COMPUMAG-FLORIANOPOLIS 2009, 2. QUASI-STATIC FIELDS, C. TIME AND FREQUENCY DOMAIN
1
Homogenization of Form-Wound Windings in Frequency and Time Domain Finite
Element Modelling of Electrical Machines
Johan Gyselinck 1 , Patrick Dular
1
2,3 ,
Nelson Sadowski 4 , Patrick Kuo-Peng 4 , Ruth V. Sabariego
2
Dept. of Bio-, Electro- and Mechanical Systems (BEAMS), Université Libre de Bruxelles (ULB), Belgium
2
Dept. of Electrical Engineering and Computer Science (ACE), University of Liège, Belgium
3
Fonds de la Recherche Scientifique, F.R.S. – FNRS, Belgium
4
GRUCAD/EEL/CTC, Universidade Federal de Santa Catarina, Brazil
In this paper the authors deal with the FE modelling of eddy-current effects in form-wound windings of electrical machines
using a previously proposed general frequency and time domain homogenization method. By way of demonstration and validation,
a real-life 1250 kW induction machine with double-layer stator winding is considered. The skin and proximity effects in one stator
conductor (copper bar) are first quantified by means of a simple low-cost FE model, leading to complex and frequency-dependent
coefficients for the homogenized winding (reluctivity for proximity effect and conductivity or resistance for skin effect). These complex
coefficients are subsequently translated into real-valued and constant coefficients that allow for time-domain homogenization when
introducing a limited numer of additional degrees of freedom in the FE model. All results obtained with the homogenized model
(considering one conductor or a complete slot) agree well with those produced by a brute-force approach (modelling and finely
discretizing each conductor).
Index Terms— Finite element methods, magnetic fields, time and frequency domain, homogenization, electrical machine, windings
I. I NTRODUCTION
Multi-turn windings in electromagnetic devices may be
subjected to considerable skin and proximity effects, leading
to higher losses and hot spots, and possibly affecting the
global characteristics of the device. In principle these effects
can be taken into account in a FE simulation of the device
by modelling and finely discretising each separate conductor
(with additional electrical circuit equations to connect the socalled massive conductors [1]). For most real-life applications
the huge computational cost of such a brute-force approach
cannot be justified. Most often the eddy-current effects are thus
simply ignored in the resolution stage of the FE simulation
(considering winding regions with uniform current density, socalled stranded conductors [1]). In this case the eddy-current
losses may be estimated a posteriori.
Frequency-domain homogenization methods for windings
have recently been proposed in [2][3]. For the winding type in
hand (e.g., round wires with hexagonal-like packing or rectangular conductors with rectangular packing), an elementary FE
model is used for determining frequency-domain coefficients
regarding skin and proximity effect [3]. These frequencydependent coefficients can next be straightforwardly translated
into constant time-domain coefficients and associated differential equations thanks to the introduction of a limited number
of additional unknowns for the homogenized winding (current
components for the skin effect, and induction components for
the proximity effect) [4]. The number of additional unknowns
required and the additional computational cost depend on the
frequency content of the application and in particular on the
conductor size (e.g. radius in case of conductors of circular
cross-section) to skin depth ratio (considering the highest
relevant frequency).
So far the abovementioned frequency and time domain homogenization methods have been mainly applied to inductorlike devices, having a multi-turn multi-layer winding, and in
which a 2-D flux pattern produces the dominant proximity
effect [3], [4]. In this paper we consider a 1250 kW induction
machine and in particular its form-wound stator winding
Manuscript received December 23, 2009. Corresponding author: J. Gyselinck (e-mail: johan.gyselinck@ulb.ac.be).
[5][6]. Compared to previous applications, the flux situation
is a priori simpler, the slot leakage flux being essentially
governed by a 1-D differential equation and allowing for an
approximate analytical approach.
However, when conductors are connected in parallel, with
furthermore transposition from one slot to the next, as it is
the case in the considered induction machine, the homogenization of the winding, following a general approach, is less
straightforward.
We"#!therefore
in this
on the
time$,!and
frequency
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series connection of all conductors. This allows to consider only one conductor (for determining the homogenization
coefficients) and subsequently one slot. A brief discussion
on the possible parallel connection and transposition of the
conductors follows.
