Design and Control of Axial-Flux Brushless DC Wheel Motors for Electric Vehicles—Part I...

7/31/2019 Design and Control of Axial-Flux Brushless DC Wheel Motors for Electric VehiclesPart I Multiobjective Optimal Des

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004 1873

Design and Control of Axial-Flux Brushless DCWheel Motors for Electric VehiclesPart I:Multiobjective Optimal Design and Analysis

Yee-Pien Yang, Member, IEEE, Yih-Ping Luh, and Cheng-Huei Cheung

AbstractWe have applied multiobjective optimal designto a brushless dc wheel motor. The resulting axial-flux perma-nent-magnet motor has high torque-to-weight ratio and motorefficiency and is suitable for direct-driven wheel applications.Because the disk-type wheel motor is built into the hub of thewheel, no transmission gears or mechanical differentials arenecessary and overall efficiency is thereby increased and weight isreduced. The dedicated motor was modeled in magnetic circuitsand designed to meet the specifications of an optimization scheme,subject to constraints such as limited space, current density, flux

saturation, and driving voltage. In this paper, two different motorconfigurations of three and four phases are illustrated. Finite-ele-ment analyses are then carried out to obtain the electromagnetic,thermal, and modal characteristics of the motor for modificationand verification of the preliminary design. The back-electromotiveforces of prototypes are examined for control strategies of currentdriving waveforms.

Index TermsAxial-flux wheel motor, electric vehicle, magneticanalysis, optimal design.

I. INTRODUCTION

A

GROWING interest in electric vehicles (EVs) has driven

researchers and engineers to develop more efficient and

reliable power systems under the pressure of the protection of

natural environment. Traditional power systems for EVs are

composed of batteries, electric motors with drives, and trans-

mission gears to wheels. Each subsystem converts chemical,

electrical, or mechanical energy into different forms, consuming

energy through the dissipation components of windage and fric-

tion. It is quite essential for engineers to look for an approach

to improve the overall efficiency of electric vehicles, and hence

to increase their driving range. In addition to new battery tech-

nologies, new concepts for the design of motors and their op-

timal driving pattern have attracted substantial attention for the

improvement of overall efficiency and reliability of EVs.

Among the various researchers, Chang [1] provided anexperts survey and concluded that induction motor drives were

preferred for EV propulsion purpose, due to their low cost, high

reliability, high speed, established converter and manufacturing

technology, low torque ripple/noise, and absence of position

Manuscript received December 18, 2002; revised March 16, 2004. This workwas supported by National Science Council of Taiwan, R.O.C., under ContractNSC90-2212-E002-218.

The authors are with the Department of Mechanical Engineering, NationalTaiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TMAG.2004.828164

sensors. However, the permanent-magnet brushless dc motor

featured compactness, low weight, and high efficiency, and

therefore provided an alternative for EV propulsion. Several

permanent-magnet motors have been developed for EVs to

fulfill the special requirements, such as high power density,

high efficiency, high starting torque, and high cruising speed.

These motors can be classified as indirect-driven [2] and di-

rect-driven motors [3][5]. The latter, also called wheel motors

or hub-in motors, are directly mounted inside the wheels, thuseliminating transmission gears or mechanical differentials

with their associated energy loss. The reduction of mechanical

components in transmission chains or gears not only improves

the overall efficiency, but also reduces the weight of the vehicle.

