energies
Article
Limitations and Constraints of Eddy-Current Loss
Models for Interior Permanent-Magnet Motors with
Fractional-Slot Concentrated Windings
Hui Zhang * and Oskar Wallmark
Department of Electric Power and Energy Systems, KTH Royal Institute of Technology, SE-100 44 Stockholm,
Sweden; [email protected]
* Correspondence: [email protected]; Tel.: +46-7-671-32-355
Academic Editor: K.T. Chau
Received: 2 January 2017; Accepted: 11 March 2017; Published: 16 March 2017
Abstract: This paper analyzes and compares models for predicting average magnet losses in interior
permanent-magnet motors with fractional-slot concentrated windings due to harmonics in the
armature reaction (assuming sinusoidal phase currents). Particularly, loss models adopting different
formulations and solutions to the Helmholtz equation to solve for the eddy currents are compared
to a simpler model relying on an assumed eddy-current distribution. Boundaries in terms of
magnet dimensions and angular frequency are identified (numerically and using an identified
approximate analytical expression) to aid the machine designer whether the more simple loss model
is applicable or not. The assumption of a uniform flux-density variation (used in the loss models) is
also investigated for the case of V-shaped and straight interior permanent magnets. Finally, predicted
volumetric loss densities are exemplified for combinations of slot and pole numbers common in
automotive applications.
Keywords: automotive applications; concentrated windings; eddy current losses; fractional-slot
windings; interior permanent-magnet motors
1. Introduction
In interior permanent-magnet motors (IPMs), the permanent magnets (PMs) are embedded into
the rotor. Compared to rotors with surface-mounted PMs, the resulting flux concentration effect
in IPMs significantly reduces the induced PM eddy-current losses caused by the changing air-gap
permeance due to the slot openings. Further, a not insignificant magnetic saliency can be realized,
which contributes with a reluctance torque component and thereby increases the torque density.
These are two main reasons why the IPM is a common machine topology when targeting automotive
applications, with recent examples including [1–7].
A fractional-slot concentrated winding (FSCW) enables very short end-winding lengths and,
thereby, potential improvements in terms of torque density. However, depending on the combination of
the number of stator slots Qs and poles p, the resulting harmonic content in the air-gap magnetomotive
force (MMF) caused by the stator currents can be substantial. During the last decade, efforts were put
into identifying suitable combinations of Qs and p where the impact of the stator MMF harmonics is
as small as possible [8–10]. Particularly, for rotors with surface-mounted PMs, a number of models to
quantify the induced eddy-current losses in the PMs based on the harmonic content in the stator MMF
were developed [11–14]. Today, the combinations of Qs and p with the lowest harmonic content have
been identified, and FSCWs are adopted in automotive applications (see, e.g., [15] and the references
in [16]).
Energies 2017, 10, 379; doi:10.3390/en10030379
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Energies 2017, 10, 379
2 of 19
However, while the above loss models indeed are useful, they are most suitable for comparing
the relative change in eddy-current losses for different combinations of Qs and p rather than accurately
predicting the losses for given PM dimensions. Particularly, the effect of segmentation of the magnets
and the impact of the skin effect is challenging. Additionally, these models target surface-mounted
PMs rather than IPMs (a typical FSCW-IPM motor with Qs = 12 slots and p = 8 poles is depicted
in Figure 1). Having a good approximation of the resulting PM losses is important for the machine
designer since relatively small losses in the PMs can result in excessive temperatures due to the
difficulty of transferring the resulting heat across the air gap.
Figure 1. Sample fractional-slot concentrated winding (FSCW)-interior permanent-magnet motor (IPM)
with Qs = 12 slots and p = 8 poles.
Essentially, predicting the losses in the PMs can be done accurately using three-dimensional (3D)
finite-element (FEM)-based simulations (as demonstrated in [17]). However, the approach must still be
considered relatively time consuming, although efforts have been made to reduce the computation
times [18]. Recently, the relatively simple analytical loss model in [19] was proposed considering
FSCWs with surface-mounted PMs and including the effect of both axial and tangential segmentation
of the PMs. This model uses the stator MMF harmonic content as input, but neglects the impact of the
skin effect. A more advanced model, incorporating the effect of axial and tangential segmentation, as
well as the skin effect, is adopted in [20] where analytical solutions of the Helmholtz equation with
an imposed source term are adopted. A different model (also accounting for segmentation and the
skin effect) is adopted in [21], where solutions to the Helmholtz equation with a prescribed boundary
surface current are used.
Contributions and Outline of the Paper
An overall aim of this work is to provide a link between models describing the harmonic content in
FSCWs due to the stator MMF with corresponding, sufficiently accurate PM eddy-current loss models
for IPMs. Particularly, the loss models in [20] (here designated Model B) and [21] (here designated
Model C), adopting solutions to different formulations of the Helmholtz equation to solve for the
eddy currents, are compared to the considerably less complex loss model in [19] (here adapted to
IPMs and designated Model A). It is demonstrated that Model B and Model C, though considerably
different in terms of implementation complexity, predict very similar results (in good agreement with
corresponding 3D-FEM simulations) for the PM dimension and rotor speed intervals typically found
in automotive applications. Further, limits (in terms of frequency and PM dimensions) where Model A
(the least complex loss model) is applicable are identified numerically and using an approximate
analytical expression. This limit is not straight forward since the rectangular shape of the PM segments
results in complex eddy-current reaction fields. Key concepts regarding FSCWs are included as an
Appendix representing a complete description of how the uniform variation of the flux density in
the PMs can be analytically predicted. The assumption of a uniform flux-density variation (used
in the loss models) is also investigated for the case of V-shaped and straight interior permanent
Energies 2017, 10, 379
3 of 19
magnets. An extensive numerical evaluation using 3D-FEM-based simulations is carried out to verify
model assumptions and and the conclusions made regarding whether each loss model is applicable or
not. The comparison with 3D-FEM results represents a solid evaluation metric since 3D-FEM-based
models of PM eddy-current losses previously have been demonstrated to yield good agreement
with corresponding experimental results (see, e.g., [17,21]). Further, the extent of a corresponding
experimental evaluation would be extremely expensive in order for a realization where a sufficiently
large number of parameters (including PM, stator and rotor dimensions) would be varied.
The paper is outlined as follows. In Section 2, the loss models considered are briefly reviewed
where the loss model in [19] (here designated Model A), but adapted to IPMs, is presented. In Section 3
(being the major part of this paper), model constraints when applied to IPM rotor geometries and the
model limitations for Model A due to eddy-current reaction fields are identified. These constraints
and limitations are also verified with presented 3D-FEM-based models. Finally, predicted magnet
losses and temperature risk indicators are tabulated for combinations of Qs and p that are commonly
considered in automotive applications in Section 4, and concluding remarks are given in Section 5.
2. Review of Eddy-Current Loss Models
For axially short PMs and assuming a sinusoidal flux-density variation, the predicted volumetric
loss density pm (W/m3 ) (presented in textbooks, e.g., [22,23]) can be expressed as:
pm =
2 B2
σm ων2m lm
νm
24
(1)
where σm is the conductivity of the PMs, lm the axial length of the PM and ωνm and Bνm are the angular
frequency and magnitude of the imposed flux density, respectively.
However, the assumption of axially short PMs is not valid for practical axial segmentation lengths,
and (1) cannot be directly applied. Generally, assuming constant µr and neglecting displacement
currents (i.e., ∂D/d∂t ≈ 0), the magnetic field in the PMs follows the Helmholtz equation, which can be
expressed as:
∇2 H = jσm µ0 µr ωνm H.
(2)
Once a solution to (2) is determined, the current density (and associated losses) can be found from:
J = ∇ × H.
(3)
In both (2) and (3), bold symbols denote vector fields, and the bar above denotes a phasor
(complex) quantity. Solutions to (2) are used for loss Model B and Model C reviewed below.