II. G EOMETRY OF STATOR SLOTS AND CONDUCTORS
Figure 1 shows the geometry of the 50 Hz 1250 kW threephase six-pole squirrel-cage induction machine [5]. The twolayer stator winding of the machine is distributed in 72
rectangular and fully-open slots (slot width ws = 14 mm, total
slot height 80 mm), each slot comprising 2 × 9 copper bars
of rectangular cross-section (conductor height hc =! 3.3 mm,
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conductor
width wc = 10.6 mm) [5].
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2D FE model of 1250 kW induction motor (half cross-section) [6]
!
4.1.2 Voltage supply
The2+'!vertical
insulation
space
two conductors
of
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are separated by 4.9 mm (see Fig. 2). The
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lamination stack length is l = 810 mm.
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COMPUMAG-FLORIANOPOLIS
QUASI-STATIC
C. TIME
AND
FREQUENCY
DOMAIN
COMPUMAG-FLORIANOPOLIS
2009,
2. QUASI-STATIC
FIELDS,
C. TIME
AND
FREQUENCY
DOMAIN
lamination
stack length 2009,
is l 2.=
810
mm. FIELDS,
50 Hz, in phase
of conductors are separated by 4.9 mm (see Fig. 2). The
lamination stack length is l = 810 mm.
50 Hz, in phase
50 Hz, in quadrature
2 2
10 kHz, in phase
10 kHz, in phase
10 kHz, in quadrature
Fig. 3. Skin-effect flux pattern at 50 Hz and 10 kHz, with flux component
in phase
andininquadrature
quadrature with the imposed
net in
current
50 Hz,
10 kHz,
quadrature
Fig.Fig.
3. 3. Skin-effect
and 10
10kHz,
kHz,with
withflux
fluxcomponent
component
Skin-effectflux
fluxpattern
pattern at
at 50 Hz and
phase
andininquadrature
quadraturewith
with the
the imposed
imposed net
in in
phase
and
net current
current
RAC/RDC
RAC/RDC
a = 0 on the complete boundary (effecting bav = 0). See the
flux patterns in Fig. 3.
a
= 0joule
on thelosses
complete
(effecting
0). See the I)
av =amplitude
The
at boundary
frequency
f (current
The
joule
losses
at frequency
f b(current
amplitude I)
fluxunder
patterns
in conditions
Fig. 3.
and
DC
(constant
current
I
and
and under DC conditions (constant current I uniform
and uniform
Thedensity,
joule losses
frequency
fcan(current
amplitude
current
i.e. i.e.
noatskin
effect)
be
expressed
as PI)as
=
current
density,
no
skin
effect)
can
be
expressed
2
2
andI under
DCP conditions
current Fig.
I and
uniformthe P =
RAC
/2
=
R
I (constant
, Irespectively.
4 shows
2 and
2
DC
R
I
/2
and
P
=
R
,
respectively.
Fig.
4
shows
AC density, i.e. no skinDC
current
effect)
be expressed
P =the the
ratio
RAC
/R
as a function
of can
hc /δ,
obtained as
with
fine model
homogenized model
2RAC DC
2
ratio
/R
as
a
function
of
h
/δ,
obtained
with the
cFig. 4 shows the
RAC I /2 FE
and model.
PDC= RThis
, respectively.
DC I figure
elementary
also shows
the analytical
fine model
homogenized
model
Fig. 2.
2. FE
FE mesh
andand
in in
quadrature,
elementary
FE
model.
This
figure
also
shows
the
analytical
ratio
RAC /RDC
as
a
function
of
h
/δ,
obtained
with
the
Fig.
mesh and
and flux
fluxlines
lines(flux
(fluxcomponent
componentininphase
phase
quadrature, approximation
is based on thec 1D diffusion problem
respectively, with
same
imposed 5050Hz
allall
1818
conductors),
finefine elementary FEwhich
respectively,
withFEthe
the
same
Hzcurrent
currentin in
conductors),
model.
figure also
shows
analyticalproblem
approximation
whichThis
is net
based
onin the
1Dthediffusion
Fig. 2.
