IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 9, SEPTEMBER 2011
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A Review of the Design Issues and Techniques for
Radial-Flux Brushless Surface and Internal
Rare-Earth Permanent-Magnet Motors
David G. Dorrell, Senior Member, IEEE, Min-Fu Hsieh, Member, IEEE, Mircea Popescu, Senior Member, IEEE,
Lyndon Evans, David A. Staton, Member, IEEE, and Vic Grout, Senior Member, IEEE
Abstract—This paper reviews many design issues and analysis
techniques for the brushless permanent-magnet machine. It reviews the basic requirements for the use of both ac and dc machines and issues concerning the selection of pole number, winding
layout, rotor topology, drive strategy, field weakening, and cooling.
These are key issues in the design of a motor. Leading-edge design
techniques are illustrated. This paper is aimed as a tutor for motor
designers who may be unfamiliar with this particular type of
machine.
Index Terms—Analysis, brushless permanent-magnet (PM)
motors, design, internal PM (IPM), torque.
I. I NTRODUCTION
T
HERE ARE MANY excellent books on the design of
brushless permanent-magnet (PM) motors. Examples of
well-known and established texts are given in [1]–[5], while
more recently, there have been tutorials at leading international
conferences with accompanying Course Notes Texts [6]. These
cover dc and ac motors and mostly cover the design of ferritemagnet machines although rare-earth machines are also covered. Materials are discussed in a variety of specialized texts;
these include magnets [7], [8], steels [9], [10], and insulation
systems [11]. Noise is also covered by several texts [12], [13].
This list is far from comprehensive, and there are many other
monographs that cover specialist aspects of electric motor
operation that are relevant to brushless PM motors. There is
still relatively little on the thermal design of electrical machines
in terms of texts although the number of technical papers
is increasing; illustrations of this are [14] and [15], while
Manuscript received April 14, 2010; revised July 23, 2010; accepted
October 1, 2010. Date of publication October 28, 2010; date of current version
August 12, 2011.
D. G. Dorrell is with the School of Mechanical, Electrical and Mechatronic
Systems, University of Technology Sydney, Sydney, N.S.W. 2007, Australia
(e-mail: ddorrell@eng.uts.edu.au).
M.-F. Hsieh is with the Department of Systems and Naval Mechatronic
Engineering, National Cheng Kung University (NCKU), Tainan 701, Taiwan
(e-mail: mfhsieh@mail.ncku.edu.tw).
M. Popescu, L. Evans, and D. A. Staton are with Motor Design Ltd.,
SY12 9DA Shropshire, U.K. (e-mail: mircea.popescu@motor-design.com;
lyndon.evans@motor-design.com; dave.staton@motor-design.com).
V. Grout is with the Centre for Applied Internet Research, Glyndwr University, LL11 2AW Wrexham, U.K. (e-mail: v.grout@glyndwr.ac.uk).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2010.2089940
[16]–[18] show coupled electromagnetic and thermal considerations in PM machines. In recent years, there have been many
papers that cover various aspects of the electromagnetic design
on rare-earth PM motors; for instance, [19]–[25] show recent
papers on PM-motor design in a variety of situations.
The aim of this paper is not to highlight particular design
aspects of one form of brushless PM motor but rather to give
an overview of many of the factors dictating option selection
and design solutions. Therefore, in this paper, the key design
points related to the design of brushless rare-earth PM machines are outlined and solutions are discussed. Techniques for
analysis are outlined, and these should be useful to a machine
designer who is unfamiliar with this particular type of machine.
Section II will consider electromagnetic and structural issues, while Section III will discuss thermal considerations.
Section IV will put forward analysis techniques. Design examples are included in the discussions.
II. I NITIAL E LECTROMAGNETIC D ESIGN C HOICES
In this section, some basic design choices are discussed.
These are necessary at the outset of the design procedure.
A. Radial or Axial Flux?
Generally, most PM motors are of the radial-flux type. The
reason for this is that fabrication is straightforward and established, using slotted stators with standard round radial laminations, and the electrical loading can be maximized because
of the use of the slots. However, there are good examples of
using axial-flux machines, and the design of these machines
is discussed in [26]. In these machines, the windings tend to
be air-gap windings (although they can have teeth [27]) which
can limit the amount of copper that can be used and, hence,
can limit the amount of loading possible. The windings tend
to be specially formed and shaped, and often, Torus windings
are used; Mendrela et al. [27] review different options for
this type of machine. Axial-flux machines are often used as
motors although they have many advantages (usually related
to their low armature reactance) in the area of generation
[28], particularly in wind generation [29]. However, axial-flux
applications can still be considered as niche, and the focus of
this paper will be on radial-flux laminated motors since these
constitute the majority of brushless PM motors.
0278-0046/$26.00 © 2010 IEEE
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 9, SEPTEMBER 2011
TABLE I
TYPICAL TRVs [1]
an engineering approach to sizing the rotor. The sheer stress
allows a more fundamental stress-limiting calculation, as shown
in the Appendix, based on the electric loading and the air-gap
flux-density limits. As can be seen in Table I, there is a wide
variation in TRV—a median for first-pass design of a larger,
efficient, and well-designed rare-earth magnet machine would
be about 40 kN · m/m3 , which is a sheer stress of 2 N/cm2 .
B. Ratings, Motor Classes, and TRV
C. AC or DC Control?
The rating of a machine is important and will dictate the size
and design demands for a machine. The torque-per-unit-rotor
volume (TRV) is a useful guide to how “good” a machine is.
The TRV is related to the tangential stress by
Brushless PM motors generally fall into two classes: ac and
dc. There are different requirements when designing them, and
this is related to the back-EMF waveform and the rotor-position
sensing. Consider a three-phase operation. For ac operation, the
phase current will be sinusoidal, and there is a 180◦ conduction
for each inverter leg using a pulsewidth-modulation strategy
with a position encoder. For dc control, the current waveform is
trapezoidal with 120◦ conduction with three Hall effect probes
usually used to detect the switching positions. Hence, an ac
machine requires sinusoidal back EMF generated by the PM
rotor, while the dc machine requires a more trapezoidal backEMF waveform. Some machines have back-EMF waveforms
that are intermediate and can be used with either ac or dc control. Generally, dc motors are suitable for power drives which
can tolerate some torque ripple and do not require substantial
field weakening at higher speeds, while ac motors are more
suitable for servo drives where smooth operation and extended
field weakening are required. DC control can offer a higher
power density, and this is illustrated in the Appendix. The
characteristics for DC and AC operations can be summarized.
The following are the characteristics of dc operation:
1) full-pitched and concentrated windings for trapezoidal
back EMF;
2) higher power density;
3) Hall effect probes to detect the correct current switching
positions (low cost);
4) suitable for power drives.
The following are the characteristics of ac operation:
1) distributed and fractional-slot windings for sinusoidal
back EMF and smooth operation;
2) better control and extended field weakening;
3) shaft encoder to control current (high cost);
4) suitable for servo drives and drives requiring excellent
field-weakening capabilities.
There are several strategies to make the machines sensorless
(no Hall probes or shaft encoder) although the norm in industrial applications is still to use position feedback.
