Skin effect and Proximity Losses in High Speed
Brushless Permanent Magnet Motors
Mircea Popescu
David G. Dorrell
Motor Design Ltd
Elsmere, UK
mircea.popescu@motor-design.com
University of Technology Sydney
Sydney, Australia
david.dorrell@uts.edu.au
Abstract—This paper investigates eddy current and proximity
losses in windings of different high speed brushless permanent
magnet machines. Several papers have already addressed this
but in this paper different topologies and winding connections
are investigated to give a comparative study. This is done
different operating speeds and loadings. This is important
because more machnes are being developed to address the
requirements of applications such as automotive electric
drives. These often operate at the thermal limits of the machine
at high current density during the transient torque cycling.
winding will be tight so that strands will not change position
between coil sides. This can generate higher losses.
I. INTRODUCTION
There is an increasing use of variable high-speed
brushless permanent magnet machines. Traditionally it was
considered that machines with air-gap windings were most
liable to eddy current losses, hence the use of Litz wire in
these machines, which is expensive. However, with the
development of high performance rare-earth magnet
machines, which have very wide speed ranges, then even
windings in slots can be prone to additional ac copper losses.
These machine are typically found in applications such as
hybrid and electric vehicles. Parallel strands are often used in
addition to Litz wire [1] but this can lead to circulating
currents in some circumstances [2], and if many strands are
used then this can reduce the fill factor although the cooling
efficiency can be improved.
In addition, when the parallel paths of a coil turns are
located in different parts of the slot they will experience a
proximity effect with circulating current between strands and
also eddy currents in individual strands since there may be
alternative flux linkage between them from the rotor magnets
and electric loading leakage and mutual coupling. This
should be differentiated from skin effect in windings as
described in [3]. The differences are noted in [4] and [5];
Fig. 1 illustrates these in a similar manner where individual
effects on strands (skin effect) and bundles of parallel strands
(proximity) are considered. Even with thin wire to eliminate
skin effect, proximity effects will exist. In [3], it was
illustrated that the turns can be divided into bundles of
parallel wires (strands-in-hand) where the series turns are
stacked in the slot and the coils spans only one slot; the end
Figure 1. Skin and proximity effects on coils.
There are several papers that address losses in motor
windings. Several use analytical calculations [6]-[12] while
others use hybrid finite element analysis (FEA) and
analytical methods or do comparisons of the two [13]-[19].
Some of these papers illustrated that more recently it has
been possible to use time-stepped FEA to calculate the losses
in individual conductors. There are now publications that
only use time stepped FEA in the analysis since there is now
a confidence in the modeling solution [2][3][20][21].
There is a drive to use an integrated design solution to
motor design as illustrated in [22]. Increasing computational
power means that individual conductors can be simulated
and calculated. This can also include shell modelling of the
conductor bundling to assess heat transfer [23]. The work in
this paper assumes constant temperature but it can be
included in an iterative simulation scheme when coupled
with a thermal simulation.
In this paper the variation of losses is investigated in
different machine topologies with different connections and
positioning of conductors in slots. This is for different
loadings and speeds. This is relevant to many modern
applications. The modeling use Cedrat Flux 2D [24]
although similar packages could be utilized.
II.
MOTORS DESIGNS
The analysis is carried out on two different machines.
The cross sections of these are given in Fig. 2 They represent
a high speed gas turbine generator and a hybrid electric
vehicle drive similar to a Toyota Prius machine.
conductors are the bottom of the slot and one has the
conductors at the top. This machine will have substantial
resin content for insulation and thermal conduction of the
copper loss.
The winding arrangement for the gas turbine machine in
the Flux 2D model is shown in Fig. 3. Two phases are kept
as simple bulk conducting regions with the resistance set to
the DC value. One phase has the coil broken down into
individual conductors as illustrated. This allows the ac losses
to be assessed and calculated and direct comparison made
between the ac and dc losses. This is a valid simulation
method because the machine is current fed and it can be seen
that the machine is fed by two current sources in the three
phase connection. A third source is not needed. Magnet eddy
current loss is not considered because this is not the focus of
this paper.
“Go” conductors in one slot
“Return” conductors in one slot
Current
sources
Phase B winding
Phase C winding
Figure 3. Flux 2D connection for gas turbine machine
Figure 2. Different motor topologies
The gas turbine generator design is shown in Fig. 2(a).
This runs up to a speed of 150000 rpm. This is essentially a
generator but to help starting it can be used in motoring
mode. There are 18 conductors in each slot as illustrated.
