386 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 3, SEPTEMBER 2003
Analytical Model for Permanent Magnet Motors With
Surface Mounted Magnets
Amuliu Bogdan Proca, Member, IEEE, Ali Keyhani, Fellow, IEEE, Ahmed EL-Antably, Member, IEEE,
Wenzhe Lu, Student Member, IEEE, and Min Dai, Student Member, IEEE
Abstract—Thispaperpresentsananalyticalmethodofmodeling
permanent magnet (PM) motors. The model is dependent only on
geometrical and materials data which makes it suitable for inser-
tion into design programs, avoiding long finite element analysis
(FEA) calculations. The modeling procedure is based on the calcu-
lation of the air gap field density waveform at every time instant.
The waveform is the solution of the Laplacian/quasi -Poissonian
field equations in polar coordinates in the air gap and takes into
account slotting. The model allows the rated performance calcula-
tionbutalsosucheffects as coggingtorque,rippletorque,back-emf
formprediction,some of whichareneglected in commonly used an-
alytical models.
Index Terms—Analytical models, finite element analysis,perma-
nent magnet synchronous motor design.
I. INTRODUCTION
P
ERMANENT magnet synchronous motors (PMSM) have
been around for many years, especially for low power ap-
plications, such as servomotors and alternators. Their main ad-
vantage, over comparable motors, is the absence of the excita-
tion winding.
The design of permanent magnet motors requires a series of
iterativecomputations basedonthe selectionof differentconfig-
urations. These include the choice of geometrical dimensions,
materials, parameter calculations, etc. The designer needs to as-
sume certain dimensions and materials and then calculate the
performance of the designed motor. The performance is then
comparedwith thedesired specification.If thespecificationsare
not satisfied, the designer has to modify the design to improve
the performance of the motor. Most designers use empirical re-
lations for motor parameters or simplified analytical models in
the early stages of the design and then use finite element anal-
ysis (FEA)in second stagesof the designfor betterperformance
evaluation.
The air gap magnetic fielddensity provides valuable informa-
tion in evaluating motor performance. Knowledge of the field
density will not only allow rated performance calculation but
also calculation of such effects as cogging torque, ripple torque,
Manuscript received November 22, 2002. This workwas supported in part by
Delphi Automotive Systems and the National Science Foundation under Grant
ECS-9625662.
A. B. Proca is with Solidstate Controls Inc., Columbus, OH 43085 USA
(e-mail: bproca@solidstatecontrolsinc.com).
A. Keyhani, W. Lu, and M. Dai are with the Department of Electrical Engi-
neering, The Ohio State University, Columbus, OH 43210 USA (e-mail: key-
hani.1@osu.edu; lu.140@osu.edu; dai.21@osu.edu).
A. El-Antably is with Allison Transmission, Indianapolis, IN 46250 USA.
Digital Object Identifier 10.1109/TEC.2003.815829
back-emfshape,etc. Although thecommonmethod ofobtaining
the air gap field density waveform is FEA, it is time-consuming
even on powerful computers and it is difficult to be inserted in
an iterative design procedure. It is therefore desirable to use al-
ternative methods to evaluate the air gap field density. Previous
efforts were concentrated mainly on evaluating the field density
of the permanent magnets. Miller [4] proposes a method of ana-
lyticallycalculating theEMF shape from geometrical databased
on BLv formula. The method is based on the single tooth coil
rather then a full pitch coil. Sebastian [5] calculates the air gap
field density using FEA and assuming a constant airgap (based
on the fact that the rotor is skewed) corrected by Carter’s coeffi-
cient and then finds an empirical formula to describe its shape.
Then the flux is calculated by integrating on a surface as a func-
tion of the rotor angle. J. de la Ree and Boules [6] assume the
flux density in the air gap as known (from FEA) and develop
a method of studying the cogging and ripple torque for surface
mounted and buried permanent magnet motors. In [7], Boules
develops a model for the permanent magnet motor in rectan-
gular coordinates and uses an equivalent pole-arc to correct the
variation of the circumferential depth of the magnet with the ra-
dius. Other analytical models for the magnetic field density in
the air gap were presented in [8]–[13]. Their major drawback
is that they do not account for the effect of slotting and they
consider the recoil permeability of the permanent magnets to
be equal to one. The most recent method was reported by Zhu,
Howe [1]–[3] and consisted in solving Laplacian/quasi -Poisso-
nian field equations in polar coordinates on the air gap of the
PM machine. The effect of slotting is taken into account using
either a 1-d model (the variable being the angular position) or a
2-d model (dependent on both the angular position and the ra-
dial position). Their approach is used in the present paper, but,
rather then focussing only on the air gap field density, deriva-
tions for cogging torque, electromagnetic torque and back-emf
shape are also performed.
