A tubular linear machine with dual Halbach array

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It also helps to decrease the axial component of flux density and thus to reduce the ... The current issue and full text archive of this journal is available at ..... For these three positions, the center lines of winding A, B and C are aligned. Figure 6.
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A tubular linear machine with dual Halbach array

Tubular linear machine

Liang Yan, Lei Zhang, Zongxia Jiao and Hongjie Hu School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics (Beihang University), Beijing, China

Chin-Yin Chen

177 Received 14 January 2013 Revised 8 March 2013 Accepted 7 May 2013

National Applied Research Laboratories, Taiwan Ocean Research Institute, Taiwan, Republic of China, and

I-Ming Chen School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore Abstract Purpose – Force output is extremely important for electromagnetic linear machines. The purpose of this study is to explore new permanent magnet (PM) array and winding patterns to increase the magnetic flux density and thus to improve the force output of electromagnetic tubular linear machines. Design/methodology/approach – Based on investigations on various PM patterns, a novel dual Halbach PM array is proposed in this paper to increase the radial component of flux density in three-dimensional machine space, which in turn can increase the force output of tubular linear machine significantly. The force outputs and force ripples for different winding patterns are formulated and analyzed, to select optimized structure parameters. Findings – The proposed dual Halbach array can increase the radial component of flux density and force output of tubular linear machines effectively. It also helps to decrease the axial component of flux density and thus to reduce the deformation and vibration of machines. By using analytical force models, the influence of winding patterns and structure parameters on the machine force output and force ripples can be analyzed. As a result, one set of optimized structure parameters are selected for the design of electromagnetic tubular linear machines. Originality/value – The proposed dual Halbach array and winding patterns are effective ways to improve the linear machine performance. It can also be implemented into rotary machines. The analyzing and design methods could be extended into the development of other electromagnetic machines. Keywords Linear motor, Force output, Motor design, Parameter design Paper type Research paper

Introduction Linear machine can generate linear motions directly without rotation-to-translation conversion mechanisms, which significantly simplifies system structure and improves the working efficiency (Mazeika et al., 2009; Shigeki et al., 2009; Qidwai and The authors acknowledge the financial support from the National Nature Science Foundation of China under Grant No. 51175012, 51235002, the Program for New Century Excellent Talents in University of China under Grant No. NCET-12-0032, the Fundamental Research Funds for the Central Universities, and Science and the Technology on Aircraft Control Laboratory.

Engineering Computations: International Journal for Computer-Aided Engineering and Software Vol. 31 No. 2, 2014 pp. 177-200 q Emerald Group Publishing Limited 0264-4401 DOI 10.1108/EC-01-2013-0022

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DeGiorgi, 2010; Lin and Lin, 2011). It has been used widely for robotics (Chris and Marc, 2011; Mohamed et al., 2011; Chang et al., 2012; Kwak and Park, 2012), high-precision manufacturing (Yajima et al., 2000; Chayopitak and Taylor, 2005; Demenko et al., 2006; Zhao et al., 2008), transportation (Costamagna et al., 2012; Palka et al., 2012), medical operation (Shutov et al., 2005; Mashimo and Toyama, 2008) and aerospace industries (Manolas et al., 2000; Karunanidhi and Singaperumal, 2010). Tubular linear machines with permanent magnets (PMs) have a number of distinctive features (Mashimo and Toyama, 2008; Ragulskis, 2008; Shkolnikov et al., 2010) such as excellent servo characteristics and high power density. High magnetic flux density is extremely important for improving the force output of electromagnetic linear machines. Thus, of a number of force-increasing methods, magnet array is one effective way as it significantly influences the magnetic flux distribution in the machine, and thus the system force output. Researchers have completed much valuable work on the study of magnet patterns. For example, Nirei et al. (2000) have developed a moving-coil-type linear DC motor with 16 pieces of PMs fitted along the inner surface of the outer yoke. These pieces could be regarded as one radially magnetized cylindrical PM. Kim and Murphy (2004) have proposed a tubular linear brushless PM motor with NS-NS-SN-SN magnetization fashion. It leads to high magnetic flux and force near the like-pole region. Alternatively magnetized pattern in axial direction is employed by Wang et al. (1998) in the design of a linear reciprocating generator to produce high flux density, and by Buren and Troster (2007) for a linear vibration-driven electromagnetic micro-power generator. Wang et al. (2001) have also proposed a tubular linear machine with surface-mounted radially magnetized magnets and compared it with that with axially magnetized PMs. This pattern can significantly increase the flux density inside the machine compared with single magnetization pattern. To improve the power density, an axially magnetized linear motor with annual shaped magnets and associated magnetic field study was proposed in Wang et al. (2002). Hong et al. (2001) have designed a linear machine with two layers of PMs. The magnetic field is formulated with equivalent magnetic circuit. Halbach array is a promising magnet pattern due to its self-shielding property and the ability to create relatively strong multi-pole, sinusoidally distributed magnetic field in the internal air space of electromagnetic machines. In this study, a novel design of tubular linear machine with dual Halbach PM array is proposed. This magnetic poles arrangement can greatly increase the magnetic flux density in the machines to generate large axial force. The mathematic model of the magnetic field from magnetic vector potential is presented. By utilizing these analytical models, per-phase force output of tubular linear machine with dual Halbach arrays is obtained. Moreover, total thrust force and force ripple are formulated analytically for different phase patterns and winding arrangements. The derived analytical models are validated with numerical results. Based on the force models, the influence of structure parameters on force output and force ripple of a three-phase linear machine has been investigated. The study provides an effective tool for comprehensively understanding the operating principle and output performance of linear machines with dual Halbach arrays.