These have a diameter of 2.34 mm so that the cross section is
4.3 mm2. For this machine the conductors are series and with
forced gas or liquid cooling so that the rated conductor
density could be up to about 15 A/mm2 rms with a conductor
current is up to 64.5 A rms. This gives a peak current of 91.2
A for sinusoidal operation. Therefore three peak current
levels are considered which are within the rated values: 0, 50
and 100 A, with one transient over-current at 150 A and one
severe fault current of 200A. Two winding scenarios are
considered for the winding in the slots as shown. One has the
The Hybrid vehicle machine is shown in Fig 2(b). This is
similar to the Toyota Prius Hybrid machine although it is not
the actual machine. The 2nd Generation Prius machine
operated up to 6000 rpm and has been commonly used in
hybrid drive motor benchmarking comparisons [25]. The
later 3rd Generation machine reduced the magnet content and
increased the speed to 13900 rpm [26]. This machine lies
somewhere between the two.
In the design here each slot contains 8 conductors. It can
be connected in different configurations. As more parallel
connections are made, the voltage is reduced and the current
increased. The conductors are 3.76 mm in diameter. This is a
high value and more likely to be split up into several strands.
However, we are attempting to illustrate eddy current and
proximity issues. Thick conductors can be used if there is a
need to run fluid ducts around them for cooling. In these
types of machines the current can run very high, and up to 25
A/mm2 rms in a transient acceleration is quite possible (with
series conductors), but this is only up to the base speed. With
the conductors used (11 mm2) then the peak transient rms
current can be set to 250 A. Typically this will be at about
1/4th or 1/5th of the maximum speed because they have an
extended torque/speed range [25]. Therefore this will be at
about 3000 rpm. At maximum speed that current will be
about 70 A rms since there is a constant power range from
3000 rpm to 12000 rpm. In steady-state the current will be
substantially lower than 250 A even at low speed.
150000 rpm at 0, 50 and 150 A peak with the conductors in
the slot bottoms. It can be seen that there is little proximity
current when there is no phase current. When there is 50 and
150 A peak in the phase winding then it can be seen that
there are eddy and proximity currents in the top layer of
turns. The scales give the level and the top and bottom and
edges are high.
(a) No phase current
(b) Phase current = 50 A peak
Figure 4. Flux 2D connection for vehicle drive motor.
III.
ANALYSIS
In this section the two designs are analyzed, the two pole
machine in terms of the conductor positioning over a range
of speeds and currents and the hybrid vehicle addresses
parallel connections.
A. Two pole turbine generator.
This machine runs up to 150000 rpm. The current density
can also be high since this is a high performance gas turbine
machine that may have forced fluid cooling. The two
different positionings of the discretized coil are shown in
Fig. 2(a). Fig. 5 shows the current density distributions at
(c) Phase current = 150 A peak
Figure 5. Winding current density at 150000 rpm for different currents –
conductors in slot bottom.
In Fig. 6 the exercise is repeated with the conductors at
the top of the slots. It can be seen that now there are
proximity currents in the conductors at the top of the slot
(a) No phase current
(b) Phase current = 50 A peak
(c) Phase current = 150 A peak
Figure 6. Winding current density at 150000 rpm for different currents –
conductors in slot top.
The instantaneous Joule loss at 150000 rpm and phase
currents of 50 and 100 A peak are shown in Fig. 7. These are
compared to the dc copper loss. Obviously the dc loss will
cycle between a maximum and zero twice per current cycle
as illustrated (180 mech. deg. is half a current cycle). “dc
loss” is a misnomer to some extent. It implies there are no
eddy or proximity losses rather than carrying dc current. It
can be used to compare to the “ac loss” in the phase where
there are eddy and proximity losses. These show that there is
continuous loss in the conductors since the eddy and
proximity EMFs that drive in the currents are not necessarily
in-phase with the main back-EMF, nor do they necessarily
have the same frequency. Bear in mind that these are the
summed proximity losses from all the individual conductors
where the eddy and proximity loss currents in individual
conductors will have different phase, magnitude and possibly
frequency.
Joule loss (instantaneous) per phase [W]
even when there is no current. When there is 50 and 150 A
peak phase current then inspection of the scales relative to
Fig. 5 illustrates there is somewhat more eddy and proximity
current and loss. These would lead to conductor hot spots
particularly in the top center conductor.
I = 50 Apk, N = 150 K rpm, Bottom; 50 W
I = 50 Apk, N = 150 K rpm, Top; 88 W
I = 100 Apk, N = 150 K rpm, Bottom; 236 W
I = 100 Apk, N = 150 K rpm, Top; 343 W
I = 50 Apk, DC loss; 36 W
I = 100 Apk, DC loss; 145 W
500
450
400
350
300
250
200
150
100
50
0
0
45
90
135
Rotor rotation [mech deg]
180
Figure 7. Instantaneous joule, or copper, losses under DC condistiosn and
at 150000 rpm for condustros in slot bottom na slot top. This is per phase.