2.1. Model A: Assumed Eddy-Current Paths
The loss model presented in [19] is developed for surface-mounted rotors where the air-gap
flux-density harmonics passes over each permanent magnet segment depending on the rotor speed
and wavelength of each harmonic. For IPMs, however, the flux concentration assumed in (A11) results
only in a time-dependent variation of the flux density in the PMs. In [19], this corresponds to the case
of an infinite wavelength where the assumed eddy-current paths are illustrated in Figure 2.
From Figure 2, the flux in the assumed eddy-current path φ can be expressed as:
φ=
Z y0 Z x 0
−y0 − x 0
Bνm sin (ωνm t) dxdy.
The losses in the specific eddy-current path dPm can then be expressed as:
(4)
Energies 2017, 10, 379
4 of 19
dPm =
(∂φ/∂t)2
REC
(5)
where REC is the resistance of the eddy-current path found as:
REC =
4x 0
4y0
+
.
0
σm hm dx
σm hm dy0
(6)
Figure 2. Assumed eddy-current paths adopted in Model A (redrawn from [19]).
y0 )
The average magnet losses for an elementary current path dPm,ave (which is a function of x 0 and
is now found by integrating (5) as:
dPm,ave
ων
= m
2π
Z 2π/ων
m
0
dPm dt.
(7)
Finally, an expression for the total average magnet losses for a magnet segment is now found as:
Pm =
Z wm /2
0
dPm,ave dx 0 =
σm hm (lm wm )3 ( Bνm ωνm )2
2 + w2 )
32 (lm
m
(8)
where y0 = lm x 0 /wm is used when evaluating the integral.
Remark 1. Note that for axially short PMs (lm wm ), the volumetric loss density pm using (8) is found as:
pm =
2 B2
σm ων2m lm
Pm
νm
→
hm lm wm
32
(9)
which is not in agreement with (1). From (9), it can, hence, be expected that Model A will underestimate the
eddy current losses if the PMs are axially short.
Remark 2. Similar loss models also adopting prescribed eddy-current paths are presented in [24].
2.2. Model B: Solving the Helmholtz Equation with the Imposed Source Term
In [25], the Helmholtz equation is solved where the source field due to the armature reaction
is added explicitly rather than as an imposed boundary condition. With this source term added,
(2) becomes:
∂2 H z
∂2 H z
+
= jσm µ0 µr ωνm
∂x2
∂y2
Bνm
Hz +
µ0 µr
(10)
Energies 2017, 10, 379
5 of 19
Similar solutions are reported in [20,26] where in [20], the solution is applied on two IPMs with
distributed windings. The resulting average eddy-current losses for a magnet segment become [20]:
Pm =
32σm ων2m hm lm wm Bν2m
π2
∞
∞
∑ ∑
n 0 =1 m 0 =1
n num o
(11)
den
1
1
+ 2
2 (2n0 − 1)2
lm
wm (2m0 − 1)2
2 (2n0 − 1)2 (2m0 − 1)2
µ0 µr σm ωνm hm 2
den = π 4
+
+
2
δ + hm
w2m
lm
2
(2n0 − 1)2 (2m0 − 1)2
+
+(µ0 µr σm ωνm )2
≈ π4
2
w2m
lm
num =
(12)
(13)
where hm δ was assumed in the approximation used in (13).
Remark 3. For axially short PMs; using (11), the volumetric loss density pm is found as:
Pm
hm lm wm
2 B2
32σm ων2m lm
νm
=
π6
pm =
∞
∞
∑ ∑
n 0 =1 m 0 =1
1
(2n0 − 1)2 (2m0 − 1)4
.
(14)
Now, the double sum in (14) equals π 6 /768 (obtained using Mathematica (Mathematica is a
registered trademark of Wolfram Research, Inc., Champaign, IL, USA), which yields an expression for
pm identical to (1).
2.3. Model C: Solving the Helmholtz Equation Prescribing Boundary Surface Currents
As is well known, the magnetization of a PM can be represented using a surface-current density
with a magnitude corresponding to the remanent flux density divided by the permeability (see,
e.g., [23]). Therefore, each flux density harmonic caused by the armature reaction can, approximately,
be modeled assuming the surface current:
Js =
Bνm
cos ( jωνm t)
µ0 µr
(15)
on the outer boundaries of the magnet segment whose normals are in the xy-plane (the xy-coordinate
system is depicted in Figure A1). Equation (2) simplifies to:
∂2 H z
∂2 H z
+
= jσm µ0 µr ωνm H z
∂x2
∂y2
(16)
with the boundary condition H z = Bνm /(µ0 µr ) on the prescribed outer boundaries.
A general solution to (16) with the given boundary condition is presented in [21,27] where in [27],
also the displacement currents are accounted for. From the solution H z ( x, y), the current density is
then found as:
J = ∇×H =
∂H z
∂H z
x̂ −
ŷ = J x x̂ + J y ŷ.
∂y
∂x
(17)
The resulting expression for the average eddy-current losses for a magnet segment becomes:
Energies 2017, 10, 379
6 of 19
Pm =
8hm wm ωνm Bν2m
π 2 µ0 µr
∞
∑
0
p =1
={α} sinh (<{α}lm ) − <{α} sin (={α}lm )
m02 (<{α}2 +={α}2 )[cosh(<{α}lm )+ cos(={α}lm)]
8hm lm ωνm Bν2m ∞
+
∑
π 2 µ0 µr
q 0 =1
={ β} sinh (<{ β}wm ) − <{ β} sin (={ β}wm )
n02 (<{ β}2 +={ β}2 )[cosh(<{ β}wm )+ cos(={ β}wm)]
where m0 = 2p0 − 1, n0 = 2q0 − 1, and:
s
2
m0 π
+ jσm µ0 µr ωνm
α=
wm
s
0 2
nπ
β=
+ jσm µ0 µr ωνm
lm
(18)
(19)
(20)
Remark 4. It is further shown in [27] that using (18), the resulting volumetric loss density pm for axially short
PMs is identical to the classical expression (1).
3. Analysis and Evaluation
3.1. Loss-Model Constraints When Applied to IPMs
An analytical approach for approximating the flux-density variation Bm (θr ) in the PMs as
a function of rotor position θr is outlined in Section A.4 in the Appendix A. From Bm (θr ), the
corresponding harmonics (of order νm and with a magnitude Bνm ) can then be identified (in this
paper, this identification has been done using the fft function in MATLAB (MATLAB is a registered
trademark of The Mathworks Inc., Natick, MA, USA). However, if required, more accurate predictions
of Bνm can rapidly be obtained using two-dimensional static FEM simulations.
The three eddy-current loss models reviewed in Section 2 all assume that the flux density variation
in the PMs is uniform. For surface-mounted PMs, it is pointed out in [19] that such an assumption
holds provided that the PM width wm is significantly lower than half of the wavelength λν of the
harmonic order of most interest ν.
With the pole-cap coefficient α p as defined in Figure A1, for IPMs, the corresponding condition
can be expressed as α p /(Cp) ≤ 1/ν where C = 1 and C = 1/2 for V-shaped and straight interior PMs,
respectively. For many combinations of Qs and p, the harmonic order ν that dominates the PM losses
is less than or equal to p (i.e., ν ≤ p). Further, α p ≈ 3/4 is not uncommon in order to realize a certain
reluctance-torque component. Thereby, we obtain:
3
Cp
≤
4
ν
(21)
and it can be concluded that for V-shaped PMs, a uniform flux-density variation in the PMs can often
be assumed, whereas for straight interior PMs, this assumption is valid only if the harmonic order ν
that dominates the PM losses fulfills ν p.
3.2. Limits for Model A Due to Eddy-Current Reaction Fields
As seen, the expressions for the volumetric loss density pm given by Model A, Model B and
Model C are of increasing complexity. An interesting issue is therefore to determine the boundaries
when the simplest model (Model A) is applicable. Since Model A does not incorporate the effect of
eddy-current reaction fields, it may risk overestimating the eddy-current losses at higher frequencies.
113
3.2. Limits for Model A Due to Eddy-Current Reaction Fields
As seen, the expressions for the volumetric loss density pm given by Model A, Model B, and
Model C, are of increasing complexity. An interesting issue is therefore to determine the boundaries
116
when the simplest model (Model A) is applicable. Since Model A does not incorporate the effect
of
Energies 2017, 10, 379
7 of 19
117
eddy-current reaction fields, it may risk to overestimate the eddy-current losses at higher frequencies.
118
However, as pointed out in (9), Model A will underestimate the eddy-current losses if either lm ≪
as ≪
pointed
out inboth
(9), Model
the eddy-current
lossesbyif (14)
either
lm wm or wm
lm (where
ModelABwill
andunderestimate
Model C simplify
to (1) as shown
119 However,
and
in w
[27],
m or
l
(where
both
Model
B
and
Model
C
simplify
to
(1),
as
shown
by
(14)
and
in
[27],
respectively).
120 wrespectively).
m
m
compareModel
ModelBBand
andModel
ModelCCto
toModel
Model A
A for typical PM dimensions
ToTocompare
dimensions llmm and
andwwmm, ,the
therelative
relative
errorsε Aε A
and
ε
are
therefore
introduced
as
errors
and
ε
are
therefore
introduced
as:
|B|B
AA
|C|C
114
115
Model A
A)) − pm
ppmm((Model
m(Model B)
p
(
Model
pmm(Model B)
pm (Model A) − m
C)
= pm (Model A) − pm (Model ..
εεAA|C|C =
p
(
Model
C
)
m
pm (Model C)
=
εεAA|B|B =
(22)
(22)
(23)
(23)
In Figure
3, contours
when
ε A|εB = =0.2
In Figure
3, contours
when
0.2(representing
(representinga amodest
modestdeviation)
deviation)have
havebeen
beenplotted
plottedfor
for
A|B
122
ω
/
(
2π
)
=
300
Hz
to
ω
/
(
2π
)
=
3000
Hz
in
steps
of
300
Hz
assuming
µ
=
1.04
and
σ
=
694
kS/m
ν
ν
r
m
m
m
ωνm /(2π ) = 300 Hz to ωνm /(2π ) = 3000 Hz in steps of 300 Hz assuming µr = 1.04 and σm = 694 kS/m
123
(typicalvalues
valuesfor
forPMs
PMsofofthe
NdFeB-type).
The
regions
representing
lm ≪
≪ml
patched
as
m are
(typical
NdFeB-type).
The
regions
representing
lmw
m worm worm w
lm are
patched
124
3.
It
is
further
found
that
the
contours
for
ε
=
0.2
and
essentially
identical.
Hence,
black
in
Figure
as black in Figure 3. It is further found that the contours forAε|AC|C = 0.2 are essentially identical. Hence,
ModelBBshould
shouldpredict
predictvery
verysimilar
similarresults
resultsto
to (the
(the somewhat
somewhat more
125
Model
more complex)
complex) Model
ModelC.