mesh
andimposed
flux lines (flux
component
in phase
and in quadrature,
model (for
brute-force
approach)
and homogenized
model
(net
current,
or
alternatively
flux,
a conducting
sheet of
model (for
brute-force
and homogenized
respectively,
withapproach)
the same imposed
50 Hz currentmodel
in all 18 conductors), fine
approximation
which
is based on
the
1D in
diffusion
problemsheet of
(net
current,
or
alternatively
net
flux,
a
conducting
thickness
h
)
[7]:
c
model (for brute-force approach) and homogenized model
(net
current,hor) alternatively
net flux, in a conducting sheet of
thickness
c [7]: RAC
= <(Y ) ,
(3)
Flux and current harmonics in the machine are due to the thickness hc ) [7]:
R
AC
RRDC
Fluxcurrent
and current
harmonics
in machine
the machine
Flux and
harmonics
in the
are are
duedue
to to
thethe
AC
=
�(Y
)
,
(3)
stator and
rotor
slotting,
saturation
andand
PWM
supply
(at 22 kHz
= �(Y ) ,
(3)
RYDC
stator
andslotting,
rotor slotting,
saturation
PWM
supply
kHz where the complex number
stator and
rotor
saturation
and PWM
supply
(at (at
2 kHz
R
is
a
function
of
h
/δ:
DC
c
switching
frequency)
[6]. [6].
switching
frequency)
wherethe
thecomplex
complex
number
is a function
of hc /δ:
switching
frequency)
[6].
where
number
Y isYa function
c /δ:
� ı ofh h
7 7
Taking
as
conductivity
· 10
S/m,
thethe
skin
depth
δδ =
11++1ıı+hhcı hc
� 11++
Taking
as
conductivity
67·S/m,
10
S/m,
skin
depth
=
c�ı hc �
the
skin
depth
δ=
�
1
+
� as conductivity σσ==6σ6·=10
ı
h
pTaking
c cotanh
YY Y
== =
cotanh2 cδ . .
. (4)(4) (4)
2/(ωµσ),
at pulsation
pulsation
ω,varies
varies
between
9.2
Hz)
2/(ωµσ),
at pulsation
ω, varies
between
9.2mm
mm
(at
50
2/(ωµσ),
at
ω,
between
9.2
mm
(at(at
5050
Hz)
22 2 δδ cotanh
2 δ2
δ
δ
andmm
1.45
(at 2 kHz),
and
the
conductor
height
toskin
skin
depth
and
mm
(atmm
kHz),
andthe
theconductor
conductor
height
depth
and 1.45
(at
22kHz),
and
height
to to
skin
depth
The
analytical
solution
is
validup
upto,to,up
say,
The
analytical
solution
is apparently
apparently
valid
say,
hch/δ
c /δ hc /δ
The
analytical
solution
is apparently
valid
to,
say,
hbetween
0.35
and
2.27.
c /δ, between
ratio,
0.35and
and2.27.
2.27.
ratio, hratio,
0.35
c /δ, between
equal
to
2.
The
relative
increase
in
joule
losses
due
to
skin
equal
to
2.
The
relative
increase
in
joule
losses
due
to
skin
equal to 2. The relative increase in joule losses due to skin
effect,
i.e.
thethe
ratio
/RDC
minus
effect,
i.e.
the
ratio
RRAC
minus 1,1, isis found
foundtotovary
vary
AC
effect,
i.e.
ratio
R/R
AC /RDC minus 1, is found to vary
between0.01%
0.01%atat 50
50Hz
Hz (h
(h
=
0.35)
and
12%
at
2
kHz
III. AND
S KINPROXIMITY
AND PROXIMITY
EFFECT
IN ONE
CONDUCTOR between
c
/δ
=
0.35)
and
12%
at
2
III.
S
KIN
EFFECT
IN
ONE
CONDUCTOR
between
0.01% at 50 Hzc (hc /δ = 0.35) and 12% kHz
at 2 kHz
III. S KIN AND PROXIMITY EFFECT IN ONE CONDUCTOR
=2.27).
2.27).
c /δ
(h(h
/δ
=
c(h
A. Elementary
FE model
and frequency-domain
calculations
A. Elementary
FE model
and frequency-domain
calculations
c /δ = 2.27).