Concept design A. System overall structure The schematic structure of the proposed PM tubular linear machine is illustrated in Figure 1. The mover consists of winding phases mounted in the holders, whereas the stator is composed of dual Halbach magnet arrays that enclose the windings on internal and external sides. (1) Stator structure. The polarization pattern of each layer of the PMs is arranged in Halbach array. As indicated in Figure 2, the Halbach array consists of alternatively magnetized PMs in radial directions separated with horizontally magnetized PMs. This array offers one impressive benefit, i.e. the flux density on one side of the PMs is significantly enhanced, whereas it is self-shielded on the other side. Figure 2 shows that the magnetization direction for horizontal magnets in both layers is opposite, while the magnetization for radial PMs is the same. This arrangement leads to the increase of radial component of flux density in between the two layers, and the reduction of axial components. Because only the radial component of flux density can create axial force on the mover, the dual Halbach array could improve the axial force

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Figure 1. Schematic drawing of the linear machines with dual Halbach array

Figure 2. Polarization pattern of dual Halbach array

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output, and reduce the radial force disturbance of liner machines. The depression of axial components helps to reduce the deformation and vibration of linear machines. (2) Mover structure. Movers in electric linear machines can be classified into two types: slotted one and slotless one. The former can usually produce high force density but also yield large force ripple due to the cogging forces. Furthermore, energy loss may exist due to the eddy current produced in the soft iron. In contrast, a slotless rotor structure eliminates the tooth ripple cogging effect and thus improves the dynamic performance and servo characteristics of linear machines, at the cost of a slight down of force output. For some particular applications such as precision manufacturing and assembly, slotless mover has greater advantages than the slotted one. Therefore, in this study, we choose the slotless type for the mover design of the linear machines. The phase pattern of windings plays an important role on force output of linear machines. In this paper, three types of phases are studied. The first one is single-phase winding as shown in Figure 3. For this design, the axial width of winding phase is equal to the pole pitch as well as the winding pitch. The second type is double-phase winding structure as shown in Figure 4. In this pattern, the axial width of winding phase is half of the pole pitch. The third pattern is the three-phase winding as shown in Figure 5. In this case, the axial width of winding phase is one-third of the pole pitch. In following sections, we will discuss the influence of three winding patterns on the machine output. B. Operating principle The operating principle of PM tubular linear machine is based on the Lorentz force law, i.e. force is generated by interaction between the magnetic field of PMs and the