Generally, the current phasor from the three-phase-winding
current set should be located on the rotor q-axis unless field
weakening is used. This is used above the base speed when,
essentially, the inverter voltage has reached its maximum where
the current cannot be controlled and the maximum current
cannot be achieved. The inverter switching is advanced, and this
can be effective up to maybe 15–20 electrical degrees depending on the machine. This is shown in Fig. 1 for a small four-pole
dc-controlled machine [shown later in Fig. 6(b)]. It can be seen
that the torque range is extended from about 2500–3000 r/min.
TRV = 2σmean
(1)
where σmean is the sheer stress on the rotor (in newtons per
square meter). The sheer stress will be discussed later. Common
limits for the TRV in various machines are quoted in [1], and
these are listed in Table I. However, it can be seen that, generally, the larger and better cooled the machine, the higher the
TRV. In totally enclosed fan-cooled machines, typical windingcurrent-density levels are in the region of 5–6 A/mm2 . This
limits the electric loading and, hence, stress, which results in
a low-range TRV. Larger water- or oil-cooled machines can
push this much higher. In electric vehicle (EV) and hybrid EV
drive motors [30], the peak power rating is a transient rating
at lower speeds, and the current density during a transient (or
acceleration) period can be in excess of 20 A/mm2 for a period
of several seconds or tens of seconds. Some basic motor types
are listed in Table I although, at this stage, no distinction is
made between ac- and dc-controlled brushless PM machines.
These volumes can be used to calculate an approximate rotor
size. However, initially, a diameter has to be selected based on
the choice of pole number, magnet size, and rotor topology.
The geometry may also be dictated by the space in which the
motor has to fit. Starting with a two-pole motor geometry, the
diameter-to-axial-length ratio will be close to unity and will
increase with pole number (moving from a long cylindrical
shape to a disk shape). This is a crude sizing approximation
for radial-flux machines over a wide power range. The first key
point to remember is that the stator yoke thickness is governed
by the flux per pole (since it has to carry this); therefore,
it decreases as the pole number increases. High-pole-number
machines tend to have a much higher diameter compared with
the axial length. In totally enclosed machines, the TRV tends
to be in the range of 7–14 kN · m/m3 for small ferrite-magnet
motors, 20 kN · m/m3 for bonded Nd–Fe–B magnets, and
14–42 kN · m/m3 for rare-earth magnets, and it is hard to
increase beyond this without using a very specialized topology.
If high-energy magnets are used, then high-efficiency machines
can be designed, and also, it allows the motor to be more
compact. When Nd–Fe–B magnets are utilized, it is reasonable
to expect a peak electromagnetic efficiency of over 90% even
on smaller machines.
In terms of the sheer or tangential air-gap stress, (1) shows a
direct relationship to the TRV, as proved in [1]. The TRV gives
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Fig. 1. Field weakening (phase advance) for a small surface-magnet dc
machine.
Fig. 2. Surface- and interior-PM four-pole rotors with red and blue magnets
oppositely polarized. Gray areas denote the laminated core. Red and blue
areas are oppositely magnetized magnets. (a) Nonsalient surface-magnet rotor.
(b) Salient interior-magnet rotor.
Phase advance has the effect of weakening the main field by
rotating the current phasor so that there is a component on
the −d-axis. AC-controlled machines with internal-PM (IPM)
rotors can have much higher field-weakening capability, and the
machine used in [30] has 60◦ phase advance—this is studied
later. IPM motors can have considerable reluctance torque as
well as excitation torque. The machine in [30] is required to
have a wide field-weakening capability because the base speed
is 1500 r/min, whereas the maximum speed is 6000 r/min.
D. Choice of Rotors
There are many possible topologies for the rotor—too many
to comprehensively list here. They lie in two basic topologies.
One has surface magnets with little saliency, which are common
in dc motors as already mentioned (although they are also often
used in ac motors), while the second has embedded magnets
and considerable saliency. Fig. 2 shows some examples of
these. Many of these topologies can be simulated in the SPEED
simulation package from the University of Glasgow, U.K., and
Miller [31] lists many brushless PM-motor rotor arrangements.
For a surface-magnet nonsalient rotor, Xd = Xq , as shown
in Fig. 2(a). Embedded magnets are possible in the rotor, as
shown in Fig. 2(b). These are used in ac machines, although
they can be used in dc machines. They have q-axis saliency
(i.e., Xq > Xd ). The advantage of this is that the peak torque is
moved from the q-axis to an angle of about 100–120 electrical
degrees away from the d-axis. This means that if there is a
transient overload when the current is on the q-axis, there
Fig. 3. Phasor diagram and equivalent circuits for brushless permanent ac
machines. (a) Phasor diagram for salient-pole PM motor—the q-axis is often
taken as the vertical-axis reference (in surface-magnet rotors, Xe = Xq ).
(b) Per-phase equivalent circuit for nonsalient PM motor. (c) d–q-axis equivalent circuits for salient-pole PM motor.
will be extra torque available to bring the motor back to the
correct firing angle, preventing pole slipping. The saliency also
offers additional reluctance torque, and this is illustrated by the
example in Section IV-B.
The phasor diagram for the two types of rotor is shown
in Fig. 3(a) (assuming ac control). This is a general case in
steady state; the difference in operation is that if there is no
saliency, then Xd = Xq and the steady-state circuit in Fig. 3(b)
can be utilized. If there is q-axis saliency, then the steady-state
circuits have to be resolved into two (onto the d- and q-axes), as
shown in Fig. 3(c); this represents an IPM machine. Under lowsaturation conditions, then Xd and Xq are independent and are
functions of the d- and q-axis reluctances. However, when there
is high saturation, there is cross-coupling between the d- and
q-axis components so that Xd = f (Id , Iq ) and Xq = f (Id , Iq ).
If an extended field-weakening range (from the base speed
upward) is required, then the IPM rotor should be used. A
surface-magnet motor simply cannot cope with this range because the field-weakening capability is limited. This occurs
when the current phasor is advanced away from the q-axis so
that there is a component on the −d-axis, as shown in Fig. 3(a).
This has three effects: It can be seen that there is a negative
Xd Id phasor on the q-axis. This weakens the motor flux which
reduces the iron loss at high speed. Additionally, it reduces the
voltage requirement from the inverter supply. The third effect
is the introduction of reluctance torque in the machine. This is
shown in Fig. 3(c), which breaks down the voltages onto the
d- and q-axes. The power due to the excitation torque is 3EIq
(where E is the back EMF induced into the rotor by the IPM
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 9, SEPTEMBER 2011
Fig. 4. Pole-face slots for control of Xq in IPM rotor.
rotor), whereas the total output power, including the reluctance,
is 3E Iq [where E is the total rotor EMF due to the magnets
and q-axis saliency, as defined in Fig. 3(c)]. In Fig. 2(b), it
can be seen that it is possible for q-axis flux to flow across
the surface of a pole face which may lead to excessive Xq ,
in which case, holes or slots can be used in the pole face to
control Xq (Fig. 4). The use of the pole-face slots will also
control the cross-saturation, making the machine performance
more predictable and stable.
E. Pole-Number Selection
It is important to select the correct pole number for the
machine. DC machines tend to have lower pole numbers
(2, 4, 6, etc.), while ac motors often have higher pole numbers
(8, 12, 16, etc.), although this is not a firm guideline. Higher
pole numbers enable fractional-slot windings. The pole number
should be a function of the speed of the machine, and the
following points should be addressed.