A. Elementary FE model and frequency-domain calculations
We consider
an elementary
FE model
comprising
oneone
copWe consider
an elementary
FE model
comprising
cop2.8
an as
elementary
FE
model
comprising
one copper
bar (modelled
as a massive
conductor)
insulating
perWe
barconsider
(modelled
a massive
conductor)
and and
the the
insulating
analytical
2.8
analytical
FE model
2.4
space
around
it.
calculations
are
carried
space
it. Frequency-domain
calculations
arethe
carried
outout
per
bararound
(modelled
as Frequency-domain
a massive conductor)
and
insulating
FE model
2.4
in
terms
of
the
complex
single-component
magnetic
vector
in terms
of the
complex single-component
magnetic
vectorout
space
around
it. Frequency-domain
calculations
are carried
2
potential
(in bold),
with
adequate
current
boundary
potential
a (in
withsingle-component
adequate
current
andand
boundary
2
in
terms
of
theabold),
complex
magnetic
vector
1.6
conditions
[1].
conditions
[1].
potential a (in bold), with adequate current and boundary
1.6
The
complex
(in volt-ampères)
is calculated
from
1.2
The complex
powerpower
S (inSvolt-ampères)
is calculated
from
conditions
[1].
the
local
flux
density
b
and
the
local
current
density
j:
1.2
theThe
local
flux density
and(inthevolt-ampères)
local
density
j: from
� current is
0
1
2
3
4
5
complex
powerb S
calculated
Z l
2
2
= P +b ıand
Ql =the local
(j /σ
+ ı ω density
ν20 b ) dΩj:
,
(1) Fig. 4. Ratio R0AC /RDC as
1 a functionh2cof/δhc /δ, obtained
3
4 and
5
the local fluxSdensity
current
analytically
2
�2(j Ω/σ + ı ω ν0 b ) dΩ ,
S = P + ıQ =
(1)
with
the
elementary
FE
model
h
/!
2l and reactive
with
a Fig. 4. Ratio RAC /RDC as a function of hcc/δ, obtained analytically and
S =P and
P +Qı the
Q active
= Ω
(j 2 /σ + power
ı ω ν0 (averaged
b2 )2dΩ , over
the 4.
elementary
Ratio RFE
/RDC as a function of hc /δ, obtained analytically and
∗(1) withFig.
ACmodel
fundamental
period),
ı
the
imaginary
unit,
and
j
/2
=
jj
/2
2
Ω
with P and 2Q the active
and reactive
power (averaged over a
with the elementary FE model
∗
/2
= bbactive
/2
r.m.s.-values
squared.
∗
with
Pand
andb Q
the
and
reactive
power
fundamental
period),
ı the
imaginary
unit,
and (averaged
j 2 /2 = jjover
/2 a
2 the average
∗
2
∗
Equation
(1)
can
be
rewritten
considering
fundamental
period),
ı
the
imaginary
unit,
and
j
/2
=
jj
/2 C. Proximity effect in frequency domain
and b /2 = bb /2 r.m.s.-values squared.
j
and
flux
density
b
(averaged
over
2 current density
∗
av
av
effect in frequency
domain
(1) /2
canr.m.s.-values
be rewritten
considering the averagethe C. Proximity
andEquation
b /2 = bb
squared.
A pure proximity-effect
excitation
is obtained by imposing a
complete elementary model, i.e. copper plus insulation):
�
current
density
j
and
flux
density
b
(averaged
over
the
A
pure
proximity-effect
excitation
by and
imposing
unit
horizontal
flux
(with
a
=
0
and
=obtained
1 on lower
upper a
av
av
Equation (1) can l be rewritten considering the average C. Proximity effect in frequencyaisdomain
2 i.e. copper plus insulation):
2
completedensity
elementary
model,
boundaries,
respectively,
and
the
implicit
Neumann
condition
unit
horizontal
flux
(with
a
=
0
and
a
=
1
on
lower
and
upper
S jav
= and
(jav
/σ
ν
b
)
dΩ
,
(2)
skin + ıbω
prox
current
flux
density
(averaged
over
the
av
av
Z 2 Ω
A pure
proximity-effect
excitation
isNeumann
obtained
by
imposing a
∂a/∂n
= 0respectively,
on left and right
slot
walls) and
zero net condition
current
boundaries,
and
the
implicit
complete
elementary
model,
i.e.
copper
plus
insulation):
l define
and
complex
skin-effect
σ
� 2 the
(I
= 0).
in=Fig.
5. aand
skin ∂a/∂n
unit
horizontal
flux
(with
0walls)
and
= 1zero
on lower
and upper
= 0See
onthe
leftflux
andpatterns
right aslot
net current
S we
= thus
/σskin
+ ıω
νprox b2avconductivity
) dΩ ,
(2)
l Ω (jav
and the complex
proximity-effect
reluctivity2 νprox .
2
2
proximity-effect
νprox
can also be
the 5.
implicit
Neumann
condition
=The
0). complex
See therespectively,
flux
patternsand
in reluctivity
Fig.