Figure 3. Machine structure with single-phase winding

Figure 4. Double-phase winding pattern

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181 Figure 5. Three-phase winding structure

current-carrying conductors. The direction and magnitude of the force is determined by the cross product of current vector on segmental winding and magnetic flux density. Therefore, the radial component of flux density can produce forces on the mover in axial direction. The force created by the axial flux component and winding segments is symmetric around the machine axis, and thus vanishes in total. The stator and mover can be switched depending on what part we want to move for particular applications. Formulation of force output A. Single-phase winding pattern The magnetic flux generated by PM arrays in tabular linear machine is symmetrically distributed around the machine axis. Therefore, there are only two components of the flux density. By taking advantage of magnetic vector potential and the boundary conditions in between different materials, the flux density distribution in the three-dimensional space can be represented as a function of r and z, where r is the radial coordinate and z is the axial coordinate as shown in Figure 3: 1 X Br ¼ 2 mn ½a3n I 1 ðmn rÞ þ b3n K 1 ðmn rÞcosðmn zÞ; n¼1

Bz ¼

1 X

ð1Þ mn ½a3n I 0 ðmn rÞ 2 b3n K 0 ðmn rÞsinðmn zÞ;

n¼1

where: mn ¼ ð2n 2 1Þ=tp :

ð2Þ

I 0 ðmn rÞ, I 1 ðmn rÞ are modified Bessel functions of the first kind, and K 0 ðmn rÞ, K 1 ðmn rÞ are modified Bessel functions of the second kind, for order zero and one, respectively. a3n and b3n are coefficients of the harmonic, and their values are dependent upon the machine topology, PM material properties and geometric parameters of linear machine. Based on Lorentz force law, the axial force exerted on a winding with current density of J, is derived as: F wp ¼ 28pJ

  twp  K n sin mn z 2 ; 2 n¼1

1 X

ð3Þ

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where:  t   t  w wp K n ¼ sin mn sin mn K rn ; 2 2 Z Rb r½a3n I 1 ðmn rÞ þ b3n K 1 ðmn rÞdr: K rn ¼

ð4Þ

Ri

tp is the winding pitch, and twp is defined as: Ri ¼ Ra þ g;

ð5Þ

where Ra is the outer radius of the internal PMs, Rb is the inner radius of the external PMs, g is the size of air gap, and tw is the axial width of the phase winding per pole. The winding arrangement for a single-phase slotless tubular linear machine is shown in Figure 3. The current density in the winding is given by: pffiffiffi J ¼ 2J rms cos vt; ð6Þ where Jrms is the RMS value and v is the angular frequency of current inputs. Since the armature moves in synchronism with the AC frequency, we have:

vt ¼

pz : tp

ð7Þ

Thus, from equation (3), the force generated by a single-phase PM tubular linear machine is: 1   X pffiffiffi twp  K n sin mn z 2 F wp ¼ 28 2pJ rms cos vt: 2 n¼1

ð8Þ

As z represents the axial position of the mover, the force output of the machine is related to the motion position. The force formulation can be simplified into:   1 X pffiffiffi pz : ð9Þ K rn cosðmn zÞcos F wp ¼ 28 2pJ rms tp n¼1 B. Double-phase winding pattern The winding pattern for a double-phase slotless tubular linear machine is presented in Figure 4. The winding width is the same for the two phases. The input current densities in phases A and B are: pffiffiffi J A ¼ 2J rms cos vt; ð10Þ pffiffiffi J B ¼ 2J rms sin vt: For this pattern, winding pitch twp equals to pole pitch tp. The thrust generated by each phase is given by:

! 1  X pffiffiffi tp  pz F A ¼ 8 2pJ rms K n sin mn z 2 ; cos 2 tp n¼1 ! 1 X pffiffiffi pz K n sin mn ðz 2 tp Þsin : F B ¼ 8 2pJ rms tp n¼1

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Therefore, the total force is obtained as: F ¼ FA þ FB   1  X pffiffiffi tp  pz pz : ¼ 8 2pJ rms K n sin mn z 2 þ cos sin mn ðz 2 tp Þsin 2 tp tp n¼1

ð12Þ

It can be simplified as:   n pz ; F n cos ð2n 2 1 þ ð21Þ Þ F ¼ F1 þ tp n¼2 1 X

ð13Þ

where F1 is the constant thrust caused by the fundamental term of radial magnetic harmonic components, and Fn is the magnitude of the force ripple created by the (2n 2 1)th harmonic component. It is noticed that there is no constant term for the force generated by single-phase winding pattern. F1 and Fn in equation (13) are calculated by: pffiffiffi pffiffiffi F 1 ¼ 28 2pJ rms K 1 ; F n ¼ ð21Þn 8 2pJ rms K n n ¼ 2; 3; . . . ð14Þ respectively. The force ripple can be represented as a function of z:   1 X pz n ¼ 2; 3; . . . : F n cos ð2n 2 1 þ ð21Þn Þ Fr ¼ tp n¼2

