: the solid
line shows the rated voltage of the corresponding winding at the
excited frequency and the dashed line shows the voltage at which
the air gap flux is rated in the absence of the other stator field.
As can be seen from Fig. 4, the effect of iron saturation is not
significant below the air gap rated flux. The effect of stator slot
magnetic wedges in reducing magnetizing currents is evident.
The magnetizing inductances, Lm1and Lm2, can be obtained
from the slope of the linear region below the rated voltages.
B. Cascade Tests
The BDFM can be operated as a self-cascaded induction
machine by exciting the PW or CW and shorting the other
winding. A cascade induction machine formed from p1and p2
pole pair induction machines has characteristics which resemble
an induction machine with p1+p2pole pairs.
The rotor parameters, Lrand Rr, and turns ratio (N1/N2)
may be obtained by applying a curve fitting method to the re-
sults from BDFM cascade operation. The stator currents (both
supplied and shorted windings) and torque are used for extract-
ing the parameters. The fitting algorithm assumes fixed values
for stator winding resistances and magnetizing inductances and
finds the values for Lr,Rr, and N1/N2, which give the best fit.
Fig. 5 shows the results from the cascade tests obtained from the
experiments and FE-LN-NW and FE-NLN-W models. The cas-
cade tests have been carried out at reduced voltages, i.e., the PW
and CW were supplied at 70 V, 20 Hz and 240 V, 20 Hz, respec-
tively, due to restrictions on stator currents; hence, the extracted
parameters do not represent the effect of iron saturation.
VI. COMPARISON OF EXISTING MODELS
The equivalent circuit inductance parameters, Lm1,Lm2, and
Lr, obtained from different modeling methods and the exper-
imental tests are shown in Tables IV and V. The parameters
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ABDI et al.: EQUIVALENT CIRCUIT PARAMETERS FOR LARGE BRUSHLESS DOUBLY FED MACHINES (BDFMS) 709
(a) (b)
Fig. 4. Magnetizing characteristic of the D400 BDFM. (a) Magnetizing characteristic of the stator PW. (b) Magnetizing characteristic of the stator CW.
(a) (b)
(c) (d)
(e) (f)
Fig. 5. Cascade results and fitted curves for Experimental data, FE-LN-NW and FE-NLN-W models. (a) Torque in PW cascade test, (b) Torque in PW cascade
test, (c) Ipw in PW cascade test, (d) Ipw in CW cascade test, (e) Icw in PW cascade test, and (f) Icw in CW cascade test.
shown in Table IV are for the machine without magnetic wedges
in the stator slots. The two linear models are in close agreement,
as expected, which validates the assumptions used to obtain the
coupled-circuit model and its transformation to the equivalent
circuit model proposed by Roberts in [15]. Nevertheless, neither
of the models are able to take into account the effect of magnetic
wedges.
Table V shows the inductance parameters obtained from FE
models and experiment when magnetic wedges are considered.
There is acceptable agreement between the parameters obtained
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710 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 3, SEPTEMBER 2014
TAB LE I V
EQUIVALENT CIRCUIT PARAMETERS WITHOUT STATOR MAGNETIC WEDGES
TAB LE V
EQUIVALENT CIRCUIT PARAMETERS WITH STATOR MAGNETIC WEDGES
from the FE-NLN-W model and the experiments, considering
the measurement accuracies and the limitations in obtaining
geometrical data. There is about 15% difference in the rotor
inductance Lr, mainly due to the fact that Lrrepresents the
stator and rotor leakage inductances, and the modeling of the
leakage effects, for example in the case of winding overhang,
is difficult. The agreement between FE-NLN-W and FE-LN-W,
both using identical geometrical data, is close, which shows
that the nonlinearity of the iron circuit can be neglected due to
reduced voltage levels in the tests. It also confirms that assuming
the linear magnetic property for the magnetic wedges, i.e., a
constant permeability, gives acceptable accuracy.
The differences between the parameters obtained assuming no
magnetic wedge in stator slots, shown in Table IV, and the ones
obtained when magnetic wedges are present, shown in Table V,
are significant; both magnetizing and leakage inductances are
larger in the latter table. Though the nonlinearity of the iron
circuit may also contribute to the differences, since the tests
were undertaken at reduced voltage levels, such effects can be
neglected.
There is, therefore, a need for an analytical approach to model
the effect of magnetic wedges in the coupled circuit model, so
these effects can be considered during the design optimization
when the computational cost is important. The following sec-
tions will present the authors’ proposed approach.
VII. CALCULATION OF MAGNETIZING INDUCTANCES
The magnetizing inductances of electrical machines corre-
spond to the magnetic flux that links the stator and rotor, pass-
ing the air gap twice. The effects of the stator and rotor slotting
in the magnetizing inductance are often modelled by Carter
factors, which effectively scale the air gap length. The effect
of magnetic wedges in reducing magnetizing currents, hence
increasing magnetizing inductances, can also be modeled by
scaling the air gap length, similar to the concept of Carter fac-
tors [19]. Hence, modified Carter factors may be derived, which
can take into account the effect of both slotting and magnetic
wedges.
The Carter factor is effectively the ratio of the air gap peak
flux density, which would be the case in the absence of slotting,
and its average value when slotting is considered [20]. When
both rotor and stator have open slots, it is assumed that the
Fig. 6. Different magnetic paths from stator tooth to the rotor.
resultant Carter factor is the product of the Carter factors for the
stator and the rotor [21]. Since magnetic wedges are only used in
stator slots of the BDFM, the following presents the calculation
of the Carter factor for the stator, assuming an unslotted rotor
using the magnetic equivalent circuit method. The Carter factor
for the rotor can be calculated using the conventional methods
described, for example, in [21]. The following assumptions are
made:
1) There is no saturation in the iron circuit and in magnetic
wedges, and hence, infinite and constant permeabilities
are assumed for iron and magnetic wedges, respectively.
2) The magnetic wedges are assumed to have a rectangular
shape.
3) There are three magnetic paths between the stator tooth
and the unslotted rotor, through which magnetic fluxes
φt,φo, and φwenter the rotor, as shown in Fig. 6. Θis
considered as the total MMF that produces these fluxes.
4) The reluctance between the above mentioned magnetic
paths is infinite, and hence, the interaction between their
corresponding magnetic fluxes is negligible.
A typical variation of air gap flux density over a slot pitch
is shown in Fig. 6 in the presence of a magnetic wedge, where
Bav,Bmax, and Bmin are average, maximum, and minimum air
gap flux densities, respectively.
The reluctances associated with the magnetic paths corre-
sponding to φoand φwinclude:
Rox =x
μ0Aox
,R
oy =y
μ0Aoy
,R
og =g
μ0Aog
(3)
Rwx =x
μ0μwAwx
,R
wy =y
μ0μwAwy
,R
wg =g+d
μ0Awg
(4)
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ABDI et al.: EQUIVALENT CIRCUIT PARAMETERS FOR LARGE BRUSHLESS DOUBLY FED MACHINES (BDFMS) 711
where gis the air gap length, dis the depth of the air-filled region
above the magnetic wedge, μwis the relative permeability of the
magnetic wedge, and Ais the area through which the magnetic
flux is passing. The flux entering the rotor consists of three
fluxes: φw,φo, and φtshown in Fig. 6. Bwx,Bwy,Box, and
Boy are the flux densities associated with the reluctances Rwx,
Rwy,Rox , and Roy, respectively. Assuming Θis the MMF
producing φw,φo, and φt, and the MMF across Rwx,Rwy,Rwg
and Rox,Roy ,Rog are Θwx,Θwy,Θwg and Θox,Θoy,Θog,
respectively, the following equations can be obtained:
Θ=Θ
ox +Θ
oy +Θ
og =Θ
wx +Θ
wy +Θ
wg (5)
Θ=RoxAox Box +(RoyAoy +RogAog)Boy (6)
Θ=RwxAwxBwx +(Rwy Awy +RwgAwg)Bwy.(7)
Along the path Rox and Roy, the area through which φois
traveling can be assumed uniform, i.e., Aox =Aoy, hence:
Box =Boy.(8)
Considering the path for φw, the flux densities Bwx and Bwy
are related as
Bwx =Bwy
wo
2hw
.(9)
From (6)—(9), and since Roy and Rwy are negligibly small,
Θ=Boy
x+g
μ0
=Bwy wox
2μ0μwhw
+g+d
μ0(10)
hence
Boy =μ0Θ
x+g(11)
Bwy =Θ
wo
2μ0μwhwx+g+d
μ0
.(12)
The magnetic fluxes φoand φwcan be worked out from the
flux densities Boy and Bwy as
φo=d
0
Boy.dx =μ0Θlng+d
g(13)
φw=w2
2
0
Bwy.dx =2hwμ0μwΘ
wo
ln wow2
4hw
μw(g+d)(14)
where woand w2correspond to the stator slot opening width
and the magnetic wedge width, respectively, as shown in Fig. 6.
Considering Bmax−airgap corresponds to the magnetic flux φmax
passing through the air gap for an unslotted stator, the air gap
correction factor for taking magnetic wedges into account can
be derived as
Kc−stator =Bmax−airgap
Bmean−airgap
=φmax
φt+2(φo+φw)(15)
where
φmax =μ0ysΘ
g(16)
φt=μ0(ys−wo)Θ
g.(17)
Substituting (13), (14), (16), and (17) into (15) gives
Kc−stator =
ys
(ys−w2)+4hwg
woμwln wow2
4hw+μw(g+d)
μw(g+d)+2gln( g+d
g)
.
(18)
The rotor Carter factor kc−rotor is calculated for the D400
BDFM using the method proposed by Heller and Hamata [21]
kc−rotor =yr+8g
yr−wr+8g(19)
where yris the rotor slot pitch and wris the rotor slot opening
width. Equation (19) has empirical justification and is valid
when wo
gis not sufficiently large (less than 12) [21]. The overall
effective air gap gecan be calculated using the product of stator
and rotor Carter factors [22]:
ge=g×kc−stator ×kc−rotor.(20)
The effective air gap can be used in the calculation of the
magnetizing inductances for the BDFM. The validity of the
proposed method will be verified in Section IX using numerical
analysis.
VIII. CALCULATION OF LEAKAGE INDUCTANCES
Inductances arising from leakage effects are inductances due
to the magnetic flux not linking stator and rotor conductors.
This definition is similar to that found in [23], except that for
the purposes of this paper, the harmonic mutual inductance terms
are not considered part of the leakage terms, rather as a separate
term.
The calculation of stator and rotor winding leakage induc-
tances is greatly simplified by the calculation of the specific per-
meance which is defined as the self-inductance per unit length
of coil per turn squared. Thus, to calculate the self-inductance of
a coil due to leakage, the different permeance terms around the
length of the coil should be summed and the result multiplied by
the number of turns squared. Consequently, the per phase leak-
age inductances for the stator PW and CW, Ls−pw and Ls−cw ,
can be calculated:
Ls−pw =Cpw N2
pw λs−pw (21)
Ls−cw =CcwN2
cwλs−cw (22)
where Cpw and Ccw are the PW and CW number of coils per
phase, Npw and Ncw are the number of turns per coil, λs−pw and
λs−cw are the specific permeances, and is the stator effective
stack length. There are four major sources of leakage permeance
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712 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 3, SEPTEMBER 2014
Fig. 7. D400 BDFM stator slot details.
in induction machines [22]–[24], which are also taken into ac-
count for the BDFM: slot permeance, tooth-top permeance,
overhang permeance and zigzag permeance. Among the leakage
terms, the slot leakage in most cases has the largest contribution
in the winding leakage.
As shown in Section VI, there is a large discrepancy in the
value of L
rwhen the effects of magnetic wedges are ignored.
This is most likely due to easier paths for stator leakage fluxes
in the stator slot openings enabled by the magnetic wedges. The
other leakage terms as well as the rotor leakage are assumed to
be largely unaffected by stator wedges. The following presents
the calculations of the slot permeance when magnetic wedges
are used in stator slot openings. The slot contains two stator
windings, the PW and CW, and each winding is assumed to
have a double-layer arrangement that is a common practice in
large electrical machines. Derivations according to [23] are used
for the other permeance terms.
Fig. 7 shows the stator slot. The slot leakage for the PW is
calculated here; a similar approach can be used to derive the CW
slot leakage. The slot permeance of the PW can be obtained
by calculating the energy of the leakage magnetic field in a
slot:
Wφ=
7

i=1
1
2μivi
B2
i(h).dvi(23)
where
B(h)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
B1(h)=μ1
Θpw−Ah
hpw
w1,0<h<hpw
B2(h)=μ2
Θpw−A
w1
,h
pw <h<h1
B3(h)=μ3
Θpw−A+Θ
pw−Bh−h1
hpw
w1
,h
1<h <h2
B4(h)=μ4
Θpw−A+Θ
pw−B
w1
,h
2<h <h3
B5(h)=μ5
Θpw−A+Θ
pw−B
w2
,h
3<h <h4
B6(h)=μ6
Θpw−A+Θ
pw−B
w2+(wo−w2)h−h4
hw1
,h
4<h <h5
B7(h)=μ7
Θpw−A+Θ
pw−B
wo
,h
5<h <h6
and
dv1=dv2=dv3=dv4=.w1.dh (24)
dv5=.w2.dh (25)
dv6=w2+(wo−w2)h
hw1.dh (26)
dv7=wo.dh (27)
μ1=μ2=μ3=μ4=μ7=μ0(28)
μ5=μ6=μw(29)
where is the stator effective stack length and Θpw−Aand
Θpw−Bare the total current linkages of the PW bottom and
upper layers. All different heights and widths are specified in
Fig. 7. The energy of the PW leakage magnetic field in the slot
from (23) is
Wφ=μ0
2w1
[Θ2
pw−Ahpw
0h
hpw 2
.dh +Θ
2
pw−Ah1
hpw
dh
+h2
h1Θpw−A+Θ
pw−B
h−h1
hpw 2
.dh
+(Θ
pw−A+Θ
pw−B)2h3
h2
.dh]
+μw
2w2
(Θpw−A+Θ
pw−B)2h4
h3
dh +μ0
2w0h5
h4
dh
+μw
2(Θpw−A+Θpw−B)2h6
h5
dh
w2+(wo−w2)h−h5
hw1
.
(30)
Note that
Θpw−A=Θ
pw−B=1
2Θpw .(31)
When a short pitched winding is used, the currents in the two
CW coils in a stator slot are either in phase or have a 120◦phase
shift. In case of the latter, the term Θpw−A.Θpw−Bin (30) is
multiplied by cos 2π
3, representing the mutual influence of the
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ABDI et al.: EQUIVALENT CIRCUIT PARAMETERS FOR LARGE BRUSHLESS DOUBLY FED MACHINES (BDFMS) 713
bottom and upper currents [22]. Assuming qis the number of
slots per phase per pole and ais the number of short pitched
slots, the cos terms can be averaged as
cos β=1
2q
2q

i=1
cos γi=1
2q2acos 2π
3+2(q−a)cos0
.