C.How
HowFigure
Figure33
126
4
(see
below))
can
be
used
by
the
machine
designer
to
determine
validity
(or
its
approximation
Figure
(or its approximation Figure 4 (see below)) can be used by the machine designer to determine the
of the different loss models are exemplified in Section 3.3.
validity
of the different loss models is exemplified in Section 3.3.
121
100
300
wm (mm)
80
60
40
3000
20
0
0
20
40
60
lm (mm)
80
100
Figure 3. Contours corresponding to ε A|B = 0.2 (red curves) for ωνm /(2π ) = 300 Hz to ωνm /(2π ) =
Figure 3. Contours corresponding to ε A|B = 0.2 (red curves) for ω νm /(2π ) = 300 Hz to ω νm /(2π ) =
3000 Hz in steps of 300 Hz assuming µr =
1.04 and σm = 694 kS/m. The arrow denotes the direction of
3000 Hz in steps of 300 Hz assuming µr = 1.04 and σm = 694 kS/m. The arrow denote the direction of
increasing ωνm . Note that the contours ε A|C = 0.2 are essentially identical.
increasing ω νm . Note that the contours ε A|C = 0.2 are essentially identical.
127
Approximation of ε A|B
3.2.1. Approximation of ε A|B
Figure 3 can be useful for a machine designer since it provides boundaries in terms of magnet
Figurelm3and
129 dimensions
canwm
beand
useful
for a frequencies
machine designer
since
it provides
boundaries
inA)
terms
of
excitation
ωνm when
the simplest
loss model
(Model
predicts
130 the
magnet
dimensions
l
and
w
and
excitation
frequencies
ω
when
the
simplest
loss
model
resulting eddy-current
m lossesmwith sufficient accuracy. In [27],νa
m number of approximations to (18)
131 (Model
(ModelC)A)
the resulting
losses
with
sufficient accuracy.
In [27],
a dimensions
number of
arepredicts
presented.
However,eddy-current
when applying
these
approximation
using typical
PM
132 (l approximations
to
(18)
(Model
C)
are
presented.
However,
when
applying
these
approximation
using
and
w
)
and
a
conductivity
σ
valid
for
NdFeB-type
PMs,
it
is
found
that
none
of
the
resulting
m
m
m
133 approximations
typical PM dimensions
(l
and
w
)
and
a
conductivity
σ
valid
for
NdFeB-type
PMs,
it
is
found
that
m
mthe contours in Figurem3 sufficiently accurately.
to ε A|C reproduce
134
none
the resulting
approximations
ε |C reproduce
the contours
Figure 3 sufficiently
Inof
order
to investigate
whether εto
be approximated,
thein
dimensional
ratios ξ accurate.
and κ are
A|B Acan
128
introduced as:
max(lm , wm )
min(lm , wm )
min(lm , wm )
κ=
δskin
ξ=
(24)
(25)
introduced as
max(lm , wm )
min(lm , wm )
min(lm , wm )
κ=
δskin
ξ=
Energies 2017, 10, 379
(24)
8 of
19
(25)
p
p
where
= 2/
2/((σσmmµµ00µµrrω
ωννm)) is
is the
the classical
classical expression
expression for
for skin
skin depth.
depth. Now,
Now, by
byconsidering
considering only
only
where δδskin
skin =
m
′
′
0
0
can be approximated as
nn =
=m
m=
= 11 in
in (11),
(11), εε A
A||B
B can be approximated as:
εε A
A||B
B
135
"
"
π22
≈ π
256
256
2 2
ξξ2 κκ2
11 + ξ22
22
##
π66
π
− 1.
+
+
1024 − 1.
1024
(26)
(26)
The contours
for ε Afor
using the approximation (26) are plotted in Figure 4. Comparing Figure 3
The contours
|B =ε 0.2
A|B = 0.2 using the approximation (26) are plotted in Figure 4. Comparing
4, it4,can
be be
seen
that
(26)
represents
thethe
model
prediction
error
reasonably
well.
and
Figure
Figures
3 and
it can
seen
that
(26)
represents
model
prediction
error
reasonably
well.
100
300
wm (mm)
80
60
40
3000
20
0
0
20
40
60
lm (mm)
80
100
Figure 4. Contours corresponding to ε |B = 0.2 (red curves) using the approximate formulation of ε A|B
Figure 4. Contours corresponding to εA
A |B = 0.2 (red curves) using the approximate formulation of ε A |B
given by (26) for ω /(2π ) = 300 Hz to ω /(2π ) = 3000 Hz in steps of 300 Hz assuming µr = 1.04 and
µr = 1.04
given by (26) for ωννmm /(2π ) = 300 Hz to νωm νm /(2π ) = 3000 Hz in steps of 300 Hz assuming
σm = 694 kS/m. The arrow denotes the direction of increasing ωνm .
and
σm = 694 kS/m. The arrow denote the direction of increasing ω νm .
136
137
3.3. 3DFEM-Evaluation
3.3. 3DFEM-Evaluation
In order to verify the boundary constraints identified in Sections 3.1 and 3.2, comparisons with
corresponding
models,
implemented
usinginJMAG
(JMAG
is aSection
registered
of
In order to3D-FEM-based
verify the boundary
constraints
identified
Sections
3.1 and
3.2,trademark
comparisons
3
the
JSOL
Corporation,
Tokyo,
Japan),
are
now
presented.
In
the
3D-FEM-based
model,
the
eddy-current
139
with corresponding 3D-FEM based models, implemented using JMAG , are now presented. In the
losses have
been
obtained
by computinglosses
the average
value
of the by
Ohmic
losses in
PM segment
140
3D-FEM
based
model,
the eddy-current
have been
obtained
computing
theaaverage
value
(caused
by
the
resulting
current
density
distribution).
141
of the Ohmic losses in a PM segment (caused by the resulting current density distribution).
A three-phase
three-phase IPM
8 poles
with
additional
keykey
parameters
reported
in
142
A
IPM with
with Q
Qss==12
12slots
slotsand
andp p==
8 poles
with
additional
parameters
reported
Table
A1
in
Appendix
B
is
used
initially
for
evaluation.
The
machine
geometry
is
depicted
in
Figure
5a;
Version
February
26,
2017
submitted
to
Energies
143
in Table B.1 in Appendix B is used initially for evaluation. The machine geometry is depicted in 9 of 20
the PM5width
is w
15 width
mm, and
axial length of the PM is varied so that l = 10, 30 and 100 mm,
m=
Figure
144
(a) and
the
PM
is wthe
m = 15 mm and the axial length of the PMm is varied so that l m =
respectively.
The
rotor
speed is varied
up tospeed
9000 rpm.
145
10,
30, and 100
mm,
respectively.
The rotor
is varied up to 9000 rpm.
138
146
147
A brief review of the fundamentals of the resulting harmonic content is provided in Appendix A.
From (29), the winding periodicity is found as tper = 4. Therefore, from (32), the harmonic orders that
3
JMAG is a registered trademark of the JSOL Corporation, Tokyo, Japan.
(a)
(b)
Figure 5. Cont.
Energies 2017, 10, 379
9 of 19
(a)
(b)
(c)
Figure 5. IPM 3D-FEM models: (a) V-shaped interior PMs (wm = 15 mm); (b) V-shaped interior PMs
(w
= 30 mm);
and (c)
Straight(a)
interior
PMs (winterior
m = 40 mm).
Figure 5.m IPM
3D-FEM
models:
V-shaped
PMs (wm = 15 mm); (b) V-shaped interior PMs
148
149
150
151
152
153
154
155
156
157
158
(wm = 30 mm); (c) Straight interior PMs (wm = 40 mm).
A brief review of the fundamentals of the resulting harmonic content is provided in Appendix A.
From (A1), the winding periodicity is found as tper = 4. Therefore, from (A4), the harmonic orders that
could could
be present
in the
airairgap
to the
thearmature
armature
reaction
(assuming
sinusoidal
phase are
currents)
be present
in the
gapdue
due to
reaction
(assuming
sinusoidal
phase currents)
ν =4,4,8,8,12,
12, 16,
fromfrom
(A6), k(34),
that are multiples
of 12.
The magnitude
are ν =
16,20,
20,. .. ... ..However,
However,
k ν harmonics
= 0 for harmonics
that are
multiples
of 12. The
ν = 0 for
of
the
corresponding
harmonics
can
be
obtained
using
(A3)–(A9).
Hence,
in
the
air
gap,
the
three
magnitude of the corresponding harmonics can be obtained using (31)–(37). Hence, in the air gap,
first harmonic orders are ν = 4, 8 and 16. For the considered rotor structures, α p ≈ 0.77 and using (A5)
the three
first harmonic orders are ν = 4, 8, and 16. For the considered rotor structures, α p ≈ 0.77 and
and (A12), the first harmonic order present in a PM is νm = 12 (higher order harmonics exist, but are of
using (33) and (40), the first harmonic order present in a PM is νm = 12 (higher order harmonics exist
significantly lower magnitude and contribute only minorly to the total eddy-current losses). Hence, at
but are9000
of significantly
and
contribute only minor to the total eddy-current losses).
rpm, ωνm /(2πlower
) = 12 · magnitude
9000/60 = 1800
Hz.
Hence, at 9000 rpm, ωνm /(2π ) = 12 · 9000/60 = 1800 Hz.
RemarkSince
5. Since
PMsare
are mounted
in the
interior
of the rotor,
therotor,
resulting
losses due to the losses
Remark:
thethePMs
mounted
in the
interior
of the
theeddy-current
resulting eddy-current
effect are negligible
to the
eddy-current
due to the armature
reaction.
due toslot-opening
the slot-opening
effect is compared
negligible
compared
to losses
the eddy-current
losses
due to the armature
reaction.
3.3.1. Negligible Eddy-Current Reaction Fields
The considered
three PMReaction
dimensions
are indicated in Figure 3 as black diamonds (). As seen, for
3.3.1. Negligible
Eddy-Current
Fields
1800 Hz, all three PM dimensions fall within the set of lm and wm for which Model A should provide
165
3.3.2. Non-Negligible Eddy-Current
0.2 Reaction Fields
162
163
166
167
168
169
170
171
172
With a wider magnet width wm = 30 mm (see Figure 5 (b)), it can be seen from Figure 3 (depicted
as a blue circle (◦)) that Model A 0will overestimate the eddy-current losses for frequencies above
0
2000
4000
6000
8000 10000
(b) to rotor speeds around 8500 rpm). This is verified in Figure 7
around 1700 Hz (corresponding
4
where the losses are substantially
overestimated at speeds above (around) 8500 rpm. Compensating
Loss (W)
161
Model A as Pm / ε A|B + 1 using the approximation of ε A|B given by (26) yields a significantly better
2
agreement with the corresponding 3D-FEM results. Model B and Model C predict similar results,