S =
(jav /σskin + ı ω νprox bav ) dΩ ,
(2) (I boundaries,
2 the
estimated
basisand
of right
the same
analytical
Ω complex skin-effect conductivity σ
∂a/∂n
=on0 the
on
left
slot well-known
walls)νprox
and can
zeroalso
net be
current
The
complex
proximity-effect
reluctivity
and we thus define
skin
solution
of
1D
and
the
and
the complex skin-effect
σskin estimated
onSee
thethe
of
the problem
same
well-known
analytical
(I = 0).(4)
thebasis
fluxdiffusion
patterns
in Fig.
5. considering
and we
theB.thus
complex
proximity-effect
reluctivity conductivity
νprox .
Skindefine
effect
flux tubes
connected
indiffusion
series (left
and right)
in parallel
and the complex proximity-effect reluctivity νprox .
solution
(4)
of the 1D
problem
and and
considering
The
complex
proximity-effect
reluctivity
νprox canthe
also be
(uptubes
and down)
with
the
copper
bar:
Following the approach developed in [3], [4], a pure skin- flux
connected
inbasis
seriesof(left
and
right)
and
in parallel
�
�
�
�
estimated
on
the
the
same
well-known
analytical
−1
−1
w
w
−
w
h
s
c
i
B. Skineffect
effectexcitation is obtained by imposing a sinusoidal current (up and
down)
thec copper
νprox
=
Y diffusion
+ bar: problem
+ and considering
. (5)
0 the 1D
(4) νwith
of
the
unit amplitude, e.g., I = 1) in the bar, with condition solution
hc
hc
w
s
B. Following
Skin(ofeffect
the approach developed in [3], [4], a pure skin−1 right)
flux tubes connected
and in parallel
wc in series
ws −(left
wc and
hi −1
νprox
ν0 with the
Y +copper bar: +
. (5)
effect excitation is obtained by imposing a sinusoidal current
and=down)
Following the approach developed in [3], [4], a pure skin- (up
hc
hc
ws
(of unit amplitude, e.g., I = 1) in the bar, with condition
�
�
�
wc
ws − wc −1 reluctivity
hi �−1
effect
is obtained
by imposing
sinusoidal
a = 0 excitation
on the complete
boundary
(effecting abav
= 0). Seecurrent
the
Fig.νprox
6 shows
the relative
proximity-effect
=
ν
Y
+
+
. (5)
0
(of
amplitude,
e.g., I = 1) in the bar, with condition ν
hc hc /δ, obtained
hc
s
fluxunit
patterns
in Fig. 3.
in threewdifferent
prox /ν0 as a function of
COMPUMAG-FLORIANOPOLIS 2009, 2. QUASI-STATIC FIELDS, C. TIME AND FREQUENCY DOMAIN
COMPUMAG-FLORIANOPOLIS 2009, 2. QUASI-STATIC FIELDS, C. TIME AND FREQUENCY DOMAIN
3
3
Figure 7 shows that with n = 2 an excellent agreement is
50 Hz, in phase
10 kHz, in phase
Figure 7 shows that with n = 2 an excellent agreement is
obtained
up hto/δhc=/δ4.=The
4. approximation
The approximation
obtained
up to
with nwith
= 1 nis = 1 is
c
valid
up
to,
say,
h
/δ
=
1.
c 1.
valid up to, say, hc /δ =
2.0
50 Hz, in quadrature
reference
n=1
n=2
1.5
10 kHz, in quadrature
νprox/ν0
Fig.Fig.5. 5. Proximity-effect
at 50
50Hz
Hz and
and1010
kHz,with
withfluxflux
Proximity-effect flux
flux pattern
pattern at
kHz,
component
phaseand
andininquadrature
quadrature with
component
in inphase
with the
the imposed
imposedflux
flux
real part
1.0
0.5
shows the the
relative
proximity-effect
reluctivity
ways:Fig.
first6 considering
complex
number Y (i.e.
without
imaginary part
/ν0 as a function
of hc /δ,
obtained
in three
different
anyνprox
consideration
for the space
around
the bar),
second
adopt0.0
first considering the complex number Y (i.e. without
0
0.5
1
1.5
2
2.5
3
3.5
4
ingways:
(5), and
third using the elementary FE model. Clearly, the
any consideration for the space around the bar), second adopthc/δ
correction
for
the
insulation
on
all
four
sides
of
the
conductor
ing (5), and third using the elementary FE model. Clearly, the Fig. 7. Real and imaginary part of the relative permeability νprox /ν0 versus
is correction
required for
a goodonaccuracy
in theofconsidered
hc /δ hc /δ, obtained directly in the frequency domain (reference) and indirectly in
forgetting
the insulation
all four sides
the conductor
Fig. 7. Real and imaginary
part of the relative permeability νprox /ν0 versus
(1) and [P (2) ])
range.