ð15Þ

The maximum force ripple produced by the double-phase winding pattern can then be computed with: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 pffiffiffi uX F rm ¼ 8 2pJ rms t ðK n 2 K nþ1 Þ2 : ð16Þ n¼2k

The relative force ripple is defined as the ratio of maximum force ripple and the fundamental term, i.e.: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P1 2 F rm n¼2k ðK n 2 K nþ1 Þ ¼ k ¼ 1; 2; . . . : ð17Þ F rr ¼ F1 K1 C. Three-phase winding pattern The three-phase winding pattern of the linear machine is presented in Figure 5. The three phases are separated by two-thirds of the pole pitch in turns. Similar to the

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Figure 6. Starting position A of mover

Figure 7. Starting position B of the machine mover

double-phase pattern, a three-phase winding comprises a few coils connected in series. In three-phase winding structure, the winding pitch twp is equal to the pole pitch tp and each coil occupies one-third of a pole pitch. The current densities in the three phases are given by:   pffiffiffi 2 J A ¼ 2J rms cos vt þ p ; 3 pffiffiffi ð18Þ J B ¼ 2J rms cos vt;   pffiffiffi 2 J C ¼ 2J rms cos vt 2 p : 3 Besides, the current inputs, the force output of the linear machine is also related to the starting point of mover. Three typical starting positions of the mover are studied in this paper, i.e. position A in Figure 6, position B in Figure 7 and position C in Figure 8. For these three positions, the center lines of winding A, B and C are aligned

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185

Figure 8. Starting position C of the machine mover

with the r coordinate, respectively. In following subsection, the force output and force ripple for the three positions are formulated. (1) Starting position A. For starting position A in Figure 6, the force generated by the three phases are:   1   X pffiffiffi tp  2 F A ¼ 8 2pJ rms K n sin mn z 2 cos vt þ p ; 2 3 n¼1     1 X pffiffiffi 5tp F B ¼ 8 2pJ rms cos vt; K n sin mn z þ 6 n¼1   1   X pffiffiffi tp  2 F C ¼ 8 2pJ rms K n sin mn z þ cos vt 2 p : 6 3 n¼1

ð19Þ

Therefore, the total thrust is obtained as: F ¼ FA þ FB þ FC      1 X pffiffiffi tp  2 ¼ 8 2pJ rms K n sin mn z 2 cos vt þ p 2 3 n¼1        5 tp tp  2 cos vt þ sin mn z þ þsin mn z þ cos vt 2 p : 6 6 3

ð20Þ

It can be further simplified into:         1 X pz p pz p F ¼ F1 þ þ 2 þ F n0 sin ð2nÞ F n1 sin ð2n 2 2Þ tp 6 tp 6 n¼3k n¼3kþ1 1 X

k ¼ 1; 2 . . . :

ð21Þ

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F1, Fn0 and Fn1 are calculated from: pffiffiffi F 1 ¼ 26 2pJ rms K 1 ; pffiffiffi F n0 ¼ 12 2pJ rms K n ð21Þkþ1 ; pffiffiffi F n1 ¼ 12 2pJ rms K n ð21Þkþ1 :

ð22Þ

Equation (21) indicates that the force ripple generated by the third harmonic term in the radial magnetic field is equal to zero. The maximum force ripple is thus: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 uX 3 1 ð23Þ ðK n þ K nþ1 Þ2 þ ðK n 2 K nþ1 Þ2 k ¼ 1; 2 . . . : F rm ¼ t 4 4 n¼3k Therefore, the relative force ripple of the tubular linear machine is derived as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P1 2 2 n¼3k ð3=4ÞðK n þ K nþ1 Þ þ ð1=4ÞðK n 2 K nþ1 Þ : F rr ¼ K1