1) The flux in the machine should not alternate at a high
frequency; otherwise, the iron losses will be excessive,
although field weakening can be used at higher speed to
limit the iron losses (see example later).
2) Flux frequency = Rotor rotational frequency × pole-pair
number.
3) For normal laminated steels, do not go beyond 150–
200 Hz, although at lower fluxes, it is possible to operate
successfully at maybe 400 Hz even for normal steels.
4) A two-pole PM motor can be difficult to fabricate, and
also, the end windings are long (leading to increased
losses) and the stator yoke is wide (leading to increased
machine diameter).
Popular pole numbers tend to be higher in fractional-slot ac
machines to enable distributed windings. In smaller machines,
a nine-slot eight-pole number is popular [32] although 9/6
arrangements are used and a 12/10 machine was reported in
[33]. In [34], the base slot number of 18 was used with different
rotors with 12 and 16 poles. In [35], an unusual rotor design
using consequent IPM poles (alternate magnet and iron poles)
with dovetail-shaped magnetic poles is discussed with pole
numbers varying between 6 and 14. All the machines in [32]–
[35] are ac drives.
F. Noise, Vibration, Cogging Torque, and Torque Ripple
This should not be ignored. Larger drives should be smoother
in operation; otherwise, they will cause excessive noise. The
9/8 machine reported in [32] is popular for small machines but
it creates high unbalanced magnetic pull (UMP—a net radial
force on the rotor). This makes it more unsuitable for larger
machines. The UMP is much less in a 9/6 machine. In [34],
the effects of winding harmonics on the UMP were studied;
Zhu et al. [36] followed a very similar method with more
slot/pole combinations but without the detailed method.
However, UMP is not the focus of this paper. More relevant
is the production of cogging torque due to the rotor-magnet
and stator-slot combination (which is an alignment torque)
and torque ripple due to the interaction of the magnet air-gap
flux waves with stator MMF spatial harmonics (which is an
excitation torque).
Cogging torque is an alignment torque between the stator
teeth and rotor magnets and is most prominent in surfacemagnet motors with integral slots per pole or pole pair. It is a
reluctance type of torque, and there are a variety of methods for
calculating it using analytical methods [37] and finite-element
analysis (FEA) (there are many studies of cogging using this
method, e.g., [38]). There are also several ways to improve the
cogging torque, such as skew (gradual in either stator or rotor
or using skewed axial rotor segments [38]), bifilar teeth [38],
pitching and staggered magnet spacing in a surface-magnet
rotor [39], and slot opening adjustment, and in ac machines,
fractional slotting is a standard way to reduce cogging. This
means that there is a fractional number of slots per pole, e.g.,
the 9/8 configuration aforementioned is an example of this.
Cogging torque in brushless dc machines was reviewed in [40].
Load torque ripple is a function of the interaction of the PM
air-gap flux waves with the winding MMF. This is reviewed in
[41] (which also discussed nodal vibration and noise). Torque
ripple under load is often neglected in studies, with a preference
for static or mean torque. This is because accurate calculation
of torque, even by using FEA, can be difficult [42], [43]. Mean
torque can be calculated using current–flux-linkage loops [44]
(indeed, so can cogging torque [45]) although many still only
do a load calculation at one position. Torque ripple tends to
be implicit in a dc machine due to the fully pitched windings
and the need to get a trapezoidal winding. For an ac machine,
there is a greater emphasis in smooth operation so the winding
layout is more sinusoidal and torque ripple is minimized. Skew
will also help reduce the load torque ripple. Considering the
equation for stress in (1), the torque (for an unskewed machine)
will be
T (t) = L
D
2
πD
σ(y, t) dy
0
πD
D
br (y, t)Jst (y, t) dy
=L
2
(2)
0
where y is the circumferential distance around the air gap (so
that ky = θ and k = 2/D where D is the mean air-gap diameter) and L is the axial length. We can define the stator electric
loading as a stator surface current density Js (in amperes per
meter), while we can define the rotor radial flux density in the
air gap as br . The product of these at any particular point will
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give the sheer stress. The air-gap flux density due to the PM
rotors is
br (y, t) =
Brm cos (mp(ωr t − ky + φm ))
(3)
m
where m = 1, 3, 5, etc. A general phase angle is set by φm . The
synchronous rotational velocity of the rotor is ωr , and this is
matched to the supply frequency ωs (in radian per second) by
the equation ωs = pωr where p is the pole-pair number of the
machine.
In many machines, it can be assumed that the winding
is a balanced three-phase winding. However, in a fractionalslot machine, it should not be assumed so that the winding
MMF is made with a fundamental one-pole-pair harmonic with
second, third, fourth, fifth, sixth, seventh, etc., windings. The
fundamental harmonic has to be taken as two for the general
case, and harmonics are eliminated if they are zero. Hence,
assuming the current phasors are in phase with the rotor flux
Jsnw cos(ωs t − nw ky + φnw )
Js (y, t) =
nw
=
Jsnw cos(pωr t − nw ky + φnw )
(4)
nw
where nw = ±1, ±2, ±3, etc., for the general case in a threephase winding. Using (2), the product of (3) and (4) shows that
torque is a function of the product of the cosine terms when
the phase angles are equal. For the main torque, nw = mp,
where m = 1 and time variation is zero, i.e., a steady torque.
Working through the mathematics, the general case for the
torque vibration is
m±1
ftorque
= (m ± 1)fsupply |nw =±mp
(5)
Fig. 5. Example of 18-slot 8-pole ac machine with one slot skew. (a) Distribution of one phase for three-phase sine winding. (b) Half cross-section for
IPM machine. (c) Three-phase controlled sinusoidal current on rotor q-axis.
(d) Three-phase back EMF. (e) Electromagnetic torque.
TABLE II
18-SLOT 8-POLE IPM AC MOTOR EXAMPLE—OPERATING
AND G EOMETRIC P ROPERTIES
where m = 1, 3, 5, etc. This does not necessarily mean that
these torque vibrations exist. If there is no matching spatial
winding harmonic and magnet flux wave, then there is no
torque. There can be winding harmonics below the pole number, and these have no effect since there is no corresponding
magnet flux wave. This tends to mean that dc machines have
some torque vibration while ac machines tend to have winding
harmonics and flux waves that, spatially, do not match so that
there is less torque ripple. This is investigated in the next
section.
G. Winding Arrangement
There are a variety of methods for winding a brushless PM
motor depending on whether it is an ac or dc motor. The aim of
an ac winding is to obtain a sinusoidal open-circuit back-EMF
waveform. For a dc winding, it is to obtain a trapezoidal waveform. Therefore, is it appropriate to consider them separately.
Slot fill is considered in Section II-J.
1) AC Windings: Distributed windings are often utilized
in ac machines with coil pitches of one slot. An excellent
examination of this arrangement was put forward in [46]. The
correct winding for a machine is very much a function of the
pole number and slot number and whether there are one or
two coil sides per slot (concentric, lap, or concentrated round
one tooth), as discussed in [31]. Here, a simple example of an
18-slot 8-pole IPM machine is shown in Fig. 5. This is a
fractional-slot machine. The winding is illustrated for one phase
in Fig. 5(a), showing the distributed nature of the winding.
