Modeling and Design Optimization of Permanent Magnet Linear

Synchronous Motor with Halbach Array

Abstract: In this paper, an analytical analysis is presented to calculate air gap flux density distribution, thrust and efficiency in air-core permanent magnet linear synchronous motor with Halbach array based on Maxwell equations. In order to improve mean thrust, thrust ripple, magnet and copper consumption, the main design parameters of analyzed machine are optimized by using genetic algorithm in an appropriate objective function. The results show an enhancement in motor performance. Finally, we have used 2-D nonlinear time stepping finite element method to demonstrate validity of the analytical analysis and optimization method.

Keywords: Optimization, finite-element method,

analytical model, linear synchronous motor, genetic

algorithm, Halbach array.

Nomenclature

� Magnetic field intensity vector

��, �� Unit vectors in � and � direction

� Motor width

Air-gap length

�� Winding factor

� Coil packing factor

� Number of pole pairs

�� Magnet height

�� Magnet width

� Pole pitch

�� Horizontal magnetized magnet width

��� Winding pitch

���� Output power

��� Copper loss

����� Mechanical loss

�� Remanence

�� Synchronous speed

� Current density

�� Equivalent magnetization current

� Load angle

1. Introduction

Nowadays, linear motors, which can provide thrust

force directly, are more and more used in factory

automation and numerical control systems. They offer

numerous advantages over rotary-to-linear systems, in

regard to their simplicity, efficiency, positioning accuracy

and dynamic performance, in terms of both their

acceleration capability and bandwidth [1], [2]. Of the

various types of linear motor, permanent magnet linear

synchronous motor (PMLSM) with Halbach magnetized

topology exhibits an essentially sinusoidal air-gap field

distribution and a sinusoidal back-emf waveform, as well

as negligible cogging force, without employing skew or

distributed winding [3], [4]. Hence, in this paper an air-

core PMLSM with Halbach array (HA) which provides

extra high accuracy operation is employed for design

optimization.

Proper performance of PMLSMs requires

optimizations of their design and control. Design of

PMLSMs has so for been presented based on different

modeling techniques including magnetic equivalent

circuit with lumped elements, analytical method using

Maxwell equations, and finite-element method (FEM)

[5]. This paper uses analytical method for design

optimization because of it has some advantages over

other methods for preliminary design of PMLSMs which

lie in its accuracy and suitability for optimization.

Design optimization of permanent magnet (PM)

machines with vertical magnetization [6], has been

considered in many researches so far in which different

objective have been studied. Nevertheless, Halbach

topologies gained less attention. Among limited work on

the design optimization of this type of motors, a

simultaneous optimization of weight and torque has been

investigated by choosing inner and outer stator width,

height, and angle for angular direction magnetization [7].

J. Choi et al. have applied a magnet array maximizing the

tangential force to a torsional spring composed of two-

and three magnet rings [8]. Reduction of detent force has

also been studied in [9]. R. Huang et al. have minimized

the normal force by using genetic algorithm in

synchronous PM planner motor with HA [10].

Unfortunately, in recent studies, the optimization of

air-core PMLSM with HA has not been considered yet.

Therefore, in this paper, mean thrust, thrust ripple,

magnet and copper volume of an air-core PMLSM with

HA are optimized. Usually, improvement in one feature

N. Roshandel Tavana, and A. Shoulaie nroshandel@ee.iust.ir, and shoulaie@ee.iust.ac.ir

Department of Electrical Engineering, Iran University Of Science and Technology, Tehran, Iran

441

might have an adverse effect on the other one. Therefore,

a compromise is needed between these features. In order

to achieve this goal, a multi-objective optimization is

employed. First, analytical method is presented for

PMLSM with HA. An effective objective function

regarding mean thrust, thrust ripple, magnet and copper

volume is then proposed and the genetic algorithm is

used to optimize design parameters. Finally, 2-D

nonlinear time stepping FEM is carried out to evaluate

the design optimization.

2. Analysis Model

2.1 Motor Topology

Fig. 1 shows a schematic view of a double sided air-

core PMLSM with HA. The moving short primary of

motor is a three-phase air core winding. Each secondary

consists of back iron and PMs facing the primary

windings.

Fig. 1: Topology of a double sided PMLSM with HA.

2.2 Field Distribution Due to PM Source

In order to establish analytical solutions for the magnetic

field distribution in the foregoing machine topology, the

following assumptions are made:

1) the length of machine is extended to infinity.

2) the permeability of iron core is infinite.

3) linear behavior of analysis model can be assumed.

4) the permeability of PM material is equal to the

permeability of free space.

Consequently, the magnetic field analysis is confined to

two regions, viz, the airspace/winding and permanent

magnet region. Fig. 2 shows simplified model of

machine.

Fig. 2: Simplified model of machine and equivalent

magnetization current distribution.