ð24Þ

(2) Starting position B. Starting position B of the mover in the linear machine is illustrated in Figure 7. The forces generated by phases of A, B and C are:  ! 1  X pffiffiffi tp  pz 2 K n sin mn z þ þ p ; F A ¼ 8 2pJ rms cos 6 tp 3 n¼1  ! 1  X pffiffiffi tp  pz ð25Þ ; F B ¼ 8 2pJ rms K n sin mn z 2 cos 2 tp n¼1 !    1 X pffiffiffi 5 tp pz 2 cos K n sin mn z þ 2 p : F C ¼ 8 2pJ rms 6 tp 3 n¼1 The total thrust force is obtained as: F ¼ FA þ FB þ FC      1   X pffiffiffi tp  pz 2 tp  pz ¼ 8 2pJ rms K n sin mn z þ þ p þ sin mn z 2 cos cos 6 tp 3 2 tp n¼1     5tp pz 2 cos 2 p : þsin mn z þ 6 tp 3 ð26Þ It can be further simplified into:       1 1 X X pz pz F ¼ F1 þ þ k ¼ 1; 2 . . . : F n0 cos ð2nÞ F n1 cos ð2n 2 2Þ t tp p n¼3k n¼3kþ1 ð27Þ

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where F1, Fn0 and Fn1 are calculated with: pffiffiffi F 1 ¼ 12 2pJ rms K 1 ; pffiffiffi F n0 ¼ 12 2pJ rms K n ð21Þk ; pffiffiffi F n1 ¼ 12 2pJ rms K n ð21Þkþ1 :

ð28Þ

187

Equation (27) shows that the force ripple caused by the third order harmonics term of the radial magnetic field is equal to zero. The force ripple is derived as: Fr ¼

      1 X pz pz þ k ¼ 1; 2; . . . : ð29Þ F n0 cos ð2nÞ F n1 cos ð2n 2 2Þ t tp p n¼3k n¼3kþ1 1 X

The maximum force ripple is:

F rm

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 pffiffiffi uX ¼ 12 2pJ rms t ðK n 2 K nþ1 Þ2 :

ð30Þ

n¼3k

And the relative force ripple is:

F rr ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P1 2 n¼3k ðK n 2 K nþ1 Þ K1

k ¼ 1; 2 . . . :

ð31Þ

(3) Starting position C. The starting position C of the mover is shown in Figure 8. The forces generated by the phases of A, B and C are, respectively, given by:     ! 1 X pffiffiffi 5tp pz 2 F A ¼ 8 2pJ rms cos K n sin mn z þ þ p ; 6 tp 3 n¼1  ! 1   X pffiffiffi tp  pz ; K n sin mn z þ F B ¼ 8 2pJ rms cos 6 tp n¼1  ! 1   X pffiffiffi tp  pz 2 F C ¼ 8 2pJ rms K n sin mn z 2 2 p : cos 2 tp 3 n¼1

ð32Þ

Correspondingly, the total force is obtained:       1 X pffiffiffi 5tp 2 cos vt þ p K n sin mn z þ F ¼ 8 2pJ rms 6 3 n¼1           tp tp 2 þsin mn z þ cos vt þ sin mn z 2 cos vt 2 p : 6 2 3

ð33Þ

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It can be represented with a compact form as:         1 1 X X pz p pz p þ þ þ F n0 cos ð2nÞ F n1 sin ð2n 2 2Þ F ¼ F1 þ tp 3 tp 6 ð34Þ n¼3k n¼3kþ1 k ¼ 1; 2 . . . :

188

F1, Fn0 and Fn1 are obtained from: pffiffiffi F 1 ¼ 6 2pJ rms K 1 ; pffiffiffi F n0 ¼ 12 2pJ rms K n ð21Þkþ1 ; pffiffiffi F n1 ¼ 12 2pJ rms K n ð21Þkþ1 :

ð35Þ

Again, equation (34) shows that the force ripple by the third harmonic term of the radial magnetic field is zero. The total force ripple is:         1 1 X X pz p pz p Fr ¼ þ þ þ F n0 cos ð2nÞ F n1 sin ð2n 2 2Þ tp 3 tp 6 ð36Þ n¼3k n¼3kþ1 k ¼ 1; 2; . . . : The maximum force ripple is: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 uX 1 3 ðK n þ K nþ1 Þ2 þ ðK n 2 K nþ1 Þ2 : F rm ¼ t 4 4 n¼3k And the relative force ripple is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P1 2 2 ð1=4ÞðK þ K Þ þ ð3=4ÞðK 2 K Þ n nþ1 n nþ1 n¼3k : F rr ¼ K1