The rotor arrangement is shown in Fig. 5(b). The machine
was modeled using the SPEED software package PC-BDC [47]
from the University of Glasgow, U.K., and the machine data are
given in Table II; this gives the operating point data together
with various geometrical and winding data. There is one statorslot skew in this machine which helps form the back EMF into
a very sinusoidal wave, as shown in Fig. 5(d), so that the torque
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 9, SEPTEMBER 2011
Fig. 7. Interaction of back EMF and current in dc machine, illustrating torqueproducing region in waveforms.
there are three slots per pole which is not accounted for in the
waveforms.
It is also necessary to consider the torque-producing region of
the waveforms. This is shown in Fig. 7. If the back-EMF wave
is too narrow, then there is torque ripple when the back EMF
is multiplied by the current. In addition, the dc machine used
Hall probes, and if they are only slightly out of position, then
there will be considerable torque ripple. This was investigated
in [39].
3) Delta Connection: Delta connection is not recommended
in a brushless PM machine. If there is any third time harmonic
in the phase back EMF, then this will induce a circulating zeroorder current in the mesh, as shown in Fig. 8. This will cause
excessive current and copper losses and potential burnout of the
winding.
Fig. 6. Comparison of idealized short-pitched and fully pitched windings
in a 12-slot 4-pole dc machine. The windings are one phase of a balanced
three-phase set in each case. (a) Short-pitched coils (two-third pitching).
(b) Fully pitched concentrated coils. (c) Trapezoidal 120-electrical-degree
three-phase current set. (d) Three-phase back EMF with short-pitched windings. (e) Electromagnetic torque with short-pitched winding. (f) Three-phase
back EMF with fully pitched windings. (g) Electromagnetic torque with fully
pitched winding.
is smooth, as shown in Fig. 5(e). The efficiency is only 85%,
but at 6000 r/min with eight poles, the frequency in the iron is
400 Hz. This may require high-grade aerospace steel, although
this was not used in this instance (Losil 800 was used), and
therefore, the iron loss dominated the loss components.
2) DC Winding: DC machines require a different winding
strategy with the aim of obtaining a trapezoidal back-EMF
waveform. This will interact with the trapezoidal current (with
120◦ conduction period) to produce a smooth torque. This
requires fully pitched concentrated windings. Fig. 6 shows the
winding layout for one phase of a three-phase set for a 12-slot
4-pole machine. The first simulation uses a short-pitched distributed winding, while the second uses a fully pitched concentrated winding. The waveforms illustrate the torque production
and the fact that there is inherent torque ripple with the shortpitched winding. This is very much an idealized waveform. The
back EMF usually has some distortion to produce ripple, and
this arrangement would have substantial cogging torque since
H. Magnet Selection and PC
The type of magnet used will have a great effect on the
motor performance and cost. The increased cost of high-energy
magnets may be offset by the fact that less magnet material
is required and the motor will be more compact. Typical
remanent magnetism and recoil permeability values at 25 ◦ C
for various magnets are listed Table III. Further details are
put forward in [7] and [8]. The nonlinear characteristics of the
specific magnets should be inspected. The magnets should not
be used in the nonlinear area, as shown in Fig. 9, and sufficient
design tolerance should be built in so that the magnets are not
demagnetized even under overload. The operating point can be
found by calculating the permeance coefficient (PC) and also
the electric loading effects. For ferrite-magnet motors, a PC of
at least eight is usually required, but for rare-earth magnets, this
can be lower since the magnets are much stronger and linear.
The PC can be improved by the use of a narrow air gap and
shorter flux path lengths and wide teeth and stator yoke. Lower
flux-density levels also improve the PC.
The thermal performance of the magnet material also has
to be considered, as shown in Fig. 10. While this paper is
mostly concerned with rare-earth magnet machines, it is worth
considering ferrite-magnet material for completeness. The
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Fig. 8.
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Zero-order 3rd time harmonics in delta-connected brushless PM motor. (a) 3-phase current and 3rd time harmonics. (b) Circulating zero-order set.
TABLE III
TYPICAL MAGNET DATA
Fig. 9. Second quadrant operation for ferrite (grades 1 and 5) and Nd–Fe–B
(Crumax 2830) magnets.
ability to withstand demagnetization for a magnet is dependent
upon the magnet temperature and the magnet type. The typical
values of temperature coefficient for the magnet intrinsic coercivity Hcj are as follows:
1) ferrite: +0.4%/◦ C;
2) Sm–Co: −0.2%/◦ C to −0.3%/◦ C;
3) Nd–Fe–B: −0.6%/◦ C to −0.11%/◦ C.
Ferrite is worse at lower temperatures due to the negative
temperature coefficient, whereas rare-earth magnets are worse
at higher temperature. Ferrite magnets have a nonlinear
region which can be easily moved into with overload and
overtemperature operation. The following points summarize
the discussion for ferrite magnets.
1) Ferrite magnets need a good magnetic circuit and a low
reluctance; otherwise, their load line will not be steep
enough and the operating point will be close to the
nonlinear region.
2) The slope of the load line is equal to negative PC when
the x-axis is scaled by μ0 .
3) PC = (magnet thickness×air-gap area)/(air-gap length×
magnet area). The PC can be used to set the magnet
thickness.
4) Air-gap area ≈ magnet area for surface magnet.
Fig. 10. Ferrite and rare-earth magnet thermal considerations. (a) Ferritemagnet example. (b) Rare-earth magnet example.
5) Therefore, the magnet thickness has to be considerably
greater that the air-gap length.
6) Hence, a lot of magnet material is required.
To summarize the discussion for rare-earth magnets:
1) The PC does not need to be as high when using rareearth magnets so that less material is required (which is
necessary since it is more expensive), and again, the PC
can be used to set the magnet thickness.
2) They have high energy, and handling can be difficult
when magnetized.
3) Premagnetizing may be required.
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Most magnet steels will saturate between 1.5 and 1.7 T
(where the knee points of the B/H curves are in Fig. 12). The
sheer stress can be maximized in high-performance machines
by increasing the flux using cobalt–iron alloys. These alloys
can have a knee point above 2 T [64]; however, they tend to be
very expensive and applicable to premium-cost applications.
Manufacturing affects the iron loss. The properties of the
steel are affected by punching and cutting. If a complicated
lamination shape is used, the properties will be affected. Worn
lamination punches will tend to lead to increased iron losses
with lamination edges having burr, which causes shorting between laminations and increased eddy-current loss. For an IPM
rotor, a very fine cut across the surface can remove a lot of the
burr and improve iron loss.
J. Insulation Systems, Slot Fill, and Mechanical Aspects of
Rotor Structure
Fig. 11. Illustration of demagnetization of rare-earth magnet with thermal
overload. Red dots illustrate points after permanent demagnetization.
4) It is possible to demagnetize the magnets under thermal
stressing, as shown in Fig. 11.
I. Steel Selection and Iron Loss
The two basic properties of interest are the B/H curve and the
iron loss in the steel. The B/H curve sets the flux levels possible
in the machine and the degree of saturation, while the iron
loss is important to the machine efficiency. The loss calculation
is often done by using a modified version of the Steinmetz
equation to obtain hysteresis and eddy-current loss [48].
This loss-calculation method is used in the SPEED modeling software used in this paper, and the equation utilized
for the watts-per-cubic-meter iron loss in [31] and [47] is obtained from
P = Ch f
a+bB
Bpk pk
+ Ce1
dB
dt
2
(6)
where there are various coefficients necessary for accurate
calculation. The loss calculation is really an estimate and only
good if the material loss data are accurate (often, they are
not). Lookup iron-loss tables are often utilized rather than the
implementation of a complicated equation set, and these are
given in [9]. As an example of the effect of steel, consider
the ac motor design in Section II-G1. The material used in this
example was Losil 800/65, and the iron loss was calculated to
be 623 W. The material can be replaced with Transil 35, which
has a lower flux density for a given MMF, as shown in Fig. 12.
However, it is a low-loss steel, as shown in the comparison in
Fig. 12, so that the iron loss is now 122 W. Loss is often a
function of the amount of silicon in the steel. Increasing the
amount of silicon (up to a maximum of 3% [9]) can reduce the
loss in the steel. Reference should be made to manufacturer’s
data. The thickness of the lamination also makes a significant
difference to the eddy-current loss. For instance, for Transil
330, at 1.5 T and 50 Hz, 0.35-mm laminations have a loss of
2.9 W/kg, while 0.5-mm laminations have 3.15 W/kg [1].
Insulation systems have been standardized and graded by
their resistance to thermal aging and failure. Four insulation
classes are in common use, as set by the National Electrical
Manufacturers Association (NEMA), U.S., and these have been
designated by the letters A, B, F, and H, as shown in Table IV.
The temperature capabilities of these classes are separated from
each other by 25 ◦ C increments. The temperature capability of
each insulation class is defined as the maximum temperature at
which the insulation can be operated to yield an average life
of 20 000 h. A maximum temperature rise is also set. There
have been new classifications introduced in 2009 (although not
yet extensively adopted) which correspond to the traditional
classifications; the new equivalent International Electrotechnical Commission classes are also quoted.
In terms of low-voltage machines with random-wound coils,
the system will consist of a slot liner into which the coil is
inserted. The coil will be formed from enameled copper wire,
and the coil will be automatically wound in situ, or automatically or manually inserted as a complete coil. There may be top
wedges to lock the coil into the slot, and if there are two coil
sides in the slot, then there may be a phase separator. The stator
may be dipped in an epoxy-resin-type varnish with the aim of
impregnating deep into the slot. This varnish has two functions.
It will fill and set so that the winding is not loose in the slot,
which will prevent vibration damage. It will also provide good
thermal conduction from the coil to the core, which is necessary
for effective cooling. Loose windings in slots are not a good
manufacturing solution. If the stator is not dipped in resin, then
it is often trickled as a hot solution down into the slots in order
to secure the coils. Different insulation systems are described
in [11].
If the wire is too thick for winding the coil, then wind
with multiple strands and connect in parallel. These are often
described as “strands in hand” and should not be confused
with parallel windings, where complete coils are connected in
parallel.
The fill factor is the ratio of the copper in a slot to the
slot area. A common mistake made is to assume a fill factor
that cannot be realized. There is a slot liner, and there may be
wedges which will occupy slot space. Also, the conductors are
round and have an enamel insulation coating so that there will
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Fig. 12. Comparison of B/H and frequency/iron-loss curves for Losil 800/65 and Transil 35 steels (B = 1.7 T for loss data).
TABLE IV
INSULATION CLASSIFICATIONS [NEMA MG 1-2006]
(AMBIENT BELOW 40 ◦ C)
4) What is the duty cycle?
5) How effectively can we cool the machine?
The latter two points will affect the thermal rating of the
machine, and this is addressed in the next section.
III. C OOLING AND T HERMAL I SSUES
be space even when tightly packed. Therefore, high fill factors
should be approached with caution. For instance, automotive
alternators are low voltage (12 V) and often have very few turns
of very thick wire. Manufacturers often work with a maximum
slot fill of 30% or less.
Many machines have environmental considerations that require the stator and/or rotor to have a protective can which
can be conducting (for instance, from stainless steel [49])
or nonconducting. These cans can add eddy-current loss to
the machine and lengthen the air-gap length so that the cans
can be accommodated. However, they can add considerable
mechanical stability to the rotor and help retain the magnets
on the surface of the rotor. Both surface magnets and IPM
motors have structural issues with retaining the magnets and
pole faces (in IPM rotors). The mechanical stresses in an IPM
rotor were described and discussed in [50], while the use of
retaining sleeves in a high-speed surface-magnet rotors was
highlighted in [51] and mechanical retention of magnets was
further discussed in [52]. The mechanical integrity of a rotor
may restrict the maximum speed of a machine and also the
possible maximum rotor diameter.
The losses in the machine can be split up into copper, iron,
and mechanical losses. Some of these losses can be difficult to
assess. For instance, there will be eddy-current losses in surface
magnets due to slotting [53] and possibly proximity losses in
conductors if they are air-gap windings or even thick conductors
[54]. However, these are normally low; Yamazaki [55] gives a
good account of the loss distribution in an IPM motor.
K. Sizing-Issues Summary
The sizing of a machine can be a complex matter. To summarize the issues, the following points should be considered.
1) Is there a restriction on length or diameter?
2) Is it in an environment that is sensitive or hazardous?
3) What are the application torque requirements?
There is a strong requirement for more energy-efficient motors. Improved thermal design can lead to a cooler machine with
reduced losses. Copper loss is a function of winding resistance
and, therefore, is a function of temperature. Rare-earth PM
flux reduces with increased temperature. The size of a motor
is ultimately dependent upon the thermal rating. The motor
components that are limited by the temperature are wire or
slot liner/impregnation, bearings (life), magnet (loss of flux
and demagnetization limit), plastic cover (low melting point),
encoder, and housing (safety limit).
The temperature of the winding insulation has a large impact
on the life of the machine. Many companies use curves such as
that shown in [56] to estimate motor life, and these are related
to the insulation classifications in Table IV.
Magnets are usually isolated from the main heat sources
so that they are protected from severe transient overloads.
The windings are most susceptible to transient overloading.
However, rare-earth magnets (Sm–Co and Nd–Fe–B) exhibit
local eddy-current losses as heat sources, which are difficult
to estimate or measure. Hence, there is a much longer time
constant for magnets compared with windings although it is
essential to know the magnet temperature for transient and
demagnetization calculation.
In this section, traditional thermal designs will be outlined,
and then, modern techniques will be reviewed.
A. Traditional Thermal-Sizing Methods
Traditional thermal sizing uses a single parameter, which is a
thermal resistance, as shown in Fig. 13(a), for the housing heattransfer coefficient. In addition, the winding current density
and specific electric loading are considered. Traditional thermal
modeling tends to be empirical with data obtained from the
following:
1) simple rules of thumb, e.g., for a totally enclosed machine, a conductor current density of 5 A/mm2 and a heattransfer coefficient [Fig. 13(b)] of 12 W/m2 /◦ C;
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C. Cooling Types and Methods
Fig. 13. Traditional thermal modeling using single thermal resistance and single heat-transfer coefficient. (a) Thermal resistance from winding to ambient.
(b) Heat-transfer coefficient.
TABLE V
TYPICAL CURRENT DENSITY AND HEAT-TRANSFER COEFFICIENTS
2) tests on existing motors;
3) competitor catalogue data for similar products.
These methods can be inaccurate. A single parameter fails
to describe the complex nature of motor cooling, and there is
poor insight into which aspects of the thermal performance of a
motor need to be focused upon. Table V lists typical values for
the current density and heat-transfer coefficient.
B. Modern Thermal Design Techniques
There are two options for modern thermal design. These
are lumped-circuit analysis (network analysis) [14], [15], [18],
[57]–[59] and numerical analysis using FEA and computational
fluid dynamics [16]. While computational fluid dynamics gives
more accurate solutions for particular examples, it can be time
consuming to set up the model. In the design office, the lumpedcircuit analysis is more useful for faster and more interactive
design procedures. It can be linked into electromagnetic design,
as illustrated in [18] where the thermal package Motor-CAD
from Motor Design Ltd., U.K., [60] is linked with the SPEED
software [47]. In the examples put forward in this paper, these
environments are used. A typical lumped circuit from MotorCAD is shown in Fig. 14; the literature has several examples
of this type of circuit as developed by many researchers (e.g.,
[14]–[18], which are, by no means, comprehensive). When
there is a high temperature gradient, more nodes are required
so the slot is modeled as a multishell structure, as shown in
Fig. 14(b). The accuracy of the circuit model in Fig. 14(a) very
much depends on the accuracy of the lumped parameters; if one
is substantially inaccurate, then it can affect the temperatures
of the surrounding nodes. Therefore, the components have to
account for the heat flow in terms of the conduction, convection,
and radiation. Several aspects of the model are manufacturing
dependent as well as material dependent. For instance, the thermal conductivity of the coil is a function of the impregnation of
the resin.
Motor-CAD covers several thermal networks including a
range of cooling types that represent the standard methods of
motor cooling.
1) Natural convection (TENV): This is very common with
many housing design types.
2) Forced convection (TEFC): There are many fin channel
design types, and fans are commonly fitted to industrial
drives.
3) Through ventilation: This utilizes rotor and stator cooling
ducts.
4) Open end-shield cooling.
5) Water jackets: There are many design types (axial and
circumferential ducts), and they can be for either stator
or rotor.
6) Submersible cooling.
7) Wet rotor and wet stator cooling: This is common for
pumping.
8) Spray cooling.
9) Direct conductor cooling using slot water jacket.
10) Conduction: Internal conduction and the effects of
mounting.
11) Radiation: Both internal and external.
Hence, there are many ways to implement effective motor
cooling.
IV. M OTOR D ESIGN T ECHNIQUES AND E XAMPLES
Modern design techniques usually use detailed analytical
algorithms and electromagnetic FEA methods to analyze a
design. While the SPEED package already mentioned used
analytical calculations, sometimes, detailed calculations require
FEA, such as to obtain accurate cogging torque and load torque
in IPM motors with phase advance. A finite-element bolt-on
package can be used for this [61]. This arrangement is not
unique; many finite-element packages now feature spreadsheet
and initial calculation tools to enter data for an initial motor design. In this section, some additional motor-analysis techniques
will be highlighted and design and analysis examples put will
be forward.
A. Current–Flux-Linkage Loops (I–Psi Diagrams)
The mean torque can be obtained in a brushless PM machine in a similar way to the switched reluctance by forming
a current-against-flux-linkage loop (I–Psi). This method was
detailed in [44] and [45]. The area enclosed (W ) is equal to
the work done during the rotation so that the torque is then the
work done divided by the distance moved. For a machine with
m pole pairs and n phases, the electromagnetic torque is
m
× W.
(7)
Te = n ×
2π
For a balanced machine, each phase will trace out the same
loop with area W . By using the example with the short-pitched
machine in Section II-G2, with both sine- and square-wave
excitation, the loops are shown in Fig. 15. The mean torque for
the dc control is 1.0 N · m, while for ac control, it is 0.87 N · m.
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Fig. 14. Thermal circuits and winding model of machine. (a) Lumped thermal model (part model) with heat sources, thermal resistances, and thermal
capacitances—surface-magnet rotor. (b) Multilayer winding representation when there is a high temperature gradient. Traditional winding for random-wound
coils and 54% slot fill.
Fig. 15. Comparison of I–Psi loops for dc and ac controls.
The peak current for both simulations was 15 A, and the same
short-pitched winding in Fig. 6(a) was utilized. Interestingly,
in the Appendix, the theoretical ac/dc control rating ratio was
calculated to be 1.5. Here, by simply changing from sineto square-wave control, the torque increases by 1.15. If the
winding is fully pitched for the dc control, then the torque is
1.07 so that the ratio is 1.23. However, the rms current with
the dc control is higher. Using the same rms currents and fully
pitched winding in the dc simulation gives a torque ratio of
1.07. These results were obtained in the SPEED PC-BDC and
PC-FEA environments.
B. Frozen Permeability Method
This method is a very powerful tool for separating out the
different torque components due to excitation and reluctance
Fig. 16. Prius PM-motor cross section in SPEED PC-BDC—this shows two
magnets per pole and high saliency.
torques [62]. This technique is used in an FEA, and many
packages allow this function. To summarize, using a magnetostatic model, a full nonlinear solution is carried out, and the
total torque can be obtained from this solution. The saturated
magnetic permeances are then locked. If the magnets are then
“switched off” (by setting the remanent magnetism Br to zero)
and the solution restarted with the locked permeances from the
full solution, then the reluctance torque can be calculated. This
reluctance torque includes the saturation effects from the full
solution. An example is shown in Fig. 16, which is a SPEED
simulation of the Toyota Prius machine in [30]. This machine
operates at a high phase advance to allow for a very wide
field-weakening range (from 1500 to 6000 r/min) and relies on
substantial reluctance torque. This is an eight-pole machine.
The peak current occurs at the base speed of 1500 r/min.
This is a transient point, and the current density (over
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Fig. 17. One-pole machine from static FEA solution. Peak flux density in
teeth is about 2.10 T. Load current is 190.9 A on the q-axis (1500 r/min).
20 A/mm2 at 190.9 A) and flux densities are high if the current
is maintained on the q-axis, as shown in Fig. 17. The frozen
permeability method was implemented at 1500 and 6000 r/min,
and the results are shown in Figs. 18 and 19. It can be seen
that the torque peaks between 30◦ and 50◦ phase advance. With
60◦ phase advance, it was found that the base speed current
could be reduced to 141.1 A at 1500 r/min for a required
maximum torque of about 300 N · m. Comparison of Fig. 18,
where the current level is much higher, with Fig. 19 shows
different curve shapes for both the excitation and reluctance
torques. This illustrates the effect cross-saturation can have on
the performance, as discussed earlier.
C. Efficiency Plots
Efficiency is becoming a more important factor in machine
design and is indeed crucial in many designs. Computational
design solutions are becoming increasingly rapid, and it is now
possible to scan a range of operating points and produce a
plot of the efficiencies over a 2-D array of torques and speeds.
In [30], measured efficiency plots were used to illustrate the
motor operation, and these can be obtained from simulations
too. Fig. 20 shows the efficiency plot for the machine in the
previous section using SPEED PC-BDC. For a brushless PM
motor, there are several parameters that can be set. In this
case, at each load point, the phase angle advance was set at
0◦ , 30◦ , and 60◦ , and the current varied until the correct torque
was obtained. The highest efficiency was then selected as the
operating point. The selected phase angle is shown in the top
chart, while the efficiencies are shown as colored regions and
contour lines in the bottom plot.
D. Fractional-Slot Design-Size Rationalization
Here, an example is put forward for the rationalization of
a motor design by consideration of the thermal design [63].
The existing motor has 50 mm of active length (core length), a
130-mm-long housing with a traditional lamination, and overlapping windings. The new motor still has 50 mm of active
length; however, the housing is now only 100 mm long. It
produces 34% more torque for the same temperature rise. The
machine uses segmented-lamination nonoverlapping windings
(one-slot pitch concentrated coils). In order to optimize the new
design, an iterative mix of electromagnetic and thermal analysis
was performed. Extensive thermal modeling was carried out.
The new design is shown in Fig. 21. Both arrangements had an
80-mm diameter; however, the traditional design had 18 slots
and 6 poles [Fig. 21(b)] and overlapping windings, while the
new design has concentrated windings and a 12-slot 8-pole
layout [Fig. 21(c)]. This illustrates that the slot/pole combination is flexible for a particular application. The traditional
winding only had a 54% slot fill but the new arrangement and
the techniques that can be applied to manufacture it (precision
bobbin wound) means that this was increased to 82% in the new
design.
Potting/impregnation material improvement was also possible. The new design has a k factor of 1 W/m/◦ C, whereas
previous materials have a k factor of 0.2 W/m/◦ C. This gave
a 6%–8% reduction in winding temperature (with respect to
Celsius scale). A potted (encapsulated in resin) end-winding
design showed a 15% reduced temperature compared with that
of the previous nonpotted design. Vacuum impregnation can
eliminate air pockets. The new design here shows 9% decrease
in temperature in a perfectly impregnated motor compared with
the one with 50% impregnation.
All these design and manufacturing improvements lead to a
much improved thermal performance for the new motor design.
This means it can be more highly rated, and so, the size can be
reduced by a reduction in the active axial length.
V. C ONCLUSION
This paper has described the design philosophy for dc and ac
PM machines. It goes on to discuss many of the modern-day
analysis techniques that can be used to assess the performance
of a machine. Many of the techniques are illustrated with
examples, and the need to consider the electromagnetic design,
thermal analysis, and manufacturing techniques in conjunction
is stressed. This paper will be very useful to an electrical machine designer who requires more detailed information about
the steps necessary to analyze and improve a motor design of
this ilk.
A. Further Literature
There are many sources of design method information from
many researchers. In terms of further texts, [65] gives a treatise
specific to PM motor design, while general ac machine design
and operation are considered in [66] and [67], which can
be very helpful in terms of winding theory and practice and
other aspects of machine operation. The technology is rapidly
developing due to new material design refinement. There are
continuing developments of algorithms that are aimed at the
automated and precise design of an electrical machine; [68]
and [69] are illustrations of these, and a literature review would
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Fig. 18. Separation of torque at 1500 r/min with 190.9-A loading—variation of current phase with respect to q-axis.
Fig. 19. Separation of torque at 6000 r/min with 35.4-A loading—variation of current phase with respect to q-axis.
Fig. 21. Design renationalization using concentrated one-tooth windings and
T-piece stator sections. (a) New design manufactured and T-piece stator.
(b) Previous design. (c) New design.
[70] is a further example in addition to the text in [13] and
technical publications [17] and [36].
B. Commercial Design Tools
Fig. 20. Efficiency plot for PC-BDC simulations using phase angles of 0◦ ,
30◦ , and 60◦ .
highlight further examples. This paper has not considered noise
and vibrations; however, these are important. There are several
papers on this subject as applied to brushless PM motors, and
The work in this paper often uses various commercial software products as the working environments while discussing
the fundamental design concepts. The products are not necessarily unique, and a designer should consider trying different
products in order to assess their suitability and even developing
their own design software using the large body of scientific
algorithms and design and analysis techniques already published. In terms of alternatives, there are other notable examples
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peak of the flux density wave is limited by the steel saturation characteristics. The analysis here makes that assumption.
Hence, under ac control, only the fundamental of the air-gap
flux density wave should be considered, together with the main
current density wave. This is for a distributed winding, and a
three-phase winding is assumed. The mean stress is then
σmean =
Bpk(fund) Jrms
Bpk(fund) Jpk
√
=
2
2
(A1)
where the stator current density can be estimated from a sinusoidal spatial variation on the stator surface [Fig. 22(a)] so that
Jrms =
Fig. 22. Air-gap flux density and stator surface current density for ac and dc
motors. (a) B and J for ac machine. (b) B and J for dc machine.
such as RMxprt from Ansoft (Ansys), U.S. This uses first-pass
analytical calculations to feed into Maxwell FEA. Infolytica
Corporation, Canada, has developed MotorSolve BLDC (and
other packages) for template-based design which feeds into
the MagNet FEA package. Cedrat Group, France, uses Flux
and Motor Overlays to specify template geometries for motor
simulation in Flux2D and 3-D, and indeed, SPEED can feed
into this package. JSOL Corporation, Japan, has developed the
JMAG FEA package, and this also has Motor Template (similar
to Motor Overlays) and JMAG-Studio and JMAG designer
can be accessed through CAD Link. This package also has a
SPEED link. The FEA package Opera from Cobham, U.K.,
(formerly Vector Fields) has application-specific tools for frontend design of rotating machines. These examples illustrate
a commonality between many packages; these tend to allow
easy geometry, material, and control setup for faster motor
design. Many packages now link to standard mechanical CAD
packages so that geometries can be imported and initial design
calculation can be done before resorting to more complex and
slower FEA solutions.
The aforementioned list is far from comprehensive but represents a global cross section of examples; many companies
and specialists have developed their own in-house design tools,
as already suggested as an option. The market is continually
changing, hence the recommendation for trial of products.
A PPENDIX
The maximum mean sheer stresses can be estimated for
brushless dc and ac machines in order to compare their torque
densities. Consider Fig. 22. The dc machine has a trapezoidal
waveform for the current density if the winding is fully pitched
and 120◦ conduction exists, while the ac has low harmonic
content and the current is sinusoidal. The idealized stress
waveforms are shown for both control strategies, and approximate stress calculations can be derived to illustrate that the dc
machine has a higher theoretical mean stress.
AC Control—Flux Density Limited by Peak of Fundamental
Sinusoidal Flux Wave: In an IPM motor, the flux density in
the air gap can be shaped for smoother operation. This is
particularly important in a servo system. Ideally, the air-gap
flux wave would be sinusoidal for low torque ripple, and the
AC
Nph Irms
3KW
×
.
2
D
(A2)
The mean air-gap diameter is D, the number of series phase
AC
winding turns is Nph , the fundamental winding factor is KW
,
and the winding current (assuming no parallel winding) is Irms .
The 3/2 factor is valid for a three-phase sinusoidal current set.
Assuming the winding factor is unity, then from (A1) and (A2)
σmean =
6Bpk(fund) Nph Irms
3Bpk(fund) Nph Irms
√
.
= 0.55
πD
2 2D
(A3)
AC Control—Fully Pitched Surface-Magnet Rotor: If we
assume that the air-gap wave is trapezoidal (and a full square
wave with 180-electrical-degree pitch), then the air-gap flux
will be limited by the peak of the trapiziodal wave, as in
Fig. 1(b); a Fourier analysis of a fully pitched trapezoidal wave
gives a peak fundamental ratio where
Bpk(fund) =
4
Bpk(trap) .
π
(A4)
Hence
σmean =
6Bpk(trap) Nph Irms
4 3Bpk(trap) Nph Irms
√
.
= 0.7
π
πD
2 2D
(A5)
DC Control: Assuming trapezoidal flux density and current
density with a 120-electrical-degree pulsewidth
σmean =
2
× Bpk(trap) Jpk .
3
(A6)
Again, assuming trapezoidal current density, this can be related
to the phase current by
DC
×
Jpk = KW
2Nph Ipk
6Nph Ipk
DC
= KW
.
×
2/3 × πD/2
πD
(A7)
For a trapezoidal current waveform with a width of 120 electrical degrees [Fig. 22(b)], the rms current is
2
Ipk .
Irms =
(A8)
3
Putting (A5) into (A7) gives
2 3 6Bpk(trap) Nph Irms
×
σmean =
3 2
πD
6Bpk(trap) Nph Irms
.
= 0.82
πD
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(A9)
DORRELL et al.: REVIEW OF DESIGN ISSUES AND TECHNIQUES FOR PERMANENT-MAGNET MOTORS
Comparison of Stresses: Comparing (A3)–(A9) shows that,
for a given phase current (whether sinusoidal or trapezoidal),
dc control gives higher stress density than ac control for a given
peak flux density in the ratio 0.82/0.55 = 1.5. However, dc
control tends to give more torque ripple and is more suitable
for power drives. If a surface-magnet rotor is used, then (A4)
can be compared with (A9), and this time, the theoretical stress
limits are in the ratio 0.82/0.7 = 1.17, which is much closer.
Relationship Between DC Link Voltage and Power Conversion: Assume that the machines operate with unity power
factor. In a three-phase ac machine where the phase voltages
and currents are sinusoidal and where there is 180◦ conduction
in the inverter, the voltages and currents can be related to each
other where Idc = Ipk and Vdc = 3Vpk /2. Therefore
Vdc Idc = 3
Vpk Ipk
= 3Vrms Irms .
2
[17]
[18]
[19]
[20]
[21]
[22]
(A10)
For a dc machine, where the waveforms are trapezoidal and
where there is 120◦ conduction in the inverter, Idc = Ipk and
Vdc = 2Vpk . The rms-to-peak values are
2
2
Vrms =
Vpk and Irms =
Ipk
(A11)
3
3
[23]
[24]
[25]
so that the relationship between the dc link and ac rms values is
Vdc Idc = 2Vpk Ipk = 3Vrms Irms .
(A12)
[26]
[27]
Comparing (A10) and (A12) shows that the same relationship
holds whether it is ac or dc.
[28]
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David G. Dorrell (M’95–SM’08) is a native of
St. Helens, U.K. He received the B.Eng. (Hons.)
degree in Electrical and Electronic Engineering from
The University of Leeds, Leeds U.K., in 1988,
the M.Sc. degree in Power Electronics Engineering
from The University of Bradford, Bradford, U.K., in
1989, and the Ph.D. degree from The University of
Cambridge, Cambridge, U.K., in 1993.
He has held lecturing positions with Robert
Gordon University, Aberdeen, U.K., and the University of Reading, Berkshire, U.K. He was a Senior
Lecturer with the University of Glasgow, Glasgow, U.K., for several years. In
2008, he took up a post with the University of Technology Sydney, Sydney,
Australia, where he was promoted to Associate Professor in 2009. He is
also an Adjunct Associate Professor with National Cheng Kung University,
Tainan, Taiwan. His research interests cover the design and analysis of various
electrical machines and also renewable-energy systems with over 150 technical
publications to his name.
Dr. Dorrell is a Chartered Engineer in the U.K. and a Fellow of the Institution
of Engineering and Technology.
Min-Fu Hsieh (M’02) was born in Tainan, Taiwan,
in 1968. He received the B.Eng. degree in mechanical engineering from National Cheng Kung University (NCKU), Tainan, in 1991 and the M.Sc. and
Ph.D. degrees in mechanical engineering from the
University of Liverpool, Liverpool, U.K., in 1996
and 2000, respectively.
From 2000 to 2003, he served as a Researcher
with the Electric Motor Technology Research Center,
NCKU. In 2003, he joined the Department of Systems and Naval Mechatronic Engineering, NCKU, as
an Assistant Professor. In 2007, he was promoted to Associate Professor. His
area of interests includes renewable-energy generation (wave, tidal current, and
wind), electric propulsors, servo control, and electric machine design.
Dr. Hsieh is a member of the IEEE Magnetics, Industrial Electronics,
Oceanic Engineering, and Industrial Applications Societies.
Mircea Popescu (M’98–SM’04) received the D.Sc.
in electrical engineering from Helsinki University of
Technology, Helsinki, Finland, in 2004.
He has more than 25 years of experience in electrical motor design and analysis. He worked for the Research Institute for Electrical Machines, Bucharest,
Romania; Helsinki University of Technology; and
SPEED Laboratory, University of Glasgow,
Glasgow, U.K. In 2008, he joined Motor Design
Ltd., Shropshire, U.K., as an Engineering Manager.
He published over 100 papers in conferences and
peer-reviewed journals.
Dr. Popescu was the recipient of the first prize best paper awards from IEEE
Industry Applications Society Electric Machines Committee in 2002, 2006,
and 2008.
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DORRELL et al.: REVIEW OF DESIGN ISSUES AND TECHNIQUES FOR PERMANENT-MAGNET MOTORS
Lyndon Evans received the B.Sc. (Hons.) degree in
computer networks from Glyndwr University, Wales,
U.K., in 2008.
He qualified as a Television and Video Service
Engineer in 1988 and worked in this field for over
15 years before returning to study and receiving his
B.Sc.(Hons.) degree. He is a Software Developer
with Motor Design Ltd., Shropshire, U.K., in partnership with Glyndwr University, and is studying for
a research degree.
Mr. Evans is a member of The Institution of Engineering and Technology and an associate member of the British Computer
Society.
3757
Vic Grout (M’01–SM’05) received the B.Sc.
(Hons.) in Mathematics and Computing from The
University of Exeter, Penryn, U.K., in 1984,
and a Ph.D. in Communication Engineering from
Plymouth Polytechnic, Devon, U.K., in 1988
He is a Professor of Network Algorithms and the
Director of the Centre for Applied Internet Research,
Glyndwr University, Wales, U.K. He has worked in
senior positions in both academia and industry for
over 20 years and has published and presented over
200 research papers and 4 books. He is an Electrical
Engineer, Scientist, Mathematician, and IT Professional.
Mr. Grout is a Chartered Engineer and a Fellow of the Institute of Mathematics and its Applications and British Computer Society and The Institution
of Engineering and Technology.
David A. Staton (M’90) received the Ph.D. degree in
computer-aided design of electrical machines from
The University of Sheffield, Sheffield, U.K., in 1988.
Since then, he has worked on motor design and
particularly the development of motor design software at Thorn EMI; the SPEED Laboratory, University of Glasgow, Glasgow, U.K.; and Control
Techniques, U.K. In 1999, he set up a new company,
Motor Design Ltd., Shropshire, U.K., to develop a
thermal analysis software for electrical machines. He
published over 50 papers in conferences and peerreviewed journals.
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