Therefore, the governing field equations, in terms of

magnetic vector potential lead to Laplace and Poisson

equations as follows [11]:

! "#$% & 0 in region 1"#$# & 01 2� in region 2

4

(1)

where 2� & " 56 and 7 is magnetization vector of

PMLSM with HA and is given by

6 & 7��� 87��� (2)

where 7� and 7� denote the components of 7 in �

and � directions, respectively, and may be expressed as

Fourier series

7� & 9 �� sin ;<���2 = cos?<��@

A

�B%,C,…

(3)

7� & 9 �� cos ;<���2 = sin?<��@

A

�B%,C,…

(4)

where �� & EFG�HIJ and <� & �H

K . The boundary condi-

tions to be satisfied by the solution to (1) are

4L#�|�B 0 4L%�|�B & 7�; 4L%�|�BO�P & 07� 4L%�|�BQR�S #⁄ & 0; 4�%�U�B & 4�#�U�B

(5)

By solving (1), the tangential and normal components

(�� and ��) of the flux density produced by the PMs in

the air gap is provided from the curl of $# as follows:

�%�?�, �@ & 9 <�VW%�X�Y� 0 Z%�XO�Y�[A

�B%,C,…cos?<��@

�%�?�, �@ & 9 <�VW%�X�Y� 8 Z%�XO�Y�[A

�B%,C,…sin?<��@

(6)

where W%� and Z%� are given by

W%� & ; \YIJ#�Y�]PY?^P_`_^S@R�Y= · b;sin ;

�YKc# = 8

cos ;�YKc# == X#�Y�P 8 2 sin ;�YKc

# = X�Y�P 8sin ;�YKc

# = 0 cos ;�YKc# =d (7)

Z%� & X#�Y?QR�S@ · We� (8)

2.3 Thrust and Efficiency Prediction

The thrust force exerted on the armature, resulting

from the interaction between the winding current and the

PM field, is given by

442

f & g 2 5 h ijk

(9)

Assuming that each coil side on the armature occupies

areas bounded by �% & � 0 �� 2⁄ , �# & � 8 �� 2⁄ ,

�% & and �# & 8 �� . The total thrust force exerted

on the one coil side may be obtained from the following

integration [12]:

l & 2g g m���%�nQR�S #⁄

Q

�RKS #⁄

�OKS #⁄i� i�

(10)

which may be written as

l & 9 o� sin<��A

�B%,C,…

(11)

where o� is given by

o� & 4�� ��� sin ;<���2 =g VW%�X�Y�

QR�S #⁄

Q8 Z%�XO�Y�[ i�

(12)

Therefore, the total force l%�� exerted on a phase

winding comprising a number of series connected coils, is

obtained as

l%�� & 9 l� cos<� ;� 0 ���2 =

A

�B%,C,…

(13)

where l� is defined as thrust constant of the qth

harmonic, and is given by

l� & 2�o� sin ;<����2 =

(14)

For a three-phase machine carrying balanced sinusoidally

time-varying currents, the total thrust force is obtained

from

lC�� & lr 8 lF 8 ls (15)

lC�� & C#l% sin ;HK � = 89 C

#l� cos ;<�R%� 0A�Bt,%%,…

<�KSc# 8<�� = 89 C

#l� cos ;<�O%� 0A�Bu,%C,…

<�KSc# 8<�� = (16)

As will be evident from (16), the mean thrust and

normalized total thrust ripple is given by

lvw� & 32l% sin ;

y� � =

(17)

zz{ &|} l�#A

�Bt,u,%%,%C…l%

(18)

In this kind of motor, iron loss is negligible due to lack of

iron in moving part. The essential part of electrical loss is

the copper loss. Therefore, motor efficiency is given by

~ & �������� 8 ��� 8 �����

(19)

where ���� & lvw� · ��.

3. Optimal Design Using Genetic Algorithm

Some of the main design parameters of motor are selected

as variables whose values are determined through an

optimization procedure. A two-pole PMLSM with HA

shown in Fig. 1 is selected as the basis for optimization.

The geometric parameters of the motor are listed Table I.

In this paper, design variables are motor width, magnet

height and width of horizontal magnetized magnet in

PMLSM with HA. The fixed variables are pole pitch,

pole pairs, primary windings current density and coil

width. To have a more realistic design, some constraints

are applied to design variables listed in Table II.

Table I. Specification of Initial Motor.

Parameter Unit Value

Number of

Coil/Phase/Pole

- 1

� - 1

� mm 42

�� mm 14

�� mm 7

mm 1.6

�� mm 3.5

�� mm 8

� mm 72

�� m/s 4.2

����� w 0.04����

Table II. Design Variables and Constrained Conditions.

Parameter Unit Min Max

�� mm 2 5

�� mm 0 21

�� mm 5 10

mm 0.8 2

� mm 70 74

lvw� N 84 86

To obtain an optimal design considering motor thrust

force, PM volume, copper volume and normalized force

ripple, the fitness function is defined as follows:

ll & lvw���?�%, … , ��@zz{�]?�%, … , ��@ · j\���?�%, … , ��@ · j����?�%, … , ��@

(20)

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where �%,…,�� are design variables. The parameters

%,…,E are chosen by the designer to determine the

relative importance of thrust, PM volume, copper volume

and thrust ripple in optimization. Maximization of ll

fulfills simultaneously all objectives of optimization.

Such an objective function provides a higher degree of

freedom in selecting appropriate design variables. The

genetic algorithm is employed to search for maximum

value of ll.

The genetic algorithm provides a random search

technique to final a global optimal solution in a complex

multidimensional search space.

A genetic algorithm with parameters listed in Table III

is employed to search for optimal design. Fig. 3 shows

the flowchart of genetic algorithm. In this paper, the

Roulette wheel method is used for selection and at each

step elite individual is sent directly to the next population

[13], [14].

Table III. Genetic Algorithm Parameters.

Parameter Value

Mutation rate 0.2

Selection rate 0.5

Population size 50

Number of generation 500

Fig. 3. Flowchart of genetic algorithm.

The values of %,…,E in general dependent on the

designer's will and the requirement of the motor

application, here three sets of power coefficient are used

to optimize the motor. In the first step, thrust force to

thrust ripple ratio is maximized by using % & 1, # & 1

and C & E & 0. In the second step, due to magnet is

more expensive than copper, % & 1, # & 0, C & 1.5

and E & 1.1 are chosen for optimization. Eventually, in

the third step, more emphasis is placed on the

minimization of copper and PM volume rather than the

minimization of thrust ripple by choosing % & 1,

# & 0.5, C & 1.1 and E & 1.6. The results of optimi-

zation for dimensions of motors are listed in Table IV.

Fig. 4 shows the enhancement of fitness function

during the optimization process in these three optimi-

zation steps.

Table IV. Specification of Optimal Designed Motors.

Variable Unite Optimal

motor 1

Optimal

motor 2

Optimal

motor 3

% - 1 1 1

# - 1 0 0.5

C - 0 1.5 1.2

E - 0 1.1 1.6

�� mm 3.6 3.1 3.3

�� mm 11 3.1 10.4

�� mm 10 7.6 7.6

mm 2 0.8 0.8

� mm 70 70 70

Fig. 4. Improvement of fitness function.

Fig. 5. Flowchart of FEM.

4. Design Evaluation

The design optimizations in this work were carried out

based on the analytical model of the machine presented in

Section II. Therefore, validity of the design optimizations

greatly depends on the accuracy of the model. However,

the model is obtained by some simplifications such as

ignoring saturation and considering an infinite motor

length. Thus, it is necessary to evaluate the extent of

model accuracy. In this Section, 2-D nonlinear time-

stepping FEM is employed to validate the model. It is

supposed that the motor controlled by using a current-

controlled inverter. The relative movement is taken into

account in the FEM by using time-stepping analysis and

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Fig. 6. Magnetic flux lines in PMLSM motor.

Fig. 7. Flux density distribution in PMLSM motor.

Lagrange multiplier method [6], [15]. The forces are then

calculated using local virtual work method. A flowchart

of the FEM is shows in Fig. 5. A graphical representation

of flux lines and flux density distribution in the analyzed

motor are depicted in Figs. 6 and 7, respectively.

Comparison between analytically predicted and FE

calculated open-circuit distributions of the normal flux

density component as function of � position is depicted in

Fig 8. It is seen that the FEM accurately verifies the

analytical method.

The results of mean thrust, thrust ripple, efficiency PM

and copper volume for initial and optimal motors are

shown in Table V. It is seen that good agreement is

achieved in mean thrust for results of FEM and analytical

prediction.

Table V. performance of Initial and Optimal Motors.

Performance (Unit) Initial

motor

Optimal

motor 1

Optimal

motor 2

Optimal

motor 3

Thrust (N) Analytical

FEM

84.7

85.35

85.95

87.98

84.04

85.1

84.05

85.12

Thrust ripple (N) FEM 1.46 0.27 3 0.31

PM volume (Cm3) 21.17 21.17 18.23 19.4

Copper volume (Cm3) 48.38 58.8 44.69 44.69

Efficiency 81.1 78.11 81.92 81.92

The optimized motors thrust value obtained by FEM is

shown in Fig. 9. It is seen that thrust ripple reduces

effectively in optimized motor 1. In fact, the optimized

motor 1 experiences a thrust ripple less than about six

times of one for the initial motor, while the mean thrust

increases almost 2.6 N. But as shown in Table V, with the

same magnet volume, copper volume increase 22% in

optimized motor 1 with respect to the initial motor.

In the optimized motor 2, the PM and copper are used

effectively in thrust production. Therefore, in comparison

with the initial motor, the PM and copper consumption

reduce about 14% and 8%, respectively. But with almost

the same thrust mean, thrust ripple increases 105%.

A multi-objective optimization is aimed to achieve high

thrust, low magnet and copper volume and low thrust

ripple in optimized motor 3. The results presented in table

II show that multi-objective optimization provides a

Fig. 8. Normal component of flux density distribution as

a function of � position.

Fig. 9. Thrust force as a function of armature position at rated current.

design with almost the same thrust, 79% decrease in

thrust ripple and 8% decrease both in PM and copper

volume, with respect to the initial motor.

5. Conclusion

A multi-objective design optimization method was

applied on air-core permanent magnet linear synchronous

motor with Halbach array to achieve high developed

thrust, reduced magnet and copper volume, and low

thrust ripple simultaneously. The analytical analysis

based on Maxwell equations is derived to predict air gap

magnetic flux density distribution, thrust and efficiency.

Motor dimensions were optimized using the genetic

algorithm. It is seen that in the first optimization step, the

thrust ripple decreases up to 81%. in the second

optimization step, the magnet and copper volume

decrease 14% and 8%, respectively. Eventually, in the

third optimization step, thrust ripple, magnet and copper

volume decrease 79%, 8% and 8%, respectively. The

results of design optimizations are then verified by a 2-D

nonlinear time stepping finite element method.

References

[1] A. boldea and S. Nasar, Linear Electromagnetic Devised. New

York: Taylor & Francis, 2001.

[2] J. Wang and D. Howe, “Design optimization of radially

magnetized, iron-cored, tubular permanent-magnet machines and

drive systems,” IEEE Trans. Magn., vol. 40, no. 5, pp. 3262-3277,

Sep. 2004.

445

[3] Z. Q. Zhu et al., “Performance of Halbach magnetized brushless

AC motors,” IEEE Trans. Magn., vol. 39, no. 5, pp. 2992-2994,

Sep. 2003.

[4] Z. Q. Zhu and D. Howe, “Halbach permanent magnet machines

and applications: a review,” IEE Proc.-Electr. Power Appl., vol.

148, no. 4, pp. 299-308, Jul. 2001.

[5] S. Vaez-Zadeh and A. H. Isfahani, “Enhanced modeling of linear

permanent magnet synchronous motors,” IEEE Trans. Magn., vol.

43, no. 1, pp. 746-749, Jan. 2007.

[6] S. Vaez-Zadeh and A. H. Isfahani, “Multi-objective design

optimization of air-core linear permanent magnet synchronous

motors for improved thrust and low magnet consumption,” IEEE

Trans. Magn., vol. 42, no. 3, pp. 446-452, Mar. 2006.

[7] M. W. Hyun et al., “Optimal design of a variable stiffness join

using permanent magnets,” IEEE Trans. Magn., vol. 43, no. 6, pp.

2710-2712, Jun. 2007.

[8] J. S. Choi and J. Yoo, “Design of a Halbach magnet array based on

optimization techniques,” IEEE Trans. Magn., vol. 44, no. 10, pp.

2361-2366, Oct. 2008.

[9] S. M. Jang, S. H. Lee and I. K. Yoon, “Design criteria for

reduction of permanent magnet linear synchronous motors with

Halbach array,” IEEE Trans. Magn., vol. 38, no. 8, pp. 3261-3263,

Sep. 2002.

[10] R. Huang, J. Zhou and G. T. Kim “Minimization design of normal

force in synchronous permanent magnet planner motor with

Halbach array,” IEEE Trans. Magn., vol. 44, no. 6, pp. 1526-1529,

Jun. 2008.

[11] S. M. Jang and S. H. Lee, “Comparison of two types of PM linear

synchronous servo and miniature motor with air-cored film coil,”

IEEE Trans. Magn., vol. 38, no. 5, pp. 3264-3266, Sep. 2002.

[12] J. Wang, G. W. Jewell and D. Howe, “A general framework for

the analysis and design of tubular linear permanent magnet

machines,” IEEE Trans. Magn., vol. 35, no. 3, pp. 1986-2000,

May. 1999.

[13] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms. Wiley

Interscience, 2004.

[14] N. Bianchi and S. Bolognani, “Design optimization of electric

motors by genetic algorithms,” IEE Proc.-Electr. Power Appl.,

vol. 145, no. 5, pp. 475-483, Jul. 1998.

[15] D. Rodger, H. C. Lai and P. J. Leonard, “Coupled element for

problems involving movement,” IEEE Trans. Magn., vol. 26, no.

2, pp. 548-550, Mar. 1990.

446