ð37Þ

ð38Þ

Simulation and validation of force model Based on the derived mathematical models, the force outputs for tubular linear machines with single-phase, double-phase and three-phase winding patterns are simulated. In addition, numerical results are employed to validate the analytical force model, so that it can be used for the performance analysis and structure design purpose. A. Force variation for single-phase machine The force output with respect to the axial motion for a single-phase linear machine is illustrated in Figure 9. The maximum force is produced when the coil is exactly aligned with the radially magnetized PMs, whereas the minimum one occurs when the coil is aligned with the axially magnetized PMs. The pitch of the magnets is the same as that of coils, which leads to the force variation period consistent to tp and the positive peak values in Figure 9. It is found that the constant component in the total force of single-phase machines is equal to zero. In addition, numerical results are computed with Ansoft to compare with the analytical models. It shows that the two sets of data

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Figure 9. Force output generated by single-phase machine

fit with each other closely. Therefore, the derived force model can be employed for parameter analysis and design of the electromagnetic tubular linear machines. B. Force variation for double-phase machine The force created by double-phase machine is shown in Figure 10. Similarly, the maximum force happens when the coil is exactly aligned with the radially magnetized PMs (Figure 4). It is because the radial component of the magnetic field and the current input arrive the maximum values at this position. The constant component is found in the total force as indicated by the dotted line in the figure. The period of force variation is determined by magnetization and winding patterns. As the distance between two coils is equal to half of the period of magnetization, the force output varies

Figure 10. Force output generated by double-phase machine

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in a period of tp/2. Similarly, the numerical results are compared with the analytical model, which indicates that the analytical model fits with the numerical results well and could be employed for parameter design of the linear machines.

190

C. Force variation for three-phase machine The force variation for three-phase linear machine is also simulated with analytical models, including force output for starting position A in Figure 11, starting position B in Figure 12 and starting position C in Figure 13. It is found that, the force output of

Figure 11. Force output for starting position A

Figure 12. Force output for starting position B

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Figure 13. Force output for starting position C

a three-phase machine varies in a period of tp/3 regardless its starting position. The variation period is determined by the winding pitch. When the mover moves a distance of tp/3, the force output created by winding phases changes in turn but the total force is equal to the previous one. For starting position B, the maximum force results from the interaction of the peak current and a maximum magnetic flux density in ktp, where k is an integral number. Specifically, when phase B winding is aligned with the radially magnetized PMs, the force output reaches the maximum value. However, for other two starting positions A and C, the current input and the flux density cannot arrive the maximum values simultaneously wherever the mover is. Therefore, the force output of starting position B is much larger than the rest two cases. Due to the symmetric winding phases, the force variation for A and C are similar with each other. Based on the above simulation results and corresponding analysis, we may have information as follows: (1) For the same electrical inputs and magnetic field, the force output of a three-phase machine is double-phase machine by 6.0 percent. And the force of a single-phase machine is higher than that of the three-phase one by 12.6 percent. (2) For three-phase machines, the starting position B can produce a high force output that is almost twice of those of other two positions. (3) For the three-phase machines, the third order harmonics of the magnetic field does not cause any force ripple. And for single-phase machines, there is no constant component in the force output. Hence, the three-phase machines have a lower force ripple compared to other two types, which is favorable for a good dynamic performance of linear machine. In general, a three-phase tubular linear machine has a better performance than other two peers considering the force capability and the possible instability caused by force ripple. Furthermore, integrated power modules are widely available for three-phase

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brushless machines, which greatly benefits the design and commercialization of this type of linear machines. In subsequent discussion, we will mainly focus on the study of three-phase tubular linear machines.

192

Parameter design of three-phase machine By using the validated analytical models of total force and force ripple, the parameter design of three-phase tubular linear machine can be carried out. In this study, the parameter design targets to maximize the force output and reduce the force ripple to less than 5 percent in a limited machine volume. A. Geometric parameters of the linