IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 1, JANUARY 2007
51
Optimal Design and Control of a Wheel Motor
for Electric Passenger Cars
Yee-Pien Yang and Down Su Chuang
Department of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C.
An optimal design and control technology of a wheel motor is proposed for small electric passenger cars. The axial-flux sandwich-type
disc motor is designed with a rotor embedded with neodymium–iron–boron (NdFeB) magnets and two plates of stators, and is directly
mounted inside the wheel without mechanical transmission and differential gears. Sensitivity analyses are performed to choose critical
design parameters, which are the most influential in design objectives, to maximize the driving torque, efficiency, rated speed, and to
minimize the weight of motor under various constraints of size, materials, and power sources. The optimal driving current waveform is
proven to be the same as the fundamental harmonic of the back electromotive force to produce maximum torque with least ripples. The
finite-element refinement results in the motor prototype with a maximum torque over 38 kg m and a corresponding torque density of
about 1.72 kg m/kg at the maximum allowable phase current of 50.25 A (rms). Two such rear driving wheels are able to drive a 600 kg
passenger car to accelerate from 0 to 40 km/h in 5 s on a 15 degree incline. This dedicated wheel motor is applicable to pure or hybrid
electric vehicles as a promising solution to the direct-driven electric vehicle.
Index Terms—Axial-flux wheel motor, electrical vehicle, optimal driving current waveform, sensitivity analysis.
I. INTRODUCTION
E
LECTRIC VEHICLES (EVs) have imperatively attracted
quite a few researchers and automobile companies in developing more efficient and reliable propulsion systems, because of the increasing concerns about natural environment and
growing shortages of petroleum resources. Traditional power
systems for EVs consist of battery, electric motors with drives,
transmission gears, and differentials to the wheels. Each subsystem converts chemical or electrical energy to mechanical
power to drive the vehicle, while consuming energy through
the dissipation components of windage, friction, magnetic hysteresis, and ohmic loss. Various designs for the EVs have been
proposed, such as those powered by hybrid resources, fuel cells,
and solar cells, while new concepts of motor designs and their
optimal driving patterns have attracted substantial attention in
the industry.
In recent years, induction motor drives are preferred for
the EV propulsion system due to their low cost, high reliability, high-speed properties, and manufacturing facilitation.
Concurrently, the permanent-magnet brushless dc motors (indirect-driven and direct-driven) featuring compactness, low
weight, and high efficiency, have become an alternative for
the EV propulsion systems [1]. The latter types of motors are
also called wheel motors or hub-in motors, which are directly
mounted inside the wheels so that the transmission gears and
differentials are eliminated with associated energy loss.
Several successful designs and/or applications of the direct
and indirect-driven motors and drives on the EV have been proposed. Chan [2] designed a five-phase radius flux brushless dc
motor with 22 NdFeB magnets on the rotor, which had a power
Digital Object Identifier 10.1109/TMAG.2006.886153
of 5 kW to drive a car of gross weight of 600 kg through transmission gears. Chen and Tseng [3] proposed a radial flux direct-driven brushless dc motor with 20 teeth on the stator and 22
magnets on the rotor. Compared with conventional three-phase
motors, it had significant improvement on the copper weight,
ohmic loss, volume, torque ripple, and cogging torque. Wijenayake et al. [4] designed, using an optimization procedure,
a 216 V disc-type permanent-magnet brushless dc motor for
a direct-driven EV, whose rated power was 100 hp and maximum speed was 3200 rpm. Patterson and Spèe [5] initiated an
axial flux permanent-magnet brushless dc motor for their solar
energy vehicle. The curb weight of their electric vehicle was
260 kg, the rated power was between 1–2 kW at 72 km/h, and
the output power was 3 kW on the 6 degree slope. A similar
disc-type motor was proposed by Eastham et al. [6], which consisted of three stator plates and four rotors in their simulations.
Alternatively, Uematsu and Wallace [7] chose the reluctance
motor for the EV, and found that the motor had to be operated
at 6000–8000 rpm for enough rated torque at 140–160 Nm and
100 kW output power. Oh and Emadi [8] investigated a commercial axial flux motor for different drive cycles of hybrid electric vehicles by varying the air gap to extend the range of operating speed and improve the efficiency by the method of hardware in the loop. An interesting double-stator starter generator
for the hybrid EV was introduced in [9]; it works as a motor at
low speeds when two stator windings are in series, while two
stators shift relatively to produce various back electromotive
forces in the generator mode. Lately, a different configuration of
an axial-flux motor was presented with a stator sandwiched by
two permanent-magnet rotors [10], but only the computer-aided
design and finite-element analysis were provided. Each of the
above researches on direct-driven wheel motor has its own specifications and configurations for their electric vehicles in terms
of gross weight, driving voltage, maximum speed, rated torque,
and power source.
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 1, JANUARY 2007
TABLE I
SPECIFICATIONS OF AN ELECTRIC CAR
TABLE II
SPECIFICATIONS OF A SINGLE WHEEL
This paper proposes a systematic optimal design methodology for the four-phase, disk-type, permanent-magnet,
axial-flux brushless dc wheel motor for electric passenger cars.
The specifications in Section II are based on the small size,
dual seats, and urban driving requirements. Section III briefly
describes a simplified two-dimensional (2-D) magnetic circuit
model for the preliminary optimal design of wheel motor.
Section IV starts with the sensitivity analysis to determine the
most influential parameters to the motor performance as the
design variables. Then, the multifunctional objective system
tool proceeds to optimize a set of cost functional subject to
the constraints on the design variables and physical properties
to get the preliminary motor geometries. In Section V, further
refinement for the reduction of weight and torque ripples is
performed by the finite-element method, where the thermal
analysis is also investigated for heat dissipation by fins over
motor cover under normal and severe driving conditions.
Finally, Section VI shows that the back electromotive force
(back EMF) provides significant contribution to the optimal
current driving pattern increasing the maximum torque as well
as the torque density. Section VII provides a summary and
conclusions.
II. SPECIFICATIONS
The specifications of a small electric passenger car are illustrated in Table I. The road load on the vehicle consists of
three components—aerodynamic drag force , rolling resistance force , and climbing force , which are expressed as
[11]
(1)
(2)
(3)
where is the air density, is the frontal area of the car,
is the aerodynamic drag coefficient, is the relative vehicle veis the rolling friction coefficient,
locity to the head wind,
is the gross mass or vehicle mass with payload, and is the
inclination angle of the road. The force required to reach the
prescribed acceleration by overcoming the road load is
(4)
Based on the specifications, the required motor torque and the
corresponding speed under various vehicle operation conditions
are calculated and summarized in Table II.
Fig. 1. Stator and rotor of wheel motor (a) assembly, (b) geometries, and
(c) series and parallel phase windings between two stator plates.
The stator and rotor assembly of the wheel motor and their
geometries are shown in Fig. 1. It is a disk type, axial flux,
permanent-magnet, brushless dc wheel motor, with a rotor disk
sandwiched between two stator plates. The stator toroids are
wound and laminated by a continuous sheet of electric steel; its
fan-shaped teeth are factorized by punching slots with varying
pitch for different radius of each layer. The fan-shaped magnets facing the stator windings contribute a main magnetic flux
flowing through two air gaps between the stator and rotor along
the axial direction. The windings on each side of the stator can
be connected in series or parallel as shown in Fig. 1(c). The
serial winding connection splits the line voltage into half on either side of the stator and is suitable for low-speed, high-torque
driving cycle, while the parallel winding connection holds the
maximum voltage that the battery source can supply and is used
for high-speed, low-torque driving cycle.
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YANG AND CHUANG: OPTIMAL DESIGN AND CONTROL OF A WHEEL MOTOR FOR ELECTRIC PASSENGER CARS
53
Fig. 3. 2-D model and magnetomotive forces.
of the magnet, and thickness . Based
current , coercivity
on the assumptions, the magnetic coenergy stored in the air gap
is expressed as
Fig. 2. Configuration of wheel motor on passenger car.
(5)
The wheel motor is directly mounted on the chassis with the
suspension and brake, and the wheel is installed on the shaft rotating with the rotor as proposed in Fig. 2. The stator is covered
with the outer case molded with heat dissipation fins to dissipate heat by natural air convection. The final shape of the wheel
motor is designed to meet the specifications through a multifunctional optimization scheme.
III. MAGNETIC CIRCUIT MODELING OF MOTOR
The optimal design of the wheel motor is to achieve a set of
prescribed objectives in terms of motor torque, torque density,
speed, and efficiency. The excitation current from stator windings and the magnetic flux from rotor magnets generate magnetomotive forces, which create the magnetic flux density distribution and magnetic energy in the air gap. The change of magnetic
energy with respect to the rotor shift produces torque distribution over the air-gap rings.
Based on the assumptions of material linearity and the
collinearity of flux and field densities, the magnetic circuit
model is used to describe the torque produced in the motor. It
is also necessary to make three additional assumptions.
1) The motor is operated in the linear range of the – curve
of the magnetic material.
2) The air-gap reluctance of the slotted stator structure is approximated by the effective air-gap length with Carter’s
coefficient.
3) The flux flows straight across the air gaps between the
stator and rotor, ignoring the fringing flux for simplified
analysis.
The three-dimensional (3-D) motor structure can be simplified to a 2-D configuration, and its two-side topology is cut into
half for facilitating the magnetic circuit analysis, as shown in
Fig. 3. The fan-shaped magnets and stator teeth are mapped into
rectangular ones as the arc is transformed to a straight line in the
2-D linear motor mode. The electromotive forces from stator
windings and magnets are expressed, respectively, as
and
, in terms of number of turns
per tooth,
where
is the air-gap length,
is the overall
magnetomotive force distribution from the rotor magnets and
is the permeability of free space, denotes
stator windings,
the rotor shift, represents the peripheral coordinate along the
, and
and
circle of the average radius
are, respectively, the inner and outer radii of the motor. The
torque resulted from the variation of the coenergy with respect
to the rotor shift is then given by
(6)
All the above functions are expressed explicitly in terms of
motor geometric dimensions, material properties, and electric
specifications of the wheel motor, as described in Appendix A.
The speed of the motor can be obtained by the electrical
equation
(7)
is the back EMF,
, and
are, respecwhere
tively, phase inductance, phase current, phase resistance, and
phase voltage. The phasor expression is
(8)
where the time differentiation is replaced by introducing the
electrical speed . For simplification, no power loss from the
power source to the mechanical output is assumed that
, where
is the average torque,
is the
number of magnets, and denotes the mechanical rotational
velocity of the motor. Then the maximum speed is obtained by
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(9)
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 1, JANUARY 2007
TABLE III
DESIGN VARIABLES FOR SENSITIVITY ANALYSIS
TABLE IV
PREDETERMINED PARAMETERS
where
IV. MULTIFUNCTIONAL OPTIMIZATION
The performance of the wheel motor is usually evaluated
based on its efficiency, weight, maximum torque, torque ripple,
and time response, which are also known as the design objective functions or performance indices, describing mechanical
and electrical dynamics in terms of motor geometries, magnetic
materials, and driving conditions. First, the sensitivity analysis
is required to determine the derivatives of the objective functions with respect to the parameters of interest, then a set of
design variables are determined. In this paper, the sensitivity
derivatives are not obtained explicitly from the equation, but are
numerically investigated. Second, the multifunctional optimization system tool is used to search for the optimal values of the
design variables that maximize the performance indices, subject to various constraints on the limitation of space and material properties.
Fig. 4. Sensitivity of torque density versus inner radius.
Fig. 5. Sensitivity of torque density versus outer radius.
A. Sensitivity Analysis
The purposes of the sensitivity analyses are as follows.
1) The designer may want to discard those design variables
with the least sensitivities of torque, speed, torque density,
torque ripple, and efficiency of the motor.
2) The designer may keep those design variables constant
with sensitivities, which are linear, or monotonic functions.
3) Only those design variables that are not included in the
above two cases are retained for the subsequent optimal
design.
Table III lists all the variables for the sensitivity analysis,
while other motor parameters are predetermined in Table IV
according to physical facts and previous design experience. It
would be a time and space consuming process to illustrate all
the sensitivity curves, though it is worth providing sufficient information for selecting critical design variables. This section
illustrates some sensitivity curves of torque density and efficiency with respect to a single variable while others are fixed at
their prescribed nominal values. The final decision of the design
variables for optimization is made by evaluating the sensitivity
Fig. 6. Sensitivity of torque density versus magnet thickness.
index, which is defined as the ratio of the variation of motor performance and the variation of design variable.
1) Torque Density: Figs. 4–9 show selected illustrations of
the sensitivity curves of torque density versus motor dimensions. It is not surprising that the torque density of the wheel
motor increases upon decreasing its inner radius and increasing
its outer radius, as shown in Figs. 4 and 5. The thicker the
magnet, the more the magnetic energy stored and the larger the
torque produced as shown in Fig. 6. The magnet fraction, defined as the ratio of the width of magnet and the rotor pitch,
affects the torque density in the way as shown in Fig. 7. Fig. 8
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YANG AND CHUANG: OPTIMAL DESIGN AND CONTROL OF A WHEEL MOTOR FOR ELECTRIC PASSENGER CARS
Fig. 7. Sensitivity of torque density versus magnet fraction.
55
Fig. 11. Sensitivity of torque density versus tooth thickness.
Fig. 12. Sensitivity of efficiency versus numbers of winding layers.
Fig. 8. Sensitivity of torque density versus fraction of slot opening.
Fig. 9. Sensitivity of torque density versus air-gap length.
Fig. 10. Sensitivity of max. torque versus thickness.
indicates that the torque density is increased by diminishing the
fraction of slot opening, which is defined as the ratio of slot
opening and pitch of the stator. Fig. 9 shows that the air gap has
minor influence on the torque density.
Additional sensitivity curves of the torque density were linear
or approximately linear with respect to design variables, such
as back iron thickness, shoe depth fraction, tooth fraction, and
tooth thickness. This is because they are irrelevant to the maximum torque, and their variations only affect the motor weight,
as shown in Figs. 10 and 11, for the sensitivities of maximum
torque and torque density with respect to tooth thickness. It is
obvious that the number of winding layers and number of turns
Fig. 13. Sensitivity index of maximum torque.
per tooth simultaneously increase the maximum torque and
motor weight, and they must reach a compromise by evaluating
the motor efficiency. An example is shown in Fig. 12, where
the efficiency curve reaches a limit as the layers of winding are
increased.
2) Sensitivity Indices: The sensitivities are investigated by
the maximum torque, torque density, torque ripple efficiency,
and maximum speed of the motor with respect to design variables. For summarizing the results and making a final decision
on the design variables, we define the sensitivity indices as the
ratio of the variation of motor performance and the variation of
design variable. For example, the sensitivity index of maximum
, where symbolizes for design
torque is denoted by
, torque
variables; the sensitivity indices of toque density
, efficiency , and maximum speed are expressed,
ripple
respectively, as
, and
.
Fig. 13 indicates that the motor torque varies significantly
with the air-gap length, number of winding layers, number of
turns per tooth, and wire diameter. However, it varies relatively
less with the inner radius, outer radius, magnet fraction, and
thickness as well as other variables in Table III where each variable has a number used to locate on the -axis of the sensitivity
index plot. Fig. 14 shows that the torque density is heavily
affected by magnet fraction, magnet thickness, stator tooth
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 1, JANUARY 2007
Fig. 14. Sensitivity index of torque density.
Fig. 15. Sensitivity index of efficiency.
Fig. 16. Sensitivity index of torque ripple.
fraction, air-gap length, number of winding layers, number of
turns per tooth, and wire diameter. In Fig. 15, the outer radius
and copper wire diameter have indispensable influence on
efficiency, where larger outer radius produces more torque and
larger wire diameter results in less copper loss. Other variables,
such as inner radius, magnet fraction and thickness, air-gap
length, number of winding layers, and turns per tooth, also
provide more influence on efficiency than the rest of the design
variables. Fig. 16 shows that the variations of magnet and slot
opening fractions are very crucial to the torque ripple. The
variation of the air-gap length is an apparent index of motor
speed as shown in Fig. 17; the air-gap length is sometimes made
adjustable to tune the motor speed under various operations [8].
From the above sensitivity analyses, we find the following
facts.
1) The inner and outer radii determine the size of the air-gap
surface between the stator and rotor of the wheel motor,
and therefore, very sensitive to torque and torque density.
Both are important design variables.
2) The magnet fraction and thickness have much influence on
all motor performances, especially the maximum torque,
torque ripple, and speed of motor; they are selected as design variables.
Fig. 17. Sensitivity index of maximum speed.
3) Stator tooth fraction has less influence on the maximum
torque, and is of little effect on efficiency, ripple, and speed.
However, the tooth fraction is an indication of the crossover
area that allows stacking windings in terms of the number
of turns and the number of layers; those determine the maximum allowable magnetomotive force to produce torque. It
is therefore selected as a design variable.
4) The tooth thickness, slot opening fraction, shoe depth fraction, and back iron thickness are not influential for torque,
torque density, efficiency, and speed, and are excluded
from the design variables.
5) The air-gap length is a compromising factor between
torque and speed. It is usually made as small as possible
to obtain a maximum torque under the constraint of speed
limit. This design variable is set at 1 mm, which is reasonable for manufacturing as the tradeoff between precision
and cost is considered.
6) The number of winding layers, number of turns per tooth,
and copper wire diameter determine the value of magnetomotive force, which produces torque but restricts speed.
They are selected as important design variables to compromise between the maximum torque and speed.
Eventually, eight variables are chosen (denoted by stars in
Table III) for the following optimal motor design. It is also worth
noting that the torque ripple was included in the sensitivity analysis but is excluded from the optimal design. The fact is that
the torque ripple is very sensitive only to the magnet fraction
and slot opening fraction of the wheel motor. In the optimization process, the 2-D model used in the optimal design does
not provide accurate values of torque ripple. Therefore, it has
not chosen an objective function in the following optimization
process, but remained for further design refinement by the finite-element method in ways of adjusting its corresponding sensitive variables.
B. Design Optimization
The compromise programming method in the multifunctional
optimization system tool (MOST) [12] is applied to search the
optimal values of design variables that maximize the following
performance indices:
Motor torque:
Motor weight:
Motor efficiency:
Motor speed:
(10)
(11)
% (12)
(13)
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YANG AND CHUANG: OPTIMAL DESIGN AND CONTROL OF A WHEEL MOTOR FOR ELECTRIC PASSENGER CARS
57
TABLE V
MOTOR CONSTRAINTS IN OPTIMIZATION
TABLE VI
OPTIMIZED MOTOR PARAMETERS AND PERFORMANCE
Fig. 18. 3-D model of magnetic flux density distribution.
The rated torque
is an implicit function of design variables, and is calculated from the magnetic circuit model by
equally spaced points of rotor shift over an electric period. Like, the core
wise, the weight of the motor , its rated speed
, and the stray loss
composed of windage, friction,
loss
noise, and other less dominant loss components are all functions of design variables. The optimizer weighs these performance indices to reach a satisfactory compromise among the design variables under the prescribed constraints, as summarized
in Table V.
The optimizer MOST can deal with real, integer, and discrete design variables simultaneously. In this design, the performance indices, design functions, and prescribed constraints
are expressed in terms of design variables, in which the number
of winding layers and the number of turns per layer are integers; the wire diameter provided by the manufacturers is discrete, while others are real. The computation flow of the gradient-based optimization algorithm in MOST is composed of:
1) initial guess of design variables; 2) calculation of gradients of
objective and constraint functions; 3) determination of the maximum descent direction and the next set of design parameters;
and 4) convergence test until reaching the final solution. Different weightings were assigned to the four performance indices
for which relative importance is addressed for the optimization,
but listed in Table VI are the three best results in terms of motor
performances.
V. FINITE-ELEMENT ANALYSIS
The above results are obtained by the optimization scheme
based on the 2-D magnetic model. This conventional magnetic
model does not usually produce precise results of optimization
due to linear assumptions and simplified 2-D motor configuration. Therefore, further investigation on the motor performance
and refinement of geometries must be made by finite-element
analysis for the improvement of dynamics and the reduction of
weight of the dedicated wheel motor, thus providing designers
advanced information for making a final decision.
From Table VI, the maximum torque of the weighting ratio
1:1:1:1 is the best among the three. The motor weight obtained
from the case of the weighing ratio 1:6:1:1 has the least value
, representing the impordue to the heaviest weighting on
tance of reducing weight. The best motor speed is also obtained
from the 1:6:1:1 weighting. However, the best torque density
happens for the 1:1:1:1 weighting. Both cases deserve further
investigation by the following finite-element analysis, which refines the motor shape to reduce its weight and torque ripple.
The magnetic analyzer Maxwell 3D of ANSOFT1 is used to
verify the motor prototype performance and to refine the motor
geometries by numerical calculation on the 3-D motor configuration. A quarter section of the motor comprising a half electric period is sufficient to model and simulate the motor performance. The finite-element mesh is automatically generated
for the calculation of magnetic flux, flux density, and torque
distributions. The boundary of the finite-element model is surrounded by air of enough thickness that its magnetic permeability is much smaller than other magnetic materials in the
magnetic loop. Fig. 18 shows the magnetic flux density distribution over a quarter section of the motor, where the flux flows
through the air gap, stator teeth, and back iron through a closed
,
magnetic loop. On the cylinder of radius
the flux density distribution curves under the maximum torque
output are illustrated in Fig. 19.
A. Reduction of Motor Weight
According to the sensitivity analysis, the torque and speed of
the wheel motor are insensitive to the back iron thickness and
tooth thickness. However, the adjustment of these variables reduces the weight before the magnetic flux is saturated. Fig. 20
1ANSOFT
is a registered trademark of Ansoft Corporation.
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 1, JANUARY 2007
TABLE VII
PERFORMANCE BEFORE AND AFTER FE REFINEMENT
Fig. 19. Magnetic flux density distribution on R.
Fig. 20. Torque versus back iron thickness curve for weighting 1:1:1:1 ( ) and
1:6:1:1 ().
Fig. 22. Torque versus shoe depth fraction.
B. Reduction of Torque Ripple
Fig. 21. Torque versus tooth thickness curve for weighting 1:1:1:1 ( ) and
1:6:1:1 ().
presents the change of torque with respect to back iron thickness. For the weighting 1:1:1:1 on the objective functions, the
optimal design of back iron thickness of 10 mm is over designed,
therefore its reduction to 8 mm is good enough to satisfy the
torque specification. The tooth thickness is also irrelevant to the
torque of the motor if it provides sufficient area for the magnetic flux without saturation under an allowable maximum current excitation. The space for stack coil windings, however, is
limited by the tooth thickness. By fixing the number of turns
of the winding, the stacking factor, defined as the ratio of the
area occupied by coils and the slot area, varies with the tooth
thickness. Fig. 21 reveals that for the weighting of 1:1:1:1, the
tooth thickness can be reduced from the optimal design of 40 to
36 mm without diminishing torque, but concurrently preserving
a permissible stacking factor and increasing the torque density
by reducing the weight. Similar evaluation refinement for the
case of weighting 1:6:1:1 is also performed as summarized in
Table VII.
The torque ripple is caused by uneven distribution of flux density in the air gap between rotor and stators, where the air-gap
length and reluctance are functions of rotor shift and contain
many high harmonics. From the motor design point of view, the
torque ripple can be reduced by properly shaping the geometry
of the motor so that the variation of air-gap reluctance becomes
smooth by evaluating the produced torque ripple. From the sensitivity analysis, the torque ripple is sensitive to the thickness
and width of the magnet, air-gap length, stator shoe depth, and
slot opening. Either increasing the thickness and width of the
magnet or the reduction of air-gap length will produce more
magnetic energy in the air gap under the stator teeth and magnets, thereby intensifying the variation of the flux density distribution. Since the optimal design has satisfied the torque specification, the values of thickness and width of the magnet and
air-gap length are preserved in the design.
The shoe depth fraction was not a design variable but predetermined at 0.11 mm. From Figs. 22 and 23, the average torque
excited by a square wave is not much affected by the shoe depth
fraction, but the torque ripple is. Furthermore, the torque ripple
is sensitive to the slot opening fraction as shown in Fig. 24. The
optimal design determines the slot opening fraction at 0.28 and
the corresponding slot opening is about 9 mm; the size of the
slot is therefore large enough for winding coils. Manufacturers
suggest a permissible slot opening value of 5 mm that the corresponding slot opening fraction can be chosen at 0.14 according
to Fig. 24 for less torque ripples.
C. Thermal Analysis and Decision Making
After the finite-element refinement, the case of the weighting
ratio 1:1:1:1 prevails in the performance of motor torque, torque
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YANG AND CHUANG: OPTIMAL DESIGN AND CONTROL OF A WHEEL MOTOR FOR ELECTRIC PASSENGER CARS
59
Fig. 23. Torque ripple versus shoe depth fraction.
Fig. 25. Heat dissipation fins on motor cover.
Fig. 24. Torque ripple versus slot opening fraction.
Fig. 26. Steady-state temperature distribution of wheel motor at rated cruising
speed of 40 km/h.
density, and torque ripple over the weighting 1:6:1:1, although
the latter comes up with less weight. The comparison of efficiency is not presented here because it was similar among the
three cases in Table VI, and it depends more on the factorization
and motor drive design. The case of the weighting ratio 1:1:1:1
is then chosen as the final design.
Finally, the thermal analysis has been carried out by designing heat dissipation fins distributed as concentric circles
over both plates of aluminum covers, as shown in Fig. 25. The
height, width, and the gap of the fin are 12, 3, and 12 mm,
respectively. By the simulation of a passenger car moving at a
cruising speed of 40 km/h, the convection coefficient is chosen
at 25 W/m K and the ambient temperature is 20 C. The
major heat source coming from the stator copper windings is
estimated to be 640 W for the rated and cruising speed, and
results in a steady-state temperature distribution as shown in
Fig. 26. The maximum temperature is 88 C around the stator
teeth made of alloy steel and the temperature on the cover
is about 82 C. The heat dissipation on the normal driving
condition is satisfactory. Without the heat dissipation fins,
the maximum temperature on the stator teeth increases up to
130 C at the steady state, which may cause demagnetization of
magnets and deteriorate motor performance. In a severe driving
condition when the car is accelerating on the 15 slope and
a 2560 W power is to be dissipated from the stator windings
through fins, the maximum temperature goes up to 100 C
after 11 min; the motor will be soon heated up over 130 C
after 11 min without the dissipation fins. The results suggest
that severe driving must be restrained or protection circuits of
the motor drive have to be properly designed for better motor
performance. Further improvement on the heat dissipation can
be achieved by introducing forced cooling systems by liquid
coolant or fans, but such an improvement will consume extra
power on the vehicle.
VI. OPTIMAL CONTROL WAVEFORM
In the process of optimal design, the motor was excited by a
square wave for simplicity. For better driving torque and efficiency, an optimal control current pattern for the wheel motor
was proved to be the same as the flux variation in the air gap between the stator and rotor [13], [14], where the optimal driving
waveform was obtained by maximizing the average torque of
the motor under a constraint on the average ohmic loss. In other
words, the stator current to produce a maximum torque must
have a phase lag of 90 electrical degrees from the permanentmagnetic flux of the rotor. Since the back EMF occurs as the
flux linkage through the coil changes, the flux variation is proportional to the back EMF and the optimal current waveform
turns out to be proportional to the back EMF.
It is well known that the torque constant of a dc motor is defined as the ratio of the produced torque and the corresponding
phase current, and is theoretically the same as the electric constant defined as the proportional coefficient of back EMF and
rotational speed of the motor. The back EMF waveform can thus
be obtained by calculating the torque constant of the motor via
the finite-element tool ANSOFT with 3-D motor model. Fig. 27
illustrates the back EMF of phase A and its corresponding spectrum, where the back EMF wave presents a bunch of high frequency harmonics causing undesirable torque ripples. Since the
amplitude of the fundamental harmonic dominates, it is then
extracted as the proposed driving current waveform with less
torque ripple.
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60
IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 1, JANUARY 2007
Fig. 27. Back EMF of phase A and its spectrum.
TABLE VIII
MOTOR PERFORMANCE WITH VARIOUS CURRENT PATTERNS
Table VIII illustrates the maximum torque and torque ripple
for different kinds of driving current patterns at the maximum
allowable phase current of 50.25 A rms. The original back EMF
pattern produces the largest torque, but its high order harmonics
yields the worst torque ripple up to 31.1%. The fundamental
harmonic of the back EMF, which has a phase lag of less than
1 degree from a pure sine wave, produces a little less torque but
the resulting torque ripple is the least. The current input of a
pure sine waveform also produces small torque ripples as the
fundamental harmonic of the back EMF does, but the latter provides 1.69 kg m more torque than the pure sine current input.
Fig. 28 shows the torque distributions produced by the input
current waveforms of square, pure sine, original back EMF and
the fundamental harmonic of the back EMF. The current excitation of the pattern of the fundamental harmonic of back EMF
increases the torque density up to 1.72 kg m/kg due to the increase of the average torque, and therefore is chosen as the optimal control waveform for the dedicated wheel motor.
Fig. 28. Torque distributions for various current waveforms.
motor under prescribed constraints. A systematic procedure
from the magnetic circuit analyses to the finite-element modification and verification constitutes a complete design of the
wheel motor. The back electromotive force of the motor prototype provides information about the optimal driving current
pattern, and its fundamental harmonic is verified to yield a least
torque ripple. The resulting prototype has a torque density of
1.72 kg m/kg that provides a maximum torque up to 38.65 kg m
torque for an optimal current input at its maximum allowable
phase current of 50.25 A (rms) with a least ratio of torque
ripple; this enables a 600 kg passenger car to accelerate from
0 to 40 km/h in 5 s up a 15 degree incline. The thermal analysis provides satisfactory information on the heat dissipation
through fins over the cover. After the prototype wheel motor
is fabricated, its performance will be tested on a dynamometer
and an optimal driving strategy will be determined by the
design of power drive and power management system.
APPENDIX A
, magnetomotive force
,
The air-gap length
, and torque
are formulated explicitly as
coenergy
functions of motor geometric and electric design variables.
A. Air-Gap Length
(A-1)
The effective air-gap length on the stator side is defined as [15]
VII. SUMMARY AND CONCLUSION
This paper presents a multifunctional optimization design
of an axial-flux permanent-magnet brushless dc wheel motor
for a small electric passenger car. The sensitivity analysis with
the magnetic circuit model provides an effective way to select
the design parameters, which are iteratively tuned through the
multiobjective optimal design process to maximize the output
torque, speed, efficiency, and torque density of the dedicated
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(A-2)
(A-3)
(A-4)
(A-5)
(A-6)
(A-7)
YANG AND CHUANG: OPTIMAL DESIGN AND CONTROL OF A WHEEL MOTOR FOR ELECTRIC PASSENGER CARS
in which is the minimum air-gap length between the stator and
the rotor, and is the slot width of the stator.
Referring to Figs. 1 and 2, the air-gap length on the rotor
; it is
over the
side is a function of magnet fraction
over
nonmagnet material part, and is approximated by
.
the width of the magnet, where its recoil permeability is
Apparently, the distribution of the air-gap length is an explicit
function of geometric design variables.
B. Coenergy and Torque
In the optimization program, both the coenergy and torque
are coded in discrete forms:
(A-8)
(A-9)
in which an electric period is divided by
equally spaced
apart while
points, each with mechanical angle
and
.
ACKNOWLEDGMENT
This work was supported by the National Science Council of
Taiwan, R.O.C., under Contract NSC92-2218- E-002-020, and
in part by NSC90-2218-E-002-053.
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Manuscript received May 27, 2005; revised September 25, 2006. Corresponding author: Y.-P. Yang (e-mail: ypyang@ntu.edu.tw).
Yee-Pien Yang (M’91) was born in Taiwan, R.O.C., in 1957. He received the
B.S. and M.S. degrees in mechanical engineering from the National ChengKung University, Taiwan, R.O.C., in 1979 and 1981, respectively, and the Ph.D.
degree in mechanical, aerospace, and nuclear engineering from the University
of California, Los Angeles, in 1988.
He was an Associate Professor in the Department of Mechanical Engineering
of National Taiwan University from 1988 through 1996, and was promoted to the
rank of Full Professor and led the Electromechanical System Research Group
since 1996. He was a visiting scholar at the University of California, Los Angeles, from 2004 to 2005, conducting research on the design and control of
electromechanical systems, biomedical engineering, and assistive tool design
for the disabled.
Down-Su Chuang was born in Taiwan, R.O.C., in 1979. He received the B.S.
degree in mechanical engineering from the National Cheng Kung University and
the M.S. degree in mechanical engineering from the National Taiwan University,
Taiwan, in 2001 and 2004, respectively.
He is currently a System Engineer at the Asia Pacific Fuel Cell Technology
Corporation, responsible for fuel cell system integration. His expertise is on the
design and control of electromechanical systems, and fuel cell system controls.
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