In Fig. 8 the Joule losses per phase are expressed for the
different values of current over the full speed range from
1000 rpm up to 150000 rpm. What should be remembered
about these losses is that they are only a small percentage of
the output power since this is a high speed and high
efficiency machine, but they do represent a substantial
amount of loss for the size of machine (it has an outer
diameter of 64 mm and a core length of 60 mm). In this case
it is not the percentage loss that is relevant, rather the actual
loss that the cooling system is able to cope with and the
individual hot spots that occur. In the figure, it can be seen
that when the turns are at the slot top there is a substantial
increase in losses and these are mostly in the conductors at
the top of the slot. This can create hot spots. In Fig. 9 the
instantaneous Joule loss is shown for center conductors at the
top and the bottom of the slot. It can be seen that in the
center top location with the coil bundle located at the top of
the slot, the conductor has very high losses and this would
create a hot spot and a weakness. Litz wire or use of thinner
wire with several strands in hand would help reduce this.
However, the exercise here is to illustrate that there can be
high eddy and proximity losses in a high speed machines
even when the slots have narrow openings as is the case here.
I = 50 A pk, Bottom
I = 50 A pk, Top
I = 100 A pk, Bottom
I = 100 A pk, Top
I = 150 A pk, Bottom
I = 150 A pk, Top
1400 I = 200 A pk, Bottom
I = 200 A pk, Top
Pmax (IPM) = Pmax (Prius)
2
⎛ Volrotor (IPM) ⎞
×⎜
⎟
⎝ Volrotor (Prius) ⎠ Lstk =constant;
Slot area increases
with square of radius
Joule loss per phase [W]
1200
( Magnet weight/Vol )rotor (IPM)
×
( Magnet weight/Vol )rotor (Prius) Same
1000
magnet
800
Base speed (IPM)
×
Base speed (Prius)
600
400
200
0
0
50000
100000
Speed [rpm]
150000
Figure 8. Variation of Joule loss at different speeds and current.
Conductor 16 (Top) 100 A pk, 150000 rpm; 27 W
Conductor 16 (Bottom) 100 A pk, 150000 rpm; 8 W
70 Conductor 4 (bottom) 100 A pk, 150000 rpm; 2 W
and the parameters for the machines are given in Table I.
This shows that the rating of the machine is about 215 KW.
The approximate operating envelope is shown in Fig. 10 and
the base speed is about 3000 rpm. The scaling exercise
carried out earlier is very much an approximation. When the
current is set to 100 A peak (70.7 A rms) the torque is about
160 Nm which gives 200 KW of power at 12000 rpm. This is
for a series connected machine. When parallel paths are used
the current is multiplied by the number of parallel paths.
TABLE I.
IPM MOTOR PARAMENTERS.
60
Parameter
50
Stator OD [mm]
Core length [mm]
Rotor OD [mm]
Rotor volume [mm2]
Magnet weight [Kg]
Magnet weight per rotor vol. [Kg/mm3]
Speed range [rpm]
Base speed [rpm]
Maximum power rating [KW]
40
30
20
10
0
0
45
90
135
180
Rotor rotation [mech deg]
Figure 9. Instantaneous Joule losses for the centre conductor 4(bottom of
slot) and 16 (top of slot).
B. 8 Pole Hybrid Electric Vehicle Drive Motor
The Hybrid electric vehicle was simulated to address the
effects of parallel paths. As already discussed previously the
machine has torque speed curve that has a narrow maximum
torque range and an extended maximum power range. Up to
the base speed these machines can operate at high current
density (possibly up to 25 A/mm2 across the bulk conductor
material) but only in the transient acceleration state for a few
seconds and they have fluid cooling.
The example machine here is similar to the Toyota Prius
although not identical. The diameter is 282 mm while the
axial length is 84 mm. This is a different size. In order to rate
the machine then a comparison was carried out. The
maximum power was estimated using
Torque [Nm] or Mech Power [kW]
Conductor Joule loss (instantaneous) [W]
(1)
IPM motor
studied
here
282
84
184
2.23 × 106
1.14
6.28 × 10-6
12000
3000
215
2nd Generation
Toyota Prius
machine
269
84
160.4
1.70 × 106
1.05
6.29 × 10-6
6000
1200
50
800
700
Power
600
Torque
500
400
300
200
100
0
0
3000
6000
9000
12000
Speed [rpm]
Figure 10. IPM motor approximate operating envelope.
The most relevant operating point is at high speed and
simulations were carried out at 12000 rpm and 100 A peak
phase current (for series connection). Fig. 11 shows the
instantaneous Joule loss in the coil represented by discrete
turns. The mean power loss is given in the legend. It can be
seen that as parallel connections are increasingly used, the
Joule loss increases. For comparison, the dc Joule loss is also
give. And it can be seen that there is a substantial increase in
copper loss whatever the connection is. When there are 8
parallel paths the loss increases by nearly ten times
compared to the dc value.