(32)
The term Θpw−A.Θpw −Bcan hence be written as
Θpw−A.Θpw −B=Θ
2
pw cos βpw .(33)
Substituting (32) and (33) into (30), the PW leakage magnetic
field energy in the slot is
Wφ=1
2Θ2
pw μ05
3+cosβpw hpw
w1
+h
w1
+h3−h2
w1
(2 + 2 cos βpw )+ ho
wo
(2 + 2 cos βpw )
+1
2Θ2
pw μwhw2
w2
(2 + 2 cos βpw )
+hw1
w2−wo
(2 + 2 cos βpw )lnw2
wo.(34)
The total PW leakage magnetic energy stored in the slot is
Wφ=1
2Lslot−leakageΘ2
pw =1
2Θ2
pw λs−pw (35)
where λs−pw is the PW slot leakage of one slot. From (34)
and (35), λs−pw can be extracted by defining three terms: self-
permeance for PW layer A λpw−A, layer B λpw −B, and mutual
permeance between these two layers λpw−M:
λpw−A=μ0hpw
3w1
+h3−hpw
w1
+ho
wo
+μwhw2
w2
+hw1
w2−wo
ln w2
wo (36)
λpw−B=μ0hpw
3w1
+h3−h2
w1
+ho
wo
+μwhw2
w2
+hw1
w2−wo
ln w2
wo (37)
λpw−M=μ0hpw
2w1
+h3−h2
w1
+ho
wo
+μwhw2
w2
+hw1
w2−wo
ln w2
wo.(38)
λs−pw is then
λs−pw =λpw−A+λpw −B+2×λpw−M×cos βpw .(39)
The slot leakage for the CW can be derived using the same
procedure as that presented for the PW as
λs−cw =λcw−A+λcw−B+2×λcw −M×cos βcw.(40)
The PW and CW leakage inductances can now be calculated
from (21) and (22).
Fig. 8. Areas covered with magnetic wedges for different cases.
TAB LE V I
DIMENSIONS AND RELATIVE PERMEABILITY OF MAGNETIC WEDGES USED FOR
VERIFICATION OF ANALYTI CAL METHODS
TAB LE V II
EQUIVALENT CIRCUIT PARAMETERS CASE A
IX. VERIFICATION OF ANALYTICAL METHODS
In order to verify the accuracy of the proposed analytical
methods in Sections VII and VIII, the magnetizing and leakage
inductances derived from the coupled circuit model embedding
the new analytical methods are compared with parameters pre-
dicted by the numerical models, i.e., FE-LN-W.
Fig. 8 shows the area in the stator slot opening which is oc-
cupied by a magnetic wedge. Three cases have been considered
to examine the accuracy of the analytical methods with respect
to wedge dimensions and permeability (see Table VI):
Case A: This is the real case in the D400 BDFM. The magnetic
wedge fills Area1and Area2and has a relative permeability of
μw=20which is an approximation of its nonlinear magnetic
characteristic provided by the manufacturer.
Case B: The magnetic wedge has identical dimensions as
Case A, but its relative permeability is assumed to be μw=10.
Case C: The magnetic wedge only fills Area1and hence
Area2is assumed unfilled, i.e., is filled by air. The relative
permeability of magnetic wedge is identical to Case A, i.e.,
μw=20.
The magnetizing and leakage inductance parameters in the
referred equivalent circuit, shown in Fig. 1, are calculated using
the coupled circuit and FE-LN-W models for Cases A, B, and
C, and are shown in Tables VII, VIII, and IX. As can be seen,
there is generally a close agreement between the parameters pre-
dicted by two models for all the three cases, hence validating the
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714 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 3, SEPTEMBER 2014
TABLE VIII
EQUIVALENT CIRCUIT PARAMETERS CASE B
TAB LE I X
EQUIVALENT CIRCUIT PARAMETERS CASE C
proposed analytical methods. The difference in the magnetizing
inductances is below 2% in all the cases, which confirms that the
proposed method to calculate the Carter factor used to obtain an
effective air gap length leads to an acceptable accuracy.