both agreeing well the corresponding 3D-FEM results at all rotor speeds considered.
0
0
(c)
Loss (W)
160
Loss (W)
164
The
considered
three (i.e.,
PM dimensions
indicated
in Figure
3 asinblack
⋄). be
As seen,
an accurate
prediction
a deviation ofare
at most
20%). This
is verified
Figurediamonds
6, where it(can
for 1800
Hz,
three
PM dimensions
fallaswithin
of and
lm and
wmC (green
for which
A should
seen
thatall
Model
A predicts
similar results
Model Bthe
(redset
lines)
Model
lines) Model
in V-shaped
Version
February
submitted
to Energies
10 of 20 6 where
interior
PMs
(wm26,
=2017
15 mm)
with
different
magnet
lm and
alsoThis
agrees
the corresponding
provide
an
accurate
prediction
(i.e.,
a deviation
oflengths
at most
20 %).
iswith
verified
in Figure
3D-FEM
results.
it can be seen that Model A predicts similar results as Model B (red lines) and Model C (green lines)
in V-shaped interior PMs (wm =
(a)15 mm) with different magnet lengths lm , and also agrees with the
corresponding 3D-FEM results. 0.4
159
15
2000
4000
6000
8000
10000
8000
10000
Figure 6. Cont.
10
5
0
0
2000
4000
6000
Speed (rpm)
(b)
Loss (W)
Energies 2017, 10, 379
4
2
10 of 19
0
Loss (W)
(c)
0
2000
0
2000
4000
6000
8000
10000
4000
6000
8000
10000
15
10
5
0
Speed (rpm)
Figure
Predictedlosses
lossesper
permagnet
magnetsegment
segment Pm as function
of of
rotor
speed
andand
for for
different
magnet
Figure
6. 6.Predicted
a function
rotor
speed
different
magnet
lengths
l
in
V-shaped
interior
PMs
(w
=
15
mm)
for
Model
A
(blue
line),
Model
B
(red
line),
Model
C C
m = 15 mm) for Model A (blue line), Model B (red line), Model
lengths lm min V-shaped interior PMs (wm
(green
line),
and
corresponding
3D-FEM
results
(diamonds
“
⋄
"):
(a)
l
=
10
mm;
(b)
l
=
30
mm;
(c)
(green line) and corresponding 3D-FEM results (diamonds “"): (a)mlm = 10 mm; (b)mlm = 30 mm; and
(c)lm
lm==100
100mm.
mm.
3.3.2. Non-Negligible
Eddy-Current
Reaction
Fields
Remark: Since the
PMs are buried
very deep
into the rotor structure, (39) represents an overly
173
174
175
simplistic approach for predicting the flux-density variation (while still being relatively uniform).
With a wider magnet width wm = 30 mm (see Figure 5b), it can be seen from Figure 3 (depicted as
Therefore, for this case only, the actual flux density Bm (θr ) is extracted from static 2D-FEM models.
a blue circle (◦)) that Model A will overestimate the eddy-current losses for frequencies above around
HzImpact
(corresponding
to rotor
speeds around
8500 rpm). This is verified in Figure 7 where the
1761700
3.3.3.
of Non-Uniform
Flux-Density
Variation
losses
are
substantially
overestimated
at
speeds
above
(around) 8500 rpm. Compensating Model A as
Following the discussion in Section 3.1, the magnet width wm for the straight IPM depicted in
177
PmFigure
/ ε A|B5+(c)1 (PM
using
the approximation
by(
(26)
significantly
better
|B given
dimensions
marked as of
theε Ared
square
) inyields
Figurea 3)
is wider than
halfagreement
of the
with
the corresponding
3D-FEM results.
Model B the
andflux-density
Model C predict
similar
results,
both
agreeing
179
dominant
harmonic’s
wavelength
λ
.
Therefore,
variation
is
not
uniform
and
the
ν
Version February 26, 2017 submitted to Energies
11 of 20
with
the corresponding
3D-FEM
resultsworsens.
at all rotor
speeds
considered.
180well
loss
prediction
accuracy for all
three models
This
is verified
in Figure 8 where neither of the
178
181
models are in agreement with the corresponding 3D-FEM results.
182
3.4. Visualization of Resulting Eddy-Current Distribution
Loss (W)
150
As is well known, the 100
time-dependent currents densities found from the time-harmonic
(complex) formulations used in Model B and Model C can be determined as
183
184
50
Jx (t) = ℜ J x cos (ωνm t) − ℑ J x sin (ωνm t)
(27)
o
n
o
n
0
(28)
J y sin (ω15000
Jy (t) = 0ℜ J y cos5000
(ωνm t) − ℑ10000
νm t ) .
Speed (rpm)
Figure 9 shows the resulting eddy-current distributions in a PM segment of the IPM depicted in
7. Predicted
losses per
per magnet
segment
Pm P
asmfunction
of rotor
speed
andspeed
for length
magnet
Figure
7. for
Predicted
losses
magnet
segment
as a function
of rotor
and llengths
for
magnet
Figure
5Figure
(b)
two different
speeds
(1000
and 15000
rpm)
with
a PM
segment
m = 60 mm.
l
=
60
mm
in
V-shaped
interior
PMs
(w
=
30
mm)
for
Model
A
(blue
lines),
Model
but
m lm = 60 mm in V-shaped interior PMs
m (wm = 30 mm) for Model A (blue lines),AModel
lengths
A,
compensated as Pm /(ε A|B + 1) (dashed blue line), Model B (red line), Model C (green line), and
but compensated
as Pm /(ε A|B + 1) (dashed blue line), Model B (red line), Model C (green line) and
corresponding 3D-FEM results (circles “◦").
corresponding 3D-FEM results (circles “◦").
400
Loss (W)
Remark 6. Since the PMs are buried very deep into the rotor structure, (A11) represents an overly simplistic
300 variation (while still being relatively uniform). Therefore, for this case
approach for predicting the flux-density
only, the actual flux density Bm (θr200
) is extracted from static 2D-FEM models.
100
3.3.3. Impact of Non-Uniform Flux-Density
Variation
Following the discussion in0Section 3.1, the magnet width wm for the straight IPM depicted in
0
5000
10000
15000
Figure 5c (PM dimensions marked as the red square
) in Figure 3) is wider than half of the dominant
Speed(
(rpm)
harmonic’s wavelength λν . Therefore, the flux-density variation is not uniform, and the loss prediction
Predicted
losses
per magnet
segment
Pm as in
function
of 8
rotor
speed
for lm =
mm
for
accuracy Figure
for all8.three
models
worsens.
This
is verified
Figure
where
neither
of60the
models
are in
the straight IPM depicted in Figure 5 (c)s (wm = 40 mm) for Model A (blue line), Model B (red line),
agreement with the corresponding 3D-FEM results.
Model C (green line), and corresponding 3D-FEM results (squares “").
185
186
187
188
189
190
191
192
Predicted distributions using Model B (red contours) and Model C (green contours). First, it is
noted that the resulting eddy-current distributions predicted by the two models are essentially
equal. Hence, both models can be expected to predict similar eddy-current losses. Further, at
1000 rpm, the eddy current distribution is very similar to what is assumed in Model A (compare
with Figure 2). However, at higher speeds, the eddy current paths during the transition between
“clockwise" and “anti-clockwise" current paths becomes more complex with additional eddies close
to corners. Corresponding 3D-FEM results are depicted in Figure 10 and, as seen, the eddy-current
distributions predicted with Model B and Model C are
qin good agreement with the 3D-FEM results.
0
0
5000
10000
Speed (rpm)
15000
Figure 7. Predicted losses per magnet segment Pm as function of rotor speed and for magnet lengths
lm = 60 mm in V-shaped interior PMs (wm = 30 mm) for Model A (blue lines), Model A but
compensated
as Pm /(ε A|B + 1) (dashed blue line), Model B (red line), Model C (green line), and
Energies 2017,
10, 379
corresponding 3D-FEM results (circles “◦").
11 of 19
Loss (W)
400
300
200
100
0
0
5000
10000
Speed (rpm)
15000
Figure
8. Predicted
losses
magnet
segment
function of
Figure
8. Predicted
losses
per per
magnet
segment
PmPm
asas
a function
ofrotor
rotorspeed
speedforforlmlm==6060mm
mmforfor the
the
straight
IPM
depicted
in
Figure
5
(c)s
(w
=
40
mm)
for
Model
A
(blue
line),
Model
B
(red
m
straight IPM depicted in Figure 5c (wm = 40 mm) for Model A (blue line), Model B (red line),line),
Model C
(green
line), and corresponding
3D-FEM
results “
(squares
(greenModel
line) Cand
corresponding
3D-FEM results
(squares
”). “").
185 Visualization
Predicted distributions
usingEddy-Current
Model B (red Distribution
contours) and Model C (green contours). First, it is
3.4.
of the Resulting
186
noted that the resulting eddy-current distributions predicted by the two models are essentially
As
is well
known,
time-dependent
currents
densities
found
from the time-harmonic
(complex)
equal.
Hence,
boththe
models
can be expected
to predict
similar
eddy-current
losses. Further,
at
formulations
in Model
B and
Model Cis can
determined
188
1000 rpm,used
the eddy
current
distribution
verybesimilar
to whatas:
is assumed in Model A (compare
187
189
190
191
192
193
with Figure 2). However, at higher
the eddy current
speeds,
paths during the transition between
J x cos
J x complex
Jx (t) = <current
sin (ωνmwith
t) additional eddies close (27)
(ωbecomes
νm t ) − =more
“clockwise" and “anti-clockwise"
paths
n o
n o
to corners. Corresponding 3D-FEM results are depicted in Figure 10 and, as seen, the eddy-current
Jy (t) = < J y cos (ωνm t) − = J y sin (ωνm t) .
(28)
distributions predicted with Model B and
Model C are
qin good agreement with the 3D-FEM results.
In Figure 11, the predicted current densities |J| =
J 2 + J 2 at 15000 rpm in selected positions ’Pa’
x
y
Figure 9 shows the resulting eddy-current distributions
in a PM segment of the IPM depicted
194
and ’Pb’ (see Figure 9 (b)) using Model A, Model B, Model C, and corresponding 3D-FEM results are
in Figure 5b for two different speeds (1000 and 15,000 rpm) with a PM segment length lm = 60 mm;
195
reported. As seen, Model B and Model C are agreeing well with the 3D-FEM based results whereas
predicted
using
B (redsignificantly.
contours) and Model C (green contours). First, it is noted
196
Modeldistributions
A overestimates
the Model
eddy-currents
that the resulting eddy-current distributions predicted by the two models are essentially equal. Hence,
197
4. PM Losses
Automotive
both
models
can be
expectedApplications
to predict similar eddy-current losses. Further, at 1000 rpm, the eddy
current distribution is very similar to what is assumed in Model A (compare with Figure 2). However,
4.1. Thermal
at198higher
speeds,Impact
the eddy current paths during the transition between “clockwise” and “anti-clockwise”
current
paths
becomes
complex with(26)
additional
eddies close
corners.
Corresponding
3D-FEM
199
or its approximation
provide necessary
inputtoon
which loss
model is suitable,
While
Figure 3 more
200
theare
associated
temperature
increase
in seen,
the PMs
(due
to the eddydistributions
currents) represents
a keywith
limiting
results
depicted
in
Figure
10,
and
as
the
eddy-current
predicted
Model
B
Version February 26, 2017 submitted to Energies
12 of 20
201
factor
for
the
machine
designer.
To
investigate
this
thermal
impact,
a
set
of
3D-FEM
thermal
and Model C are in good agreement with the 3D-FEM results.
simulations of the machine described above are carried out at 9000 rpm and rated current for different
volumetric loss densities pm in the PMs. The implemented thermal model is similar to the 3D-FEM
(b)
(a)
30
30
20
20
y (mm)
203
y (mm)
202
10
0
Pa
Pb
10
0
−10
0
x (mm)
10
−10
0
10
x (mm)
Figure9.9. Resulting
Resulting eddy
eddy current
current distribution
distribution using
and
Model
C (green
Figure
usingModel
ModelBB(red
(redcontours)
contours)
and
Model
C (green
contours) evaluated
evaluated at
a magnet
length
lm =lm60=mm
two
speeds:speeds:
(a)
contours)
at ωωννmmtt==π/4
π/4with
with
a magnet
length
60 at
mm
atdifferent
two different
1000
rpm;
(b)
15000
rpm.
Note
that
the
green
contours
are
almost
completely
covered
by
the
red
(a) 1000 rpm; (b) 15,000 rpm. Note that the green contours are almost completely covered by the red
contours which
which indicates
indicates that
thethe
same
eddy
current
distribution.
contours,
thatthe
thetwo
twomodels
modelspredict
predictessentially
essentially
same
eddy
current
distribution.
x (mm)
x (mm)
Figure 9. Resulting eddy current distribution using Model B (red contours) and Model C (green
contours) evaluated at ω νm t = π/4 with a magnet length lm = 60 mm at two different speeds: (a)
1000 rpm; (b) 15000 rpm. Note that the green contours are almost completely covered by the red
Energies 2017, 10, 379
12 of 19
contours which indicates that the two models predict essentially the same eddy current distribution.
(b)
(a)
Figure10.
10.Resulting
Resultingeddy
eddycurrent
current distribution
distribution from
π/4 with
νmνt =
from the
the 3D-FEM
3D-FEMmodel
modelevaluated
evaluatedatatωω
Figure
m t = π/4 with
a magnetlength
lengthl lm==60
60mm
mmat
attwo
two different
different speeds:
speeds: (a)
a magnet
(a) 1000
1000rpm;
rpm;(b)
(b)15,000
15000rpm.
rpm.
m
206
207
208
209
210
211
212
213
214
215
216
217
218
|J| (A/mm2 )
205
0
4.2. Losses for p and Qs Common in Automotive
Applications
0
(b)
90
180
270
360
Now, resulting PM eddy-current
losses for combinations of p and Qs often considered in
3.6
automotive applications are considered.
The harmonic content from double-layer FSCWs are
2.7
considered in this paper. For single-layer FSCWs, the harmonic content is described in [8,12,14]. The
1.8
rotor radius rr and air-gap length δ are selected identical to what reported in Table B.1 and the main
0.9
harmonic stator MMF (the ampere-turns
and winding factor k ν= p/2 product) is also kept the same
|J| (A/mm2 )
204
q
In Figure 11, the predicted current densities |J| = Jx2 + Jy2 at 15, 000 rpm in selected positions ‘Pa’
thermal
model described in [28] and the intention is to determine approximately how different values
and ‘Pb’ (see Figure 9b) using Model A, Model B, Model C and the corresponding 3D-FEM results are
ofreported.
pm correspond
to26,resulting
magnet
temperatures. The iron losses in the stator and rotor13laminations
VersionAs
February
2017 submitted
toModel
Energies C are agreeing well with the 3D-FEM based results,
of
20
seen,
Model
B and
whereas
are
computed
using
the
in-built
loss
model
in JMAG. The winding, including the end-winding part,
Model A overestimates the eddy-currents significantly.
is impregnated in Epoxy with an assumed effective thermal conductivity of 0.68 W/(m·K). The
outer surface of the stator lamination
is fixed to 60◦ C representing a water cooling jacket with a
(a)
2.4
coolant temperature typical to what found in automotive applications. A sample result from the
1.8
3D-FEM thermal simulation is depicted
in Figure 12 and the resulting PM temperatures are reported
in Figure 13. The results in Figure1.2
13 will be used below in order to provide approximate intervals of
pm which potentially can result in0.6
excessive PM temperatures in an automotive application.
0
0
90
180
ωνm t (o )
270
360
Figure
11. |J11.
| evaluated
at 15,000
rpm
(bluelines),
lines),Model
Model
B (red
lines),
Model
C (green
Figure
| J| evaluated
at 15000
rpmfor
forModel
Model A (blue
B (red
lines),
Model
C (green
and 3D-FEM
model
(black
lines)
selectedpoints:
points: (a)
lines)lines),
and 3D-FEM
model
(black
lines)
atatselected
(a)’Pa’;
‘Pa’;(b)
(b)’Pb’.
‘Pb’.
4.219PMmeaning
Losses that
in Automotive
each resultingApplications
machine design results in a similar output torque (the harmonic content
220
will be different, however, for each combination of p and Qs ). The pole-cap coefficient is selected to
4.1.
Thermal
221
α p = 3/4Impact
to yield a rotor saliency and the PM height is selected to h m = 5 mm. Further, the PM width
wm is selected
no-load flux density
in the necessary
air gap is 0.75
T. For
p = 8, 10,
12,model
and 14,
While
Figure 3soorthat
its the
approximation
(26) provide
input
on which
loss
is this
suitable,
results in wm = 14.2, 11.3, 9.5, and 8.1 mm. Resulting rotor geometries for p = 8 and p = 14 (with
the associated temperature increase in the PMs (due to the eddy currents) represents a key limiting
224
magnetic bridges and air pockets inserted) are depicted in Figure 14.
factor for the machine designer. To investigate this thermal impact, a set of 3D-FEM thermal simulations
225
The resulting PM loss densities for lm = 10 mm and lm = 30 mm are reported in Table 1
of226theand
machine
above
carried
at combinations
9000 rpm and
forindifferent
volumetric
Table 2,described
respectively.
As is
seen,
only out
a few
ofrated
p andcurrent
Qs result
sufficiently
low
loss
densities
pm losses
in thesoPMs.
The implemented
thermal
model isAlso,
similar
to the 3D-FEM
thermal
227
eddy-current
that excessive
PM temperatures
are avoided.
the increase
of PM length
model
described
[28],
the
intention
is
to
determine
approximately
how
different
values
of pm
228
from
lm = 10 in
mm
to land
=
30
mm
rules
out
all
but
two
combinations
of
p
and
Q
.
Further,
the
loss
m
s
229
Table 2 are predictions
using Model
However,
using laminations
Figure 3 or are
densities
in magnet
Table 1 and
correspond
toreported
resulting
temperatures.
The iron losses
in theC.
stator
and rotor
its approximation
(26), it can
230
bemodel
determined
that Model
A providesincluding
sufficient accuracy
(i.e., withinpart,
a
computed
using the in-built
loss
in JMAG.
The winding,
the end-winding
is
222
223
20% deviation from Model B and Model C) for all cases considered in Table 1 and Table 2.
Table 1. pm [W/cm3 ] for lm = 10 mm at 9000 rpm
Qs \p
8
10
12
14
6
4.0
4.7
N.F.
4.1
Energies 2017, 10, 379
13 of 19
impregnated in epoxy with an assumed effective thermal conductivity of 0.68 W/(m·K). The outer
surface of the stator lamination is fixed to 60◦ C representing a water cooling jacket with a coolant
temperature typical of what is found in automotive applications. A sample result from the 3D-FEM
thermal simulation is depicted in Figure 12, and the resulting PM temperatures are reported in
Figure 13. The results in Figure 13 will be used below in order to provide approximate intervals of pm ,
which potentially
can
in excessive
Version February
26, result
2017 submitted
to Energies PM temperatures in an automotive application. 14 of 20
3
Figure 12. Implemented 3D-FEM thermal model with pm = 0.1 W/cm at
rated current and 9000 rpm
Figure 12.
Implemented 3D-FEM thermal model with pm = 0.1 W/cm3 at
rated current and 9000 rpm
(winding impregnation not shown).
(winding impregnation not shown).
180
Tmag (o C)
150
120
90
60
0
0.5
1
1.5
2
pm (W/cm3 )
Figure 13. Resulting PM average temperature as function of PM volumetric loss density p at rated
m
Figure 13. Resulting PM average temperature as a function of PM volumetric loss density
pm at rated
current and 9000 rpm. The considered low, medium and high temperature ranges have been indicated
current and 9000 rpm. The considered low, medium and high temperature ranges have been indicated
using green, orange and red colors, respectively.
using green, orange and red colors, respectively.
4.2. Losses for p and Qs Common in Automotive Applications
5. Conclusions
Now,
resulting PM eddy-current losses for combinations of p and Qs often considered in
232
In this paper, are
threeconsidered.
models for predicting
average
magnet
losses
in IPMs with
FSCWsisdue
to
automotive
applications
The harmonic
content
from
double-layer
FSCWs
considered
233 paper.
induced
eddy
currents caused
by the
reaction
(assuming
sinusoidal
phase currents)
wereradius
in this
For
single-layer
FSCWs,
thearmature
harmonic
content
is described
in [8,12,14].
The rotor
234
analyzed
and compared.
Provided
that the
order ν that
less
rr and
air-gap
length
δ are selected
identical
toharmonic
what is reported
in dominates
Table A1, the
andPM
thelosses
mainisharmonic
235
or equal to to the number of poles p, it was found that that for V-shaped PMs, a uniform flux
stator MMF (the ampere-turns and winding factor k ν= p/2 product) is also kept the same, meaning
236
density variation in the PMs can typically be assumed. Boundaries in terms of PM dimensions
that237each
resulting
machinewere
design
results
similar
output
torque
(thethe
harmonic
content
will be
and
angular frequency
identified
to in
aidathe
machine
designer
whether
simplest loss
model
different,
however,
for each combination
of p and Qanalytical
is selected
asalso
α p = 3/4
s ). The pole-cap
238
(Model
A) is applicable
or not. An approximate
expressioncoefficient
to these boundaries
was
to yield
a rotor saliency,
and the
height
is Model
selected
as hmrelying
= 5 mm.
Further,formulations
the PM width
239
identified.
It was found
thatPM
Model
B and
C, while
on different
and wm is
240
solutions
of Helmholtz
equation,
provide
very
predictions
in8,
very
agreement
selected
so that the
no-load flux
density
in the
airsimilar
gap isloss
0.75
T. For p =
10,good
12 and
14, thiswith
results in
241
corresponding
3D-FEM
based
simulations.
Further,
tables
were
provided
with
resulting
volumetric
wm = 14.2, 11.3, 9.5 and 8.1 mm. Resulting rotor geometries for p = 8 and p = 14 (with magnetic bridges
loss densities pm for combinations of Qs and p commonly used in automotive applications. Finally, by
and242
air pockets
inserted) are depicted in Figure 14.
243
compiling results from previous publications, a complete description on how to analytically predict
The resulting PM loss densities for lm = 10 mm and lm = 30 mm are reported in Tables 1 and 2,
244
the magnitudes Bνm and corresponding harmonic orders νm for FSCW-IPMs was provided in the
respectively.
As seen, only a few combinations of p and Qs result in sufficiently low eddy-current
245
appendix.
231
Energies 2017, 10, 379
14 of 19
losses so that excessive PM temperatures are avoided. Furthermore, the increase of PM length
from lm = 10 mm to lm = 30 mm rules out all but two combinations of p and Qs . Further, the loss
densities reported in Tables 1 and 2 are predictions using Model C. However, using Figure 3 or its
approximation
(26),
it can betodetermined
that Model A provides sufficient accuracy (i.e., within a 20%
Version
February 26, 2017
submitted
Energies
15 of 20
deviation from Model B and Model C) for all cases considered in Tables 1 and 2.
Figure
Rotor
geometries
(shaftexcluded)
excluded) for
for:(a)
(a) pp =
= 8, ααpp ==3/4,
5 mm,
wm w
=m14.2
mm;mm;
Figure
14. 14.
Rotor
geometries
(shaft
3/4,hmhm= =
5 mm,
= 14.2
p =α14,
5 mm,
wm==8.1
8.1mm.
mm.
p = 3/4,
(b) p(b)
= 14,
hm h=m5=mm,
wm
p =α3/4,
3
Table2.1.ppm[W/cm
(W/cm3 ]) for
rpm.
Table
for llmm==10
30mm
mmatat9000
9000
rpm
m
s \p
Qs Q\p
6
6 9
9 12
1215
1518
1821
2124
2427
30
27
N.F.
30
N.F.
88
4.0
9.8
N.F.
N.F.
0.8
1.9
N.F.
N.F.
0.5
N.F
1.2
qs = 1
N.F
10
12
1414
10
12
4.7
N.F.
4.1
9.8
N.F.
6.7
N.F.
6.3
N.F.
N.F.
11.3
N.F.
2.0
N.F.
6.2
4.0
N.F.
9.1
1.0
N.F.
N.F.
2.0
N.F.
N.F.
0.5
1.2
4.6
1.37.4
N.F.
N.F.
1.0
2.1
5.6
N.F.
7.72.2
N.F.
N.F.
N.F.
0.8
N.F.
qs = 1
11.5
N.F.
12.5
qs = 1 N.F.
0.9
N.F.
1.4
N.F.
Not feasible/unbalanced winding
q s = 1 N.F.
Distributed
windings 1.5
Not feasible/unbalanced
winding
Low losses
Distributed
windings
Medium losses
High
Low losses
losses
Medium losses
Table 2. pm (W/cm3 ) forHigh
lm = 30
mm at 9000 rpm.
losses
Qs \p
246
247
248
249
250
251
252
253
254
255
8
10
12
14
6
9.8 the Swedish
9.8
N.F.
6.7
Acknowledgments: The financial support
from
Hybrid Vehicle
Center (SHC) and the STandUP for
11.3
N.F.
N.F.
N.F.
Energy research initiative is gratefully 9acknowledged.
12
1.9
4.0
N.F.
9.1
Author Contributions: This paper is 15
a resultN.F.
of the full
of all the authors. Hui Zhang and Oskar
2.0 collaboration
N.F.
N.F.
Wallmark have contributed developing
18 ideas
1.2and analytical
1.0
2.1 model.7.4Hui Zhang built the FEM models and
analyzed the data. Oskar Wallmark was
for
and
21 responsible
N.F
N.F.the guidance
N.F.
2.2 a number of key suggestions.
24 noqconflict
11.5
N.F.
s =1
Conflicts of Interest: The authors declare
of interest.
12.5
N.F.
1.4
N.F.
qs = 1 N.F.
1.5
Appendix A.
Not feasible/unbalanced winding
Distributed windings
How to predict the magnitudes Bνm and corresponding
harmonic
Low losses
(compiling results from [8,10,29,30]) is, for completeness,
described in this
Medium losses
High losses
27
30
FSCW Fundamentals
N.F.
orders νm in FSCW-IPMs
appendix.
Appendix A.1. Preliminaries
5. Conclusions
The winding periodicity tper can be expressed as [8]:
In this paper, three models for predicting average magnet losses in IPMs with FSCWs due to
induced eddy currents caused by the armature
reaction
(assuming
sinusoidal phase currents) were (29)
tper = gcd
( Qs , p/2
)
where Qs is the number of stator slots, p the number of poles, and gcd( a, b) denotes the greatest
common denominator for a and b. Introducing mph as the (odd) number of phases, the (sinusoidal)
current in in phase n, n = 1, 2, . . . , mph , can be expressed as
Energies 2017, 10, 379
15 of 19
analyzed and compared. Provided that the harmonic order ν that dominates the PM losses is less or
equal to the number of poles p, it was found that that for V-shaped PMs, a uniform flux density variation
in the PMs can typically be assumed. Boundaries in terms of PM dimensions and angular frequency
were identified to aid the machine designer to decide whether the simplest loss model (Model A)
is applicable or not. An approximate analytical expression to these boundaries was also identified.
It was found that Model B and Model C, while relying on different formulations and solutions of the
Helmholtz equation, provide very similar loss predictions in very good agreement with corresponding
3D-FEM-based simulations. Further, tables were provided with resulting volumetric loss densities pm
for combinations of Qs and p commonly used in automotive applications. Finally, by compiling results
from previous publications, a complete description on how to analytically predict the magnitudes Bνm
and corresponding harmonic orders νm for FSCW-IPMs was provided in the Appendix.
Acknowledgments: The financial support from the Swedish Hybrid Vehicle Center (SHC) and the STandUP for
Energy research initiative is gratefully acknowledged.
Author Contributions: This paper is a result of the full collaboration of all of the authors. Hui Zhang and
Oskar Wallmark have contributed by developing ideas and the analytical model. Hui Zhang built the FEM models
and analyzed the data. Oskar Wallmark was responsible for the guidance and a number of key suggestions.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A. FSCW Fundamentals
How to predict the magnitudes Bνm and corresponding harmonic orders νm in FSCW-IPMs
(compiling results from [8,10,29,30]) is, for completeness, described in this Appendix.
Appendix A.1. Preliminaries
The winding periodicity tper can be expressed as [8]:
tper = gcd ( Qs , p/2)
(A1)
where Qs is the number of stator slots, p the number of poles and gcd( a, b) denotes the greatest
common denominator for a and b. Introducing mph as the (odd) number of phases, the (sinusoidal)
current in in phase n, n = 1, 2, . . . , mph , can be expressed as:
in =
√
2I cos
hp 2
θr − αph (n − 1) − ϕ
i
.
(A2)
Here, θr is the (mechanical) rotor position; ϕ is the current phase angle angle; and αph the angular
displacement of the air-gap MMF distribution due to each adjacent phase (αph = 2π/(mph tper )).
Appendix A.2. Air-Gap MMF Distribution Due to Stator Current
Assuming that all coils belonging to each phase are series connected, the resulting air-gap MMF
Fδ can be expressed as [31]:
Fδ =
∑
ν
(m
h i
ns Qs k ν in
∑ πmph ν cos ν θs − αph (n − 1)
n =1
ph
)
(A3)
where ns is the number of turns per slot, k ν the winding factor for harmonic ν and θs is an angle
spanning the air-gap circumference (expressed in mechanical degrees). Equation (A3) is valid if
magnetic saturation can be neglected, and it is derived assuming that the current in the stator slots can
be represented as point-like sources. The harmonic orders ν that can be present in the air gap can be
expressed as [8]:
Energies 2017, 10, 379
16 of 19
ν=
(
(2k − 1)tper , if Qs /tper is even
ktper , if Qs /tper is odd
(A4)
where k is an integer. Now, sgn(ν) is introduced to denote the direction of rotation of each MMF
harmonic ν with respect to the rotor. Following from (A3) (see [29] for further details), sgn(ν) can be
expressed as:
sgn(ν) =

 +1,
 −1,
if rem
if rem
p
2p
2
− ν, mph tper
+ ν, mph tper
(A5)
Note that (A5) is applicable only to harmonics ν that can be present in the air gap (defined
by (A4)).
Appendix A.3. Winding Factor for Harmonic ν
The winding factor for harmonic ν k ν can be expressed as [8,29]:

qph αph,ν
πν


sin
sin



2
Q


s ,


αph,ν


qph sin

2 kν =
qph αph,ν
πν


sin
sin



4
Q


s ,


q
α
ph
ph,ν


sin

2
2
if qph is odd
(A6)
if qph is even
where the electrical angle between the electromotive force of two adjacent coils αph and the number of
spokes per phase in the star of slot qph are given as:
2πν
round ( Qs /p)
Q
s
= Qs / mph tper .
αph,ν = π −
qph
(A7)
(A8)
A sufficient condition for (A6) to be valid is [29]:

Qs qph


round


p



,


q
ph
round ( Qs /p) =
Qs qph

 round


2p


,

q

ph

2
if qph is odd
(A9)
if qph is even
If (A9) is not fulfilled, the winding factor can then be computed by numerically computing the
harmonic content of Fδ as in [10,30].
Appendix A.4. PM Flux Density Variations
The pole-cap coefficient α p defined in Figure A1 is an approximate measure of how much flux
occurs across a complete pole pitch that is concentrated through the PMs.
Energies 2017, 10, 379
17 of 19
Remark A1. Note that the geometries depicted in Figure A1 are simplified without the presence of eventual
to fix the
PMs. This simplification will render the predicted flux-density magnitudes Bνm17 of 20
Version magnetic
February bridges
26, 2017used
submitted
to Energies
approximate (mainly due to flux-leakage drop the magnetic bridges). If required, more accurate predictions of
Bνm can rapidly be obtained using two-dimensional static FEM simulations.
A sufficient condition for (34) to be valid is [29]:
Now, ψm (θr ) (a function of the rotor position θr ) is introduced to denote the flux in a PM pole due

to Fδ . From the geometries depicted in
s qph that ψm (θr ) can be approximated as:
Figure A1, itQfollows
261
262
263
264

round


p



, if qph is odd


q Z
p ) π/p
µ0 lm rrph (θr +α
round (Qs /pψ)m=
( θr ) =
Fδ dθs
(A10) (37)
Qq

(δ + hm ) (sθr −ph
α p )π/p
round



2p


, if qph is even

q

where µ0 is the permeability of air. 
The corresponding
flux density
in a PM Bm (θr ) can now be
ph
approximated as:
2
ψm (can
θr ) then be computed by numerically computing the
If (37) is not fulfilled, the winding factor
Bm (θr ) =
2Cl
wm
m
harmonic content of Fδ as in [10,30].
Z (θr +α p )π/p
µ0 rr
F dθ
=
Appendix A.4. PM Flux Density Variations2(δ + hm )Cwm (θr −α p )π/p δ s
(A11)
The
pole-cap
coefficient
α pV-shaped
defined and
in Figure
is anPMs,
approximate
measure
of howorders
much flux
where
C = 1 and
C = 1/2 for
straightA.1
interior
respectively.
The harmonic
in Bm (pole
θr ) (of
orderthat
νm and
with a magnitude
Bνm ) are
obtained from (A11).
acrosspresent
a complete
pitch
is concentrated
through
thenow
PMs.
A1. Magnet
dimensions
anddefinition
definition of
of rotor-cap
αp α
(note
thatthat
for the
FigureFigure
A.1. Magnet
dimensions
and
rotor-capcoefficient
coefficient
forcoordinate
the coordinate
p (note
system
fixed
to
PMs,
the
y-axis
is
directed
in
the
paper):
(a)
V-shaped
interior
PMs;
(b)
system fixed to PMs, the y-axis is directed into the paper): (a) V-shaped interior PMs;straight
(b) Straight
interior
interior
PMs. PMs.
265
266
267
268
269
270
Remark A2. When utilizing (A4) and (A5) when inserting (A3) into (A11), it can be verified that the harmonic
Remark:
Note that the geometries depicted in Figure A.1 are simplified without the presence
orders present in the PMs νm can be expressed as:
of eventual magnetic bridges used to fixate the PMs. This simplification will render the predicted
(
flux-density magnitudes Bνm approximate
to flux-leakage
ν − p/2,(mainly
if sgn(ν)due
= 1 and
ν 6= kp/α p drop the magnetic bridges). If
νm =
required, more accurate predictions νof+Bp/2,
rapidly
be
obtained
using
two-dimensional (A12)
static FEM
if sgn
(ν) = −
1 and
ν 6= kp/α
νm can
p
simulations.
whereψk is(an
integer.
Now,
m θr ) (a function of the rotor position θr ) is introduced to denote the flux in a PM pole
due to Fδ . From the geometries depicted in Figure A.1, it follows that ψm (θr ) can be approximated as
ψm ( θ r ) =
Z (θr +α p ) π/p
µ0 l m rr
F dθs
(δ + hm ) (θr −α p )π/p δ
(38)
where µ0 is the permeability of air. The corresponding flux density in a PM Bm (θr ) can now be
approximated as
Energies 2017, 10, 379
18 of 19
Appendix B. IPM Parameters
Table A1. IPM parameters.
Parameter
Value
Unit
# poles (p)
# stator slots (Qs )
# turns per slot (ns )
Rated current (I)
Rotor radius (rr )
Air-gap length (δ)
Magnet height (hm )
Magnet conductivity (σm )
Magnet relative permeability (µr )
8
12
16
97
69.25
0.75
7.51
694
1.04
A (rms)
mm
mm
mm
kS/m
-
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Gu, W.W.; Zhu, X.Y.; Li, Q.; Yi, D. Design and optimization of permanent magnet brushless machines for
electric vehicle applications. Energies 2015, 8, 13996–14008.
Lim, S.; Min, S.; Hong, J.-P. Optimal rotor design of IPM motor for improving torque performance considering
thermal demagnetization of magnet. IEEE Trans. Magn. 2015, 51, 1–5.
Ahn, K.; Bayrak, A.E.; Papalambros, P.Y. Electric vehicle design optimization: Integration of a high-fidelity
interior-permanent-magnet motor model. IEEE Trans. Veh. Technol. 2015, 9, 3870–3877.
Kato, T.; Minowa, M.; Hijikata, H.; Akatsu, K.; Lorenz, R.D. Design methodology for variable leakage flux
IPM for automobile traction drives. IEEE Trans. Ind. Appl. 2015, 51, 3811–3821.
Wallscheid, O.; Böcker, J. Global identification of a low-order lumped-parameter thermal network for
permanent magnet synchronous motors. IEEE Trans. Energy Convers. 2016, 31, 354–365.
EL-Refaie, A.M.; Jahns, T.M. Application of bi-state magnetic material to an automotive IPM starter/
alternator machine. IEEE Trans. Energy Convers. 2005, 20, 71–79.
Yang, Z.; Brown, I.P.; Krishnamurtht, M. Comparative study of interior permanent magnet, induction, and
switched reluctance motor drives for EV and HEV applications. IEEE Trans. Transport. Electr. 2015, 1,
245–254.
Bianchi, N.; Dai Pré, M. Use of the star of slots in designing fractional-slot single-layer synchronous motors.
IET Electr. Power Appl. 2006, 153, 459–466.
Skaar, S.E.; Krøvel, Ø.; Nilssen, R. Distribution, coil-span and winding factors for PM machines with
concentrated windings. In Proceedings of the International Conference on Electrical Machines (ICEM),
Chania, Greece, 2–5 September 2006; p. 346.
Bianchi, N.; Bolognani, S.; Dai Pré, M.; Grezzani, G. Design considerations for fractional-slot winding
configurations of synchronous machines. IEEE Trans. Ind. Appl. 2006, 42, 997–1006.
Aslan, B.; Semail, E.; Korecki, J.; Legranger, J. Slot/pole combinations choice for concentrated multiphase
machines dedicated to mild-hybrid applications. In Proceedings of the IEEE Industrial Electronics Society
(IECON), Melbourne, Australia, 7–10 November 2011; pp. 3698–3703.
Bianchi, N.; Fornasiero, E. Index of rotor losses in three-phase fractional-slot permanent magnet machines.
IET Electr. Power Appl. 2009, 3, 381–388.
Zheng, P.; Wu, F.; Lei, Y.; Sui, Y.; Yu, B. Investigation of a novel 24-slot/14-pole six-phase fault-tolerant
modular permanent-magnet in-wheel motor for electric vehicles. Energies 2013, 6, 4980–5002.
Bianchi, N.; Bolognani, S.; Fornasiero, E. An overview of rotor losses determination in three-phase
fractional-slot PM machines. IEEE Trans. Ind. Appl. 2010, 46, 2338–2345.
Tangudu, J.; Jahns, T. Comparison of interior PM machines with concentrated and distributed stator windings
for traction applications. In Proceedings of the IEEE Vehicle Power and Propulsion Conference (VPPC),
Chicago, IL, USA, 6–9 September 2011; pp. 1–8.
EL-Refaie, A.M.; Bolognani, S.; Fornasiero, E. Fractional-slot concentrated-windings synchronous permanent
magnet machines: Opportunities and challenges. IEEE Trans. Ind. Electron. 2010, 57, 107–121.
Energies 2017, 10, 379
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
19 of 19
Malloy, A.C.; Martinez-Botas, R.F.; Lampérth, M. Measurement of magnet losses in a surface mounted
permanent magnet synchronous machine. IEEE Trans. Energy Convers. 2015, 30, 323–330.
Zhang, P.; Sizov, G.Y.; He, J.; Ionel, D.M.; Demerdash, N.O.A. Calculation of magnet losses in concentrated
winding permanent magnet synchronous machines using a computationally efficient finite element method.
IEEE Trans. Ind. Appl. 2013, 49, 2524–2532.
Aslan, B.; Semail, E.; Legranger, J. General analytical model of magnet average eddy-current volume losses
for comparison of multiphase PM machines with concentrated winding. IEEE Trans. Energy Convers. 2014,
29, 72–83.
Paradkar, M.; Böcker, J. 3D analytical model for estimation of eddy-current losses in the magnets of IPM
machine considering the reaction field of the induced eddy currents. In Proceedings of the IEEE Energy
Conversion Congress and Exposition (ECCE), Montreal, QC, Canada, 20–24 September 2015; pp. 2862–2869.
Yamazaki, K.; Fukushima,Y. Effect of eddy-current loss reduction by magnet segmentation in synchronous
motors with concentrated windings. IEEE Trans. Energy Convers. 2011, 47, 779–788.
Chalmers, B.J. Electromagnetic Problems of A.C. Machines; Chapman and Hall: London, UK, 1965; pp. 66–87.
Sadarangani, C. Electrical Machines—Design and Analysis of Induction and Permanent Magnet Motors;
Universitetesservice: Stockholm, Sweden, 2006; pp. 543–658.
Ruoho, S.; Santa-Nokki, T.; Kolehmainen, J.; Arkkio, A. Modeling magnet length in 2-D finite-element
analysis of electric machines. IEEE Trans. Magn. 2009, 45, 3114–3120.
Krakowski, M. Eddy-current losses in thin circular and rectangular plates. Archiv für Elektrotechnik 1982, 64,
307–311.
Sinha, G.; Prabhu, S.S. Analytical model for estimation of eddy current and power loss in conducting plate
and its application. Phys. Rev. Spec. Top. Accel. Beams 2011, 14, 062401.
Mukerji, S.K.; George, M.; Ramamurthy, M.B.; Asaduzzaman, K. Eddy currents in solid rectangular cores.
Prog. Electromagn. Res. B 2008, 7, 117–131.
Nategh, S.; Wallmark, O.; Leksell, M.; Zhao, S. Thermal analysis of a PMaSRM using partial FEA and lumped
parameter modeling. IEEE Trans. Energy Convers. 2012, 27, 477–488.
Zhang, H.; Wallmark, O.; Leksell, M.; Norrga, S.; Harnefors, M.N.; Jin, L. Machine design considerations
for an MHF/SPB-converter based electric drive. In Proceedings of the IEEE Industrial Electronics Society
(IECON), Dallas, TX, USA, 29 October–1 November 2014; pp. 3849–3854.
Mayer, J.; Dajaku, G.; Gerling, D. Mathematical optimization of the MMF-function and -spectrum in
concentrated winding machines. In Proceedings of the International Conference on Electrical Machines and
Systems (ICEMS), Beijing, China, 20–23 August 2011.
Zhao, L.; Li, G. The analysis of multiphase symmetrical motor winding MMF. Electr. Eng. Autom. 2013, 2,
11–18.
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