II. P
ERMANENT MAGNET MOTOR MODEL USING THE AIR GAP
FIELD DENSITY
Fig. 1 represents the permanent magnet model used in this
paper for phase A. Similar models are used for the other two
phases. The voltage EA represent the self-inducted voltage of
phase A whereas EB and EC represent the voltages mutually
generated by phases B and C. EPM represents the voltage gen-
erated by the rotating permanent magnets fieldson phase A. The
0885-8969/03$17.00 © 2003 IEEE
PROCA et al.: ANALYTICAL MODEL FOR PERMANENT MAGNET MOTORS WITH SURFACE MOUNTED MAGNETS 387
Fig. 1. PM motor model per phase.
above voltages can be expressed as
(1)
(2)
(3)
(4)
where
is the total number of slots, is the number of
coils per phase,
is the length of the rotor, and is its speed. It
is assumed that the phase resistance
can be easily calculated
from the winding data. The leakage inductance
is approxi-
mated as in [14] and is of little interest in this study. In the first
equation,
represents the average instantaneous magnetic
field density created by coil
of phase A in slot of phase A.
Equations (2) and (3) are represented in a similar fashion. In the
fourth equation,
represents the average field density cre-
ated by the permanent magnets on slot
of phase A.
The advantages of using such a model over other existing
models is that it can predict parasitic effects in the permanent
magnetmachine,such ascoggingtorque, rippletorque, and non-
sinusoidal back-emf. To be able to use the above model, knowl-
edge of both the permanent magnet field
and the ar-
mature field (
, , ) variation as a function of angular
position is needed.
III. A
SSUMPTIONS
Certain assumptions have to be taken into account for this
model. The magnets are surface mounted and are magnetized
radially. The stator slots are either rectangular or trapezoidal as
shown in Fig. 2. Also, the following material assumptions were
used.
• The ferromagnetic material of the core has a linear B-H
curve.
• Saturation is neglected.
• The spacer between the magnets has the same perme-
ability as the magnets.
IV. P
ERMANENT MAGNET FIELD
The permanent magnet field density of the motor is affected
by slotting. Slotting is dependent on the rotor position and the
Fig. 2. Cross-section of the motor.
wave shape of the magnetic field density referred to the rotor
also changes with position. In order to obtain a position inde-
pendent mapping, an approach similar to the one in [3] was
used. The authors mapped the air gap field density to the di-
mensions of the motor assuming a slotless stator. The field den-
sity function was then multiplied with the relative air gap per-
meance function, as described in [3] for a 1-D model. Fig. 2
shows a motor section and the stator and rotor references used
throughout this paper.
Using the same variables as in the picture, the instantaneous
value of the permanent magnet field density in the air gap is
(5)
where
is the field density function for a slotless
stator,
is the angular position on the rotor surface, is the
rotor displacement, and
is the position referenced to the rotor.
is the relative air gap permeance function that accounts for
slotting and is based on the assumption that the magnetic flux
lines have semicircular paths in the slots with radii equal to the
shortest distance to the tooth edges as shown in Fig. 3.
The relative permeance function can be derived as
(6)
and
outside the slots, where isthe distance between
point where the field is evaluated and the closest tooth edge,
is the air gap length, and is the magnet depth. The instanta-
neous field density distribution of one magnetic pole
assuming a slotless stator, is calculated as in [1]. The permanent
magnet magnetization, shown in Fig. 4 for a pair of poles, is de-
composed in a Fourier series of odd terms as follows:
(7)
where
is the magnetic remanence of the permanent magnet
material,
isthe magnet pitch/pole pitch ratio, and
is the number of pair of poles.
Bysolvingthemagneticpotentialdistributionequationsinthe
air gap in polar coordinates [1], an expression for thepermanent
388 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 3, SEPTEMBER 2003
Fig. 3. Approximation of the flux line paths in a slot.
Fig. 4. Magnetization of permanent magnets.
magnetfielddistributionatthestatorsurfaceresults:(Seeequation
atthebottomofthepage)where
istheradiusatthestatorsurface,
isthe radiusatthe magnet surface,and istheradius atthe
rotorcoresurfaceand
.Forthecaseinwhich
theproduct
(fundamentalofthefielddensityofatwp-pole
motor),(8)cannotbeused.Thesolutionofthemagneticpotential
distributionequations inthe air gap yields
(9)
Fig. 5. Current sheet distribution for a coil.
V. A RMATURE FIELD
The armature field density of one phase can be modeled as
a summation of the field densities created by each coil of that
phase. For example, the armature field of a phase of a three-
phase, six-pole, and 18-slot motor (as shown in Fig. 2) can be
derived as
(10)
where the three coils sitting in slots 1 and 4, 7 and 10, and 13
and 16 all contribute to phase A flux density.
To calculate the armature reaction field density of a single
coil, the stator is first assumed slotless, and the armature field
is calculated. Then the relative permeance function is estimated
as in the permanent magnet field section. The product between
the slotless stator field and the relative permeance function will
providethe expressionof thearmaturereaction fieldfor onecoil.
For the slotless stator armature field calculation, it is assumed
that the current density sheet is uniform along the slot opening,
as shown in Fig. 5.
Using the coordinate system of Fig. 2, the Fourier series ex-
pansion of the current density sheet results in
(11)
where
is the instantaneous value of the current, is the slot
opening, and
is the coil opening angle. By solving the mag-
netic potential distribution equations in the air gap in polar co-
ordinates [2], an expression for the armature field density of a
(8)
PROCA et al.: ANALYTICAL MODEL FOR PERMANENT MAGNET MOTORS WITH SURFACE MOUNTED MAGNETS 389
coil (distributed in slots k and l) is obtained
(12)
and when slotting is considered
(13)
VI. T
ORQUE CALCULATION
The instantaneous torque can be expressed as the derivative
of the co-energy in respect to the rotor-stator position in the air
gap
(14)
The co-energy can be expressed as
(15)
where
isthe component due only to the PM field(cogging
torque),
is the component due to the interaction be-
tween the armature field and the permanent magnet field (elec-
tromagnetic torque) and
is the component due to the ar-
mature field only (reluctance torque). Assuming that the rotor
has no saliency (the filler has the same permeability as the mag-
nets), the third term in the equation is zero. The electromagnetic
torque can be calculated as the summation of the torques pro-
duced by the current-field interaction at each slot
(16)
Numerically, the equation becomes
(17)
where
is the number of samples in which the evaluation is
performed
• if is in
a phase A slot and the current has negative direction;
•
is similar for phases B and C;
•
if is outside the
slot opening.
The ripple in the electromagnetic torque has three main
causes. The first is the nonsinusoidal shape of the currents in
most brushless dc motors. The second cause is the mismatch in
shape between the back-emf shape and the current shape. The
third is given by the presence of the stator slots. Equation (17)
is able to predict all three causes. However, in our simulations,
the currents were presumed sinusoidal, and consequently, the
first component will not appear.
The cogging torque is the torque that results from the noncon-
stant airgap reluctance due to slotting (i.e. due to the tendency
of the rotor to align with the low reluctance paths). The torque
is produced by the fluxes that enter the teeth walls, as shown in
Fig. 3. Using the notations of Fig. 3, the cogging torque expres-
sion results into
(18)
Numerically, the equation becomes
(19)
where
and outside the slot opening;
and on the left side of the slot
opening;
and on the right side of the slot
opening.
VII. B
ACK-EMF CALCULATION
The back-emf is the voltage induced in the stator windings by
the variable magnetic field in the airgap. There are two common
definitions of the back emf in literature. One definition regards
the back emf as only the effectof the rotor magnetic field,where
as the other one also includes the mutually and self-induced
voltage between windings as part of the back emf. In this study,
the second definition will be used.
The back emf of one phase can be calculated as the summa-
tion of the voltages induced in each coil side of that phase
(20)
where
isthe magnetic flux in slot j. The numerical expression
390 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 3, SEPTEMBER 2003
(a)
(b)
(c)
Fig. 6. Comparison with FEA for PM field density.
of the above equation becomes
(21)
(a)
(b)
(c)
Fig. 7. Comparison with FEA for armature field density.
where if is a phase A slot and the direction of the
conductor is positive;
if is a phase A slot and the
direction of the conductor is negative;
otherwise.
VIII. C
OMPARISON WITH FINITE ELEMENT RESULTS
The analytical model developed in this paper was compared
with a FEA model. As a test case, a motor with three phases, six
poles, and 18 slots was chosen. Fig. 6 shows a comparison be-
tween the PM field density in the air gap for various dimensions
of the motor.