In the class of direct-driven motors, the axial-flux motor

competes the radial-flux motor with some advantages, such as

balanced motor-stator attractive forces, better heat removal con-

figuration, no rotor back iron, and adjustable air gap [6]. Zhang

et al. [7] compared the several axial-flux permanent-magnet

(AFPM) wheel motors for electric cars, and concluded that the

ones with interior PM seemed to be the best compromise in

terms of power density, efficiency, compactness, and capability

characteristics. Their motor configuration had a stator core withslots on both sides, which was sandwiched by two steel rotor

disks with arc-shaped poles and tangentially magnetized square

magnets. Similar design of a two-stage AFPM machine with

ironless water-cooled stator winding was presented by Caricchi

et al., where the switching from series to parallel winding

connections provided variable-speed operations [5]. Lovatt

et al. [8] addressed the solar-powered vehicle and axial-flux

in-wheel motor, which consisted of a magnetic array oriented

along the direction of the flux flow and an ironless air-gap

winding. Hredzak et al. also had their double-sided axial-flux

direct wheel motor, and the torque pulsations were eliminated

by a vector control scheme [9].

This paper proposes a systematic optimal design method-ology on the permanent magnet axial-flux brushless dc wheel

motor and its drive for EVs. Its specifications for motorcycles

are proposed according to the policy of the environmental pro-

tection agency in this country. First, the electromagnetic prop-

erties are modeled in terms of motor geometries and electrical

parameters with sensitivity analyses. Second, the preliminary

motor shapes of three and four phases are obtained by opti-

mizing a set of cost functional subjected to the constraints on the

design parameters and physical properties. Third, the finite-el-

ement analyses are performed on the electromagnetic, thermal,

0018-9464/04$20.00 2004 IEEE


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1874 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

Fig. 1. Explosive graph of the axial-flux disk-type wheel motor.

and dynamical characteristics. Finally, the prototypes are fabri-

cated and experiments are performed to provide information for

possible current driving patterns.

II. MOTOR LAYOUT AND SPECIFICATIONS

The novel design of the disk-type AFPM wheel motor pre-

sented in this paper is a prototype for electric vehicles, which

was developed by the Electro-mechanical Power System Re-

search Group at National Taiwan University, Taipei, Taiwan,

R.O.C. The graph of the wheel motor is illustrated in Fig. 1.

The rotor disk of the hub-in motor has 16 magnets, and is sand-

wiched between the two stator plates, each with 24 teeth to form

a three-phase motor. For the same stator structure, 18 magnets

in the rotor form a four-phase motor. The fractional number of

slots per pole per phase is desired so as to yield a uniform mag-netic force distribution between the stator and rotor, hence elim-

inating most of the cogging torque that usually occurs in perma-

nent-magnet motors. The main magnetic flux flows through two

air gaps between the stator and rotor along the axial direction.

The tire is installed on the outer case, which rotates with the

rotor.

The final shape of this wheel motor is designed to meet

the requested specifications of a multifunctional optimization

scheme, with various constraints, such as limited space, current

density of conductor, flux saturation, and driving voltage.

Instead of being X-connected for four-phase motors or Y-con-

nected for three-phase motors, the coils are independently

wired on stator poles, and are grouped into required phaseswithout a neutral. Since there is no neutral point for independent

winding, the driving voltage is directly applied to each phase.

Therefore, larger back-electromotive force (EMF) is induced

and higher motor speed can be reached. Other specifications

are listed in Table I.

III. MAGNETIC CIRCUIT MODEL

The torque of electric motors is produced by the rate of

change of magnetic energy stored in the air gap. The magnetic

energy comes from the magnetic field generated by the current

flowing in the wires and/or permanent magnets. Both the

TABLE ISPECIFICATIONS OF ELECTRIC VEHICLE WITH WHEEL MOTOR

Fig. 2. Dual axial flux motor topology and its 2-D configuration.

sources generate magnetic flux forming flux loops in the

magnetic materials of the motor. 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 Carters

coefficient [10].

3) The flux flows straightly 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 simpli-

fied to a two-dimensional (2-D) configuration, and its two-side

topology is cut in half for facilitating the magnetic circuit anal-

ysis, as shown inFig. 2. The stator is laminated by sheetsof elec-

tric steel, which are oriented in a stack coming out of the paper.

The fan-shaped magnets are mapped into rectangular ones as

the arc is transformed to a straight line in the 2-D linear motor

mode. Therefore, the total flux through the same area of the sur-

faces of the magnet is unchanged. Take the three-phase motor as

an example, in a section of 360 electrical degrees, the magnetic

circuit of one flux loop is composed of three teeth on each side

of the stator facing toward two permanent magnets embedded


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YANG et al.: DESIGN AND CONTROL OF AXIAL-FLUX BRUSHLESS DC WHEEL MOTORS 1875

Fig. 3. Magnetic circuit model of an electrical period for the three-phasemotor.

in the rotor. As a matter of fact, the ferromagnetic material has

very high permeability so that its reluctance can be ignored. The

magnetic circuit model for one electrical period of half-side of

themotoris depicted in Fig. 3 in terms of the air-gap reluctances,

stator magnetomotive forces, and fluxes through the magnetic

circuit. The air-gap reluctances are denoted by , and

corresponding to three stator phases, while , and

represent the reluctances of stator teeth, back iron and slot,

respectively, and the corresponding fluxes are given by ,and . The air-gap fluxes , and are calculated

by the equivalent magetomotive force of the magnets and effec-

tive air-gap length. The magnetomotive forces of slots are rep-

resented by , and , each of which contains

two phase coils.

The Kirchhoffs voltage law calculates the magnetic flux at

each branch, and the flux density of the air gap is

(1)

where is the overall magnetomotive force distribution

from stator windings and rotor magnets, is the air-gap

length, is the permeability of free space, and denotes the

rotor shift. Therefore, the torque distribution is given by

(2)

where the coenergy is expressed as

(3)

the outer and inner radii of the stator are and , respectively,

and represents the peripheral coordinate along the circle of

Fig. 4. Parameters of stator and rotor of the wheel motor magnet flux.

the average radius . It is worth noting that the

above functions used for the motor design are expressed explic-

itly in terms of the geometric dimensions and properties of mag-

netic materials of the wheel motor, as described in Appendix A.

That inspires an optimal motor designso that betterperformance

objectives are fulfilled.

IV. DESIGN OPTIMIZATION

The electrical and mechanical performances of the wheelmotor depend on its geometry and properties of magnetic mate-

rials. In the past, basic electromagnetic theories and industrial

experience are used for preliminary motor design, and the

finite-element analysis contributes to the detailed modification

and verification of the final shape of the motor. However,

performance of a motor can be enhanced by the optimal

design in terms of efficiency, weight, torque, and frequency

response. These characteristics are described by the above

electromagnetic equations, functions of motor dimensions,

material parameters, and electrical constants.

A. Sensitivity to Design Variables

In most of the efficiency optimization methods, the design

sensitivity analysis is required to determine the derivatives of the

objective functions with respect to the parameters of interest. In

this paper, the sensitivity derivatives are not obtained explicitly

from the equations. Instead, the sensitivities of the motor torque,

torque density, and efficiency to the design variables are numer-

ically investigated. The purposes of the sensitivity analyses are

as follows.

1) The designer may want to discard those design variables

with the least sensitivities of the torque, torque density,

and efficiency of the motor.

2) The designer may keep those design variables con-

stant with sensitivities which are linear, or monotonicfunctions.

3) Only those design variables not included in the above two

cases are retained for the subsequent optimal design.

In this way, the number of the design variables can be kept

minimal to expedite the design optimization of motors. On

the other hand, decision makers can make a proper shift or

modification of the final solutions according to the linearity

of sensitivity with respect to a certain motor parameter. Fig. 4

illustrates some possible design variables in geometry. Other

variables are also investigated in the sensitivity analysis, such

as number of stator poles, number of winding layers, number

of turns per layer, and copper wire diameter.


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Fig. 5. Sensitivity of motor efficiency versus number of stator poles.

Fig. 6. Sensitivity of motor efficiency versus outer radius.

According to the geometric, material, and electric specifica-

tions and constraints, a set of nominal values of design variables

is predetermined. For the three-phase motor configuration, the

sensitivity curves of the motor efficiency with respect to each

single design variable are obtained from the magnetic circuit

model, while other variables are kept at their nominal values.