(with
fitted [P
is required for getting a good accuracy in the considered hc /δ the htime
/δ,domain
obtained
directly
in the] frequency
domain (reference) and indirectly in
c
range.
the time domain (with fitted [P (1) ] and [P (2) ])
The integration of the additional flux density components
in theThe
FE equations
is developed
in [4]. flux density components
integration
of the additional
relative reluctivity
1.4
1.2
1
real part
in the FE equations is developed in [4].
0.8
0.6
IV. H OMOGENIZATION OF A COMPLETE SLOT
Y
analytical
FE model
H OMOGENIZATION
OF Adomain
COMPLETE
SLOT
We nowIV.
carry
out time and frequency
calculations
using We
a FE
complete
slot, with domain
either a calculations
fine
nowmodel
carry ofouta time
and frequency
discretisation
of each
(18 massive
or a fine
0
using a FE
modelconductor
of a complete
slot,conductors)
with either
0
0.5
1
1.5
2
2.5
3
homogenization
of
the
two
groups
of
nine
conductors
each
discretisation of each conductor (18 massive conductors) or
hc/δ
(two stranded conductors). The two meshes used (totaling
homogenization of the two groups of nine conductors each
Fig.
6.
Real
and
imaginary
part
of
the
relative
permeability
ν
/ν
versus
prox
0
Fig. 6. Real and imaginary part of the relative permeability νprox /ν0 versus 3618 first-order triangular elements versus 74) are represented
(two stranded conductors). The two meshes used (totaling
hc /δ, obtained with the elementary FE model and using analytical formulas
hc /δ, obtained with the elementary FE model and using analytical formulas in Fig. 2. All nine conductors in a group are assumed con3618infirst-order
triangular elements versus 74) are represented
nected
series.
0.4
0.2
D. Proximity effect in time domain
imaginary part
in Fig. 2. All nine conductors in a group are assumed connected in series.
A. Frequency domain
D. Proximity effect in time domain
The frequency-dependent reluctivity νprox (hc /δ) can be
complex reluctivity
The frequency-dependent
(hc /δ)
can be The
A. Frequency
domain νprox is adopted in the two homogtranslated
in an equivalent reluctivity
time-domainνprox
relation
between
enized winding regions, together with an imposed uniform
translated
in
an
equivalent
time-domain
relation
between
the instantaneous magnetic field h(t) and the magnetic flux current
The
complex
is adopted
in the
twobehomogproxresistance
density.
Thereluctivity
equivalentνAC
RAC
is to
thedensity
instantaneous
and the magnetic
flux enized
b(t) by magnetic
consideringfield
n −h(t)
1 auxiliary
flux density
regions,equations
together(ifwith
used in thewinding
electrical-circuit
any).an imposed uniform
density
b(t) by
considering
− [4].
1 auxiliary
components
b2 (t),
b3 (t), . . . , bnn (t)
A system flux
of n density
firstcurrent
Theconsidered,
equivalentthe
AC
AC is to be
Two
casesdensity.
are further
tworesistance
conductorR
groups
components
b2 (t), equations
b3 (t), . . . ,can
bn (t)
system
of n firstorder differential
then[4].
be A
written
in terms
of
used
in
the
electrical-circuit
equations
(if
any).
belonging
either
to
the
same
phase
or
to
different
phases,
T
the column
matrices
[B(t)]Tcan
= then
[b(t) be
b2 (t)
b3 (t) in
. . .]terms
and of
order
differential
equations
written
Twoeither
cases zero
are further
twobetween
conductor
and with
or 120 considered,
degree phasethe
shift
the groups
T
= matrices
[h(t) 0 0 [B(t)]
. . .]T : T = [b(t) b2 (t) b3 (t) . . .]T and belonging either to the same phase or to different phases,
the[H(t)]
column
respective
currents
(of
same
unit
amplitude,
I
=
1).
Some
flux
�
�
[H(t)]T = [h(t) 0 0 . . .]T : σµ0 h2c (n) d
and with
zero or belonging
120 degree
phase
between
the
(witheither
all conductors
to the
sameshift
phase)
are
[H(t)] = ν0 [B(t)] +
[P ] [B(t)] , (6) patterns
4 2
dt
depicted
in Fig.
2.
respective
currents
(of
same
unit
amplitude,
I
=
1).
Some
flux
σµ0 hc (n) d
[H(t)][P (n)
= ] νis0 a [B(t)]
+ dimensionless,
[P ] symmetric
[B(t)] , and(6) Using
patterns
conductors
belonging
to the
are
(1) (with
and (2)allwe
obtain the complex
power
andsame
thus phase)
the
where
real-valued,
4
dt
equivalent
depictedAC
in resistance
Fig. 2. RAC = P/(2I 2 ) and AC inductance
tridiagonal matrix.
2
LAC Using
= Q/(2ωI
) at(2)a we
given
pulsation
ω. Fig. power
8 showsand
thethus the
(1) and
obtain
the complex
In [P
the(n)
frequency
domain (6) dimensionless,
becomes
where
] is a real-valued,
symmetric
and
2 of hc /δ. There
�
�
ratios
RAC /RDC
and
LAC /LDC
as
a
function
equivalent
AC
resistance
R
=
P/(2I
)
and
AC
inductance
2
AC
h
tridiagonal matrix.
2
excellent
agreement
results obtained
the
[H] = ν0 [1] + ı σ c [P (n) ] [B] ,
(7) is an
LAC
= Q/(2ωI
) atbetween
a giventhepulsation
ω. Fig.with
8 shows
the
In the frequency domain
(6) becomes
2
brute-force
approach
and
with
the
homogenized
model.
ratios
R
/R
and
L
/L
as
a
function
of
h
/δ.
There
AC
DC
AC
DC
c
matrix, from
the approximate
with [1] the n×n identity
For
one-phase
case, e.g.,
at 50 the
Hz results
the resistance
is anthe
excellent
agreement
between
obtainedin-with the
h2c which
(n)readily calculated.
complex reluctivity
by 34%,approach
and at 2 kHz
by
a factor
of 300. Clearly,
the
,n (h
[H] = νν0prox[1]
+cı/δ)
σ can
[Pbe
] [B] ,
(7) creases
brute-force
and
with
the
homogenized
model.
By properly fitting the elements of2[P (n) ], with a sufficiently skin-effect losses are negligible compared to the proximityFor the one-phase case, e.g., at 50 Hz the resistance ingreat value for n, a good agreement between νprox ,n (hc /δ) effect losses (considering the 0.01% and 12% increases menwith
[1]
the
n×n
identity
matrix,
from
which
the
approximate
creases
by 34%, and at 2 kHz by a factor of 300. Clearly, the
and ν
(hc /δ) can be obtained up to a preset value of hc /δ. tioned above).
complexprox
reluctivity
νprox ,n (hc /δ) can be readily calculated. skin-effect losses are negligible compared to the proximityWith n = 3 the following values are obtained through
losses (considering the 0.01% and 12% increases menByfitting:
properly fitting the elements of [P (n) ], with a sufficiently B. effect
Time domain
great value for n, agood agreement between νprox ,n (hc /δ) tioned above).
0.0747
0
and νprox (hc /δ)
can be0.1085
obtained
up to a preset
value of hc /δ. The current waveform considered for the time-domain cal[P (3) ] =  0.0747 0.0596 0.0103  ,
(8) culations (current imposed in all 18 conductors) is depicted
With n = 3 the following values are obtained through
in B.
Figure
The main frequency components are the 50 Hz
Time9. domain
0
0.0103
0.0166
fitting:
The current waveform considered for the time-domain cal

0.1085 0.0747
0
culations (current imposed in all 18 conductors) is depicted
[P (3) ] =  0.0747 0.0596 0.0103  ,
(8) in Figure 9. The main frequency components are the 50 Hz
fundamental (amplitude 313.2 A) and the 1850 Hz and 2050 Hz
0
0.0103 0.0166
1.2
1
0.8
0.6
0.4
0.2
brute force
homogenized
two phases
one phase
1000
two phases
0
0.5
1
1.5
hc/δ
2
2.5
3
400
200
current (A)
10
1
1
LAC/LDC
PWM
fundamental
300
all conductors in series
ABC... ABC...
ABC... CBA...
100
Fig. 8. Relative AC resistance and inductance (with DC values as reference)
versus hc /δ for a complete slot, obtained with reference FE model (brute
force approach) and with homogenized FE model
100
0
0.1
-100
0.1
-200
1
hc/δ
-300
Fig. 11. Effect on resistance and inductance of the three parallel paths in a
single slot (considering the transposition among the two groups or not)
-400
0
Fig. 9.
4
Figure 11 shows the effect of the parallel paths inside a
single slot (one-phase case), disregarding the transposition
from one slot to the next. However, transposition inside the two
halfs of one slot is possible and clearly reduces the circulating
currents (ABC/CAB versus ABC/ABC).
one phase
RAC/RDC
RAC/RDC
500
400
300
200
100
0
LAC/LDC
COMPUMAG-FLORIANOPOLIS 2009, 2. QUASI-STATIC FIELDS, C. TIME AND FREQUENCY DOMAIN
5
10
time (ms)
15
20
PWM waveform of the imposed 50 Hz current
switching harmonics (19.5 A et 17.6 A, respectively). This
current waveform is similar to the currents shown in [5][6].
Figure 10 depicts the instantaneous joule losses in the
slot (18 conductors) during 3 ms (i.e. approximately 6 PWM
switching periods). The homogenization with n = 1 is clearly
much less accurate than the one with n = 2.
fine model
homogenized n=1
homogenized n=2
joule losses (kW)
8
VI. C ONCLUSION
Frequency and time domain homogenization methods have
been successfully applied to the form-wound winding of an
induction machine, hereby assuming a plain series connection
of the copper bars and disregarding their actual transposition in
the slots. Thanks to the transposition, circulating currents are
very small, so that the simplified analysis carried out is also
relevant for the real winding (with transposition). In case of
PWM supply, the homogenization method with one additional
degree of freedom (i.e. n = 2) produces accurate results.
6
ACKNOWLEDGMENT
4
This work was partly supported by the Belgian Science Policy (IAP P6/21), the Belgian F.R.S. – FNRS and the Brazilian
CNPq.
2
0
9
9.5
10
10.5
time (ms)
11
11.5
12
Fig. 10. Instantaneous joule losses in slot calculated with fine FE model and
homogenized FE model
V. PARALLEL CONNECTION AND TRANSPOSITION
The stator winding of the considered machine comprises
three parallel branches [6]. This means that a group of nine
conductors in a slot constitutes three effective turns with
another group of nine conductors in another slot (5/6 of a
pole pitch away in this case). As the machine has four slots
per pole and per phase, the conductors are transposed in order
to reduce effectively the unequal distribution of the current
among the three parallel paths. FE simulation of the complete
machine shows that circulating circuits are practically zero
thanks to the transposition [6]. This means that each of the
nine bars in a group carries the same current, as assumed in
the above sections.
R EFERENCES
[1] P. Lombard and G. Meunier, “A general purpose method for electric
and magnetic combined problems for 2D, axisymmetric and transient
systems”, IEEE Trans. on Magn., vol. 29, pp. 1737–1740, March 1993.
[2] A. Podoltsev, I. Kucheryavaya, and B. Lebedev, “Analysis of effective
resistance and eddy-current losses in multiturn winding of high-frequency
magnetic components,” IEEE Trans. on Magn., vol. 39, pp. 539–548,
January 2003.
[3] J. Gyselinck and P. Dular, “Frequency-domain homogenization of bundles
of wires in 2D magnetodynamic FE calculations,” IEEE Trans. on Magn.,
vol. 41, pp. 1416–1419, April 2005.
[4] J. Gyselinck, R. Sabariego, and P. Dular, “Time-domain homogenisation
of windings in two-dimensional finite element models”, IEEE Trans. on
Magn., vol. 43, pp. 1297–1300, 2007.
[5] M. Islam and A. Arkkio, “Effects of pulse-width-modulated supply
voltage on eddy currents in the form-wound stator winding of a cage
induction motor,” IET Electr. Power Appl., vol. 3, pp. 50–58, 2009.
[6] M. Islam, “Finite-element analysis of eddy currents in the form-wound
multi-conductor windings of electrical machines,” PhD thesis, Helsinki
University of Technology, to be published in 2010.
[7] J. Lammeraner and M. Stafl, “Eddy currents,” London, U.K.: ILIFFE
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