The difference in Lrwhich represents the stator and rotor
leakage inductances is around 4% to 5%, well within the limi-
tations of calculating leakage fluxes. The prediction of leakage
inductance by the coupled circuit model in all three cases is
always higher than the numerical model, showing that the ana-
lytical method overestimates the leakage inductances.
An important conclusion from the results shown in Tables VII,
VIII, and IX is that the differences in the parameters calculated
from the analytical and numerical methods are small and, for
all the three cases, are in a similar range; this confirms the ac-
curacy of the analytical methods with respect to the dimensions
and the magnetic property of magnetic wedges. Thus, the ap-
proach can be utilized during the BDFM design optimization
to specify suitable magnetic wedges for stator slot openings in
order to achieve an appropriate balance between the magnetiza-
tion requirements, reactive power management, and low-voltage
ride-through performance of the machine.
X. CONCLUSION
In this paper, the equivalent circuit parameters have been
derived for a 250 kW D400 BDFM using several methods in-
cluding the analytical and numerical models and from the exper-
imental tests. One key feature of this BDFM as compared to the
ones studied before is the use of magnetic wedges in stator slot
openings, which affect the magnetizing properties of the ma-
chine as well as reactive power management and performance
during grid low voltage faults. The use of magnetic wedges in
large electrical machines is common and is expected to be uti-
lized in large BDFMs, and hence, the approach of this paper is
important.
It was shown that the existing analytical methods are not able
to give acceptable accuracy in calculating magnetizing and leak-
age inductances for the BDFM with magnetic wedges used in
stator slots. Hence, analytical methods have been proposed to
take into account the effects of magnetic wedges. A modified
Carter factor has been derived which is used to calculate the
effective air gap length to obtain the magnetizing inductances in
the BDFM. In addition, a method to calculate slot leakages has
been presented which calculates the increase in stator leakage
inductances due to the presence of magnetic wedges. The analyt-
ical methods have been used to predict the parameter values for
the D400 BDFM and two other design variations with magnetic
wedges of different dimensions and magnetic property, and the
results have been compared to the predictions from numerical
models. It has been shown that the analytical methods lead to
a good agreement with the numerical models and hence have
sufficient accuracy to be used in the BDFM design optimization.
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Salman Abdi received the B.Sc. degree from Fer-
dowsi University, Mashhad, Iran, in 2009, and the
M.Sc. degree from Sharif University of Technology,
Tehran, Iran, in 2011. He is currently working to-
ward the PhD degree in electrical machines design
and modeling at Cambridge University, Cambridge,
U.K.
His main research interests include electrical ma-
chines and drives and wind power generation.
Ehsan Abdi (SM’12) received the B.Sc. degree from
Sharif University of Technology, Tehran, Iran, in
2002, and the M.Phil. and Ph.D. degrees from Cam-
bridge University, Cambridge, U.K., in 2003 and
2006 respectively, all in Electrical Engineering.
He is currently the Managing Director of Wind
Technologies Ltd., where he has been involved with
commercial exploitation of the brushless doubly fed
induction generator technology for wind power appli-
cations. His main research interests include electrical
machines and drives, renewable power generation,
and electrical measurements and instrumentation.
Ashknaz Oraee received the B.Eng. degree from
Kings College London, London, U.K., in 2011. She
is currently working toward the Ph.D. degree in elec-
trical machine design and optimization at Cambridge
University, Cambridge, U.K.
Her research interests include electrical machines
and drives for renewable power generation.
Richard McMahon received the BA degree in elec-
trical sciences and the Ph.D. degree from Cambridge
University, Cambridge, U.K., in 1976 and 1980,
respectively.
Following postdoctoral work on semiconductor
device processing, he became a University Lecturer
in electrical engineering at the Engineering Depart-
ment, Cambridge University, in 1989, and a Senior
Lecturer in 2000. His research interests include elec-
trical drives, power electronics, and semiconductor
materials.
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