DC loss - 24 W
Series discrete conductors - 134 W
Disrete conductors - 2 parallel paths - 164 W
Disrete conductors - 4 parallel paths - 185 W
Disrete conductors - 8 parallel paths - 205 W
500
To examine the lower speed operation then Fig. 13
illustrates that the series connection loss is 40 W when the
speed is reduced to 3000 rpm. This is still a AC/DC loss ratio
of 1.67 compared to 5.58 at 12000 rpm. Parallel connection
gives a ratio of 3.58. When the current is increased to give
maximum torque at 3000 rpm the ratio is now 1.42 for series
connection. These machines operate at margins of thermal
stability at high torque and this could be very significant;
hence, thinner conductors should be used.
450
400
Instantaneous coil loss [W]
The current density distribution is examined in Fig. 12.
Only the series and 8 parallel path connections are examined.
The legend shows that the parallel path connection has much
higher current density in the top conductor compared to the
series connection due to circulating currents between the
parallel legs. Hence the loss increases from 134 W to 205 W.
350
300
250
DC loss - 24 W
12000 rpm Series discrete conductors - 134 W
3000 rpm Series discrete conductors - 40 W
3000 rpm Parallel discrete conductors - 86 W
200
150
350
100
300
0
0
30
60
Rotor position [mech deg]
90
Figure 11. Instantaneous coil loss at 12000 rpm for different connections –
current = N × 100 A peak where N is the number of parallel paths. dc loss
included for reference.
Instantaneous coil loss [W]
50
250
200
150
100
50
0
0
30
60
Rotor position [mech deg]
90
Figure 13. Instantaneous coil loss at 3000 rpm with seres and parallel
connections compared to 12000 rpm amd DC losses.
DC loss at 250 A rms - 299 W
3000 rpm Series discrete conductors - 250 A peak - 426 W
800
(a) Current density on load with series connection
700
Instantaneous coil loss [W]
600
(b) Current density on load with 8 parallel connections
Figure 12. Current dersnity distribution at 12000 rpm with series and
parallel turns and 100 A peak load.
500
400
300
200
100
0
0
30
60
Rotor position [mech deg]
90
Figure 14. Instantaneous coil loss at 3000 rpm and high torquwe (I = 250 A
rms).
The current density is examined in Fig. 15 at 3000 rpm.
The legend illustrates that the current is more evenly
distributed at the lower speed when series connected
compared to the high speed operation. This is to be expected.
Series discrete conductors - open circuit - 3.46 W
16
Parallel discrete conductors - open circuit - 5.45 W
14
Instantaneous coil loss [W]
12
(a) Current density at 3000 rpm – Series connection
10
8
6
4
2
0
0
30
60
90
Rotor position [mech deg]
Figure 16. Instantaneous coil power loss on open circuit at 12000 rpm with
series and parallel coil turns.
(b) Current density at 3000 rpm – 8 parallel paths
(a) Current density on open circuit with series connection
(c) Current density at 3000 rpm – Series connection and high
250 A high torque current
Figure 15. Current density at low speed and high torque with series and
parallel connections.
It is also worth examining the eddy and proximity losses
in the machine when running at 12000 rpm and on open
circuit. The results of a comparison between series
connection and 8 parallel path connection is shown in Fig.
16. The loss is relatively low, even for parallel connection.
This suggests the effects of the magnets on the parallel
connections is relatively low and it is mostly eddy current
and proximity effects rather than circulating currents
between parallel paths that causes the increase in copper loss.
This is illustrated in Fig. 17 where there is a difference but
not a large difference in current distribution. The current
density is still highest in the top turns.
(b) Current density on open circuit with 8 parallel connections
Figure 17. Current density distribution on open circuit at 12000 rpm.
C. Comments
Both of these machine designs exhibit high levels of eddy
and proximity losses and they was a deliberate design flaw.
This is because they contain thick single conductors and the
aim of the paper is to illustrate these are susceptible to these
effects. In reality the conductor bundles in both machines
would be broken down into multiple strands-in-hand and
parallel paths. The 2 pole generator illustrates that even with
semi-closed slots, and copper losses that appear to be low
compared to the output power, the conductors can still have
hot spots that are potential failure points. The hybrid
machine likewise illustrates that that the thick conductors
lead to high eddy and proximity losses. It also illustrates that
parallel strands can lead to circulating currents.
IV. CONCLUSIONS
With the developments in wide speed-range drive motors
for automotive applications that are often very thermally
critical the losses in the windings of such machines has to be
carefully considered. Thermal analysis is now being broken
down into conductor-level simulation and this will lead to
greater understanding of the heat loss distribution. This will
allow for possible hot-spots to be identified and better design
to be carried out.
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