It is not surprising that the increasing number of stator poles

and larger outer radius produce larger torques and higher

efficiencies, as shown in Figs. 5 and 6. However, these param-

eters are assigned with upper limits due to constrained wheelspace for the motor. On the contrary, a smaller inner radius

of stator poles allows larger magnetic flux distribution in the air

gap to produce torque, and hence the motor efficiency as de-

picted in Fig. 7. It is also reasonable that longer air-gap length

yields larger gap reluctance, and thereby lower efficiency, as

illustrated in Fig. 8. Fig. 9 shows that the thicker rotor pro-

duces more magnetomotive force and yields higher efficiency.

However, the increasing inertia of the motor makes its efficiency

reach a limit. Larger efficiency can be produced by thicker back

iron for less magnetic density and core loss in the elec-

tric steel, as shown in Fig. 10. The motor efficiency is also

sensitive to the slot fraction , magnet fraction

, stator tooth fraction , andshoe depth fraction , as shown in Figs. 1114.

The number of winding layer , the number of turns of layer

, and the wire diameter , which affect the excitation of cur-

rent and the magnetomotive force, have considerable influence

on the efficiency, as displayed in Figs. 1517. The sensitivities

of the motor torque and torque density, defined by the generated

torque per unit motor weight, with respect to the motor param-

eters were also investigated. Different from the sensitivities on

the motor efficiency, the torque is not sensitive to the thickness

of the back iron, stator tooth fraction, and shoe depth fraction.

The design variables are thus determined by the above sensi-

tivity analyses and are listed in Table II.

Fig. 7. Sensitivity of motor efficiency versus inner radius.

Fig. 8. Sensitivity of motor efficiency versus air-gap length.

Fig. 9. Sensitivity of motor efficiency versus rotor thickness.

Fig. 10. Sensitivity of motor efficiency versus back iron thickness.

Fig. 11. Sensitivity of motor efficiency versus slot fraction.

B. Optimization

The compromise programming method in the multifunctional

optimization system tool (MOST) [11] is applied to search for


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Fig. 12. Sensitivity of motor efficiency versus magnet fraction.

Fig. 13. Sensitivity of motor efficiency versus stator tooth fraction.

Fig. 14. Sensitivity of motor efficiency versus shoe depth fraction.

Fig. 15. Sensitivity of motor efficiency versus number of winding layers.

Fig. 16. Sensitivity of motor efficiency versus number of turns per layer.

Fig. 17. Sensitivity of motor efficiency versus wire diameter.

TABLE IIDESIGN PARAMETERS

the optimal values of the design variables that maximize the

following performance indexes:

Motor torque:

(4)

Torque density:

(5)

Motor efficiency:

% (6)

The average torque is a function of design variables im-

plicitly, and is calculated from (2) by 60 equally spaced points

of rotor shift over an electrical period. The driving current is a

six-step three-phase-ON square wave. Likewise, the weight of

the motor , the ohmic loss , the core loss , and the strayloss composed of windage, friction, noise, and other less

dominant loss components, are all functions of design variables.

The general formulations of the compromise programming

method are described in Appendix B. The optimizer weighs

these performance indexes to reach a satisfactory compromise

among the design variables under the prescribed constraints.

1) The motor dimensions must be realized.

2) The permeance coefficient of permanent magnets 4.

3) The slot current density is less than 9 10 (A/m ).

4) Conductor packing factor 0.42.

5) The slot opening is 1.8 times larger than the air-gap

length.


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Fig. 18. Flowchart of optimization process.

6) The peak value of back-EMF per phase should be less

than the component of the driving voltage along the

back-EMF vector.

7) The slot opening is 0.35 times less than the slot pitch.

8) The flux density in electrical steel is less than its satura-

tion value 1.8 T.

9) The shoe depth fraction is confined between 0.25 and 0.5.

The optimizer MOST can deal with real, integer, and dis-

crete design variables simultaneously. In this design, the perfor-

mance indexes, design functions, and prescribed constraints are

all expressed in terms of design variables, in which the number

of stator poles, the number of winding layers, and the number

of turns per layer are integer design variables, while the wire

diameter provided by manufacturers is a discrete design vari-

able. The computational flow of the gradient-based optimiza-

tion algorithm in MOST is composed of the following steps.

1) Give a set of initial guess of parameters in the design space,

and evaluate the objective and constraint functions. 2) Calculate

gradients of the objective and constraint functions with respect

to each design variable. The gradients are not self-designed, butautomatically calculate gradients using forward, backward, or

central finite differences.3) Based on thevalues of functions and

gradients, determine a maximum descent direction, and the next

set of design parameters are determined. 4) If any constraint is

violated or convergence test or not satisfied, adjust the current

point and repeat steps 1)4) until a final solution is obtained.

Fig. 18 shows the flowchart of the optimization process.

The motor parameters of the three and four phases from the

optimal design and the technical data are listed in Table III, and

the detailed geometrical features of the stator tooth and the rotor

of the four-phase motor are illustrated in Fig. 19. Since the coils

are independently wired on stator poles, eight parallel coils are

TABLE IIIMOTOR SPECIFICATIONS FROM THE MAGNETIC CIRCUIT MODEL

grouped into one phase for each side of the stator of the three-

phase motor, and six parallel coils are grouped into one phase

for each side of the stator of the four-phase motor. Although the

geometric dimensions between the three-phase motor and thefour-phase motor are not very different, further investigation on

the motor performance by the finite-element method becomes

necessary for the final decision.

V. FINITE-ELEMENT MAGNETIC ANALYSIS

It is usually difficult to predict precisely the performance

of the designed motor by the conventional magnetic circuit

model. First, the proposed motor has a fractional number of

slots per pole per phase, yielding additional assumptions and

simplifications in the magnetic circuit analysis. Second, the

nonlinear characteristics of magnetic materials produce satura-

tion under overload currents in the condition when the electricvehicle is accelerating or climbing upslope. The finite-element

analyses on the preliminary design prototype become necessary

to provide detailed information on the magnetic flux and

torque distribution, steady-state temperature distribution, and

modal dynamics. These help designers investigate if additional

performance specifications are satisfactory, otherwise iterative

design and modification must be preceded.

As opposed to the magnetic circuit analysis, the finite-element

tool ANSOFT1 numerically calculates the magnetic field

of the 3-D motor configuration. The finite-element mesh is

1ANSOFT is a registered trademark of Ansoft Corporation.


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Fig. 19. Detailed geometrical features of (a) stator tooth and (b) rotor.

(a) (b)

Fig. 20. (a) Three-phase and (b) four-phase torque patterns with respect to rotor shift.

automatically generated for the calculation of magnetic flux,

flux density, and torque distributions. The boundary of the

finite-element model is surrounded by air with enough thickness,

whose magnetic permeability is much smaller than that of

magnetic materials. First, the square-wave current excitation

is assigned. The values of magnetic coenergy in the air gap

of the motor are then calculated at equally spaced rotor shift

angles, and their difference per unit angle determines the

torque as shown in Fig. 20. As summarized in Table IV,for the maximum current of 4.53 A in the conductor due to

the current density limit by its cross section, the maximum

phase currents are, respectively, 72.4 and 54.3 A for three and

four-phase motors, each with the average torque of 5.2 and

5.9 kg m. The maximum torques occur at 6.21 and 6.05 kg m,

respectively, for the three- and four-phase motors. Therefore,

the torque ripple of the four-phase motor generated by the

reluctance components for the coils and magnets is 2.7%,

which is much smaller than 18.9% of the three-phase motor.

Moreover, the four-phase motor has a larger torque constant

per phase current than the three-phase motor. The current

distribution from the battery to multiple phases also relieves

TABLE IVMOTOR PERFORMANCE BY THE MAGNETIC ANALYSIS (SQUARE

CURRENT WAVE)

the battery loading, thus increasing its life cycle. Better driving

performance for the four-phase motor can be expected.


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Fig. 21. Steady-state temperature distribution.

VI. THERMAL AND MODAL ANALYSIS

The thermal conduction analysis provided by COSMOS/M2

predicts the steady-state as well as the transient temperature

distributions in the rotor, stator, and frame of the motor. Fig. 21

shows the steady-state temperature distribution of the wheel

motor under a severe driving operation at 1000 rpm with a

high phase current of 20 A, when the heat source comes from

35 W core loss and 350 W ohmic loss. In the steady state,

the higher temperature holds near the coil at 123 C, which is

below the upper temperature limit of 180 C for the insulation

Class H. The lower temperature around 48 C happens at the

frame surface, where the natural convection coefficient is set

at 75 W/m K and the ambient temperature at 25 C. Also,the transient analysis in Fig. 22 reveals that it will take about

4 h for the motor to reach the steady-state temperature at the

constant speed operation. For the normal operation of electric

vehicles, the result of thermal analysis is quite conservative,

therefore being satisfactory for the motor design.

The natural frequencies of the motor are usually designed be-

yond the maximum speed 1700 rpm or 178 Hz to avoid un-

desirable motor resonance. In the modal analysis, the natural

frequencies are examined for the rotor combined with the outer

cases on which the tire is mounted. The boundaries are simply

supported on the bearings of the motor shaft, while the bearings

are assumed perfectly stiff. The first mode represents an axial

vibration of the motor, and the second mode illustrates a tor-sional vibration along the axial direction. Each mode has a nat-

ural frequency of 261 and 293 Hz, respectively, and resonance

may rarely occur below the maximum speed. The axial and tor-

sional vibration modes are shown in Fig. 23.

VII. PROTOTYPE AND CONTROL STRATEGY

For a limited space of the wheel motor in the hub, the

constraints on its outer diameter and thickness were specified

for the optimization. Therefore, the dimensions of the stator

2COSMOS/M is a finite-element tool of Structural Research and AnalysisCorporation.

Fig. 22. Transient response of temperature in stator and rotor (diamond: statorteeth; square: outer case).

components from the optimal design are similar between the

three-phase and four-phase motors, as shown in Table III. The

most apparent difference between the three-phase motor andthe four-phase motor is the number of magnets on the rotor.

That leads the final decision to fabricate the prototypes of three-

and four-phase motors with the same stator, but different rotors

with 16 and 18 magnets, respectively. Each magnet has the

same size; hence, the only difference of the rotors is their pole

pitch. The same stator with commutable rotors facilitates the

manufacturing process and reduces the cost of the prototype,

under allowable performance tolerance between the design and

the prototype. The lamination material of 0.35 mm AISI M19

steel is selected for the stator teeth and yokes. The magnets of

NdFeB 30SH are embedded in the rotor of the aluminum alloy

6061-T6. The shaft is made of stainless steel, and the cover of

the motor is made of the aluminum alloy. The stator, rotor, andtheir assembly are provided in Fig. 24.

Since the electric vehicles usually run in variable speed and

load conditions, the control strategy must be made for high

efficiency at high speed with enough torque for accelerating. To

achieve this goal, the drive controller must have the following

features. First, the motor is operated under an optimal driving

pattern in terms of optimal current waveform for each indepen-

dent phase. The square, triangular, or sinusoidal current pat-

terns may not be the best for the dedicated design of the wheel

motor. For most of the brushlessdc motors, the maximum torque

may be produced if the input current wave is proportional to

the back-EMF. Fig. 25 shows the back-EMF of the prototype

operated at 103 and 108 rpm, respectively, for the three- andfour-phase wheel motors. The optimal current waveform can be

obtained either by the minimization of the copper loss or by the

maximization of the motor efficiency through the optimization

process.

VIII. SUMMARY AND CONCLUSION

Dedicated disk-type axial-flux brushless three- and four-

phase dc wheel motors have been successfully designed and

fabricated. 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


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Fig. 23. Axial (left) and torsional (right) vibration modes.

Fig. 24. Stator, rotor, and motor assembly.

Fig. 25. Back-EMF of the disk wheel motor: (a) three-phase and (b) four-phase.

design to maximize the output torque, efficiency, and torque

density of the dedicated motor under prescribed constraints.

A systematic procedure from magnetic circuit analyses to

the finite-element modification and verification constitutes a

complete design of the wheel motor. For square-wave current

excitations, the four-phase motor presents satisfactory output

torque with smaller ripple compared to the three-phase motor,

and becomes a promising solution for the electric motorcycle

of required specifications. Finally, the experiments on the

back-EMF of motor prototypes give information of optimal

driving current patterns.

APPENDIX A

The distributions of the air-gap length , magnetomo-

tive force , coenergy , and torque are for-

mulated explicitly as functions of motor geometric and electric

design variables.


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Air-Gap Length:

(A1)

The effective air-gap length on the stator side is defined as [ 6]

(A2)(A3)

(A4)

(A5)

(A6)

(A7)

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. 2 and 4, the air-gap length on the rotor side

is a function of magnet fraction ; it is over the non-

magnet material part, and is approximated by over 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.

Magnetomotive Force:

(A8)

The magnetomotive force produced by each stator

winding is , and is assumed to be equally distributed

under the stator shoe, where is the number of turns on

each stator pole, and is the conductor current. Similarly, the

magnetomotive force of the magnet is approximated by

and equally distributed over each magnet, where

is the coercivity of the magnet and is its thickness.

Coenergy and Torque: In the optimization program, both the

coenergy and torque are coded in discrete forms

(A9)

(A10)

in which an electric period is divided by equally spaced

points, each with mechanical angle apart while

and .

APPENDIX B

The compromise programming [12][14] is to minimize the

distance between the ideal solution and the optimal solution that

is called the compromise solution. The distance measure used

in compromise programming to evaluate how close the set of

nondominated points come to the ideal point is the family of

matrices defined as

(B1)

subject to , in w hich are weights, and are the

minimum value and the worst value of the th objective function

respectively, is the value of implementing the design vari-

able with respect to the th objective, and is the feasible

design space. In the case of , all deviations from are

taken into consideration in proportion to their magnitudes. For

, the larger the deviation, the larger the weight in

. In the limiting case of , only the largest deviation

from the minimum objectives is considered.

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[13] Y. P. Yang and C. H. Tseng, Multiobjective optimization of complexstructures integrated with finite element software on workstations, J.Chin. Soc. Mech. Eng., vol. 16, no. 2, pp. 167179, 1995.

[14] Y. P. Yang and C. C. Tu, Multiobjective optimization of hard disk sus-

pension assemblies: Part IStructure design and sensitivity analysis,Comput. Struct., vol. 59, no. 4, pp. 757770, 1996.

Yee-Pien Yang (A92M02) was born in Taiwan, R.O.C., in 1957. Hereceived the B.S. and M.S. degrees in mechanical engineering from NationalCheng-Kung University, Taiwan, in 1979 and 1981, respectively, and the Ph.D.degree in mechanical, aerospace, and nuclear engineering from the Universityof California, Los Angeles, in 1988.

He was an AssociateProfessor in the Department of MechanicalEngineering,National Taiwan University, from 1988 to 1996. He was promoted to Professorof Mechanical Engineering and led the Electromechanical System ResearchGroup in 1996. His research interests are design and control of electromechan-icalsystems,adaptive controlof flexiblestructures, biomedical engineering, andassistive tool design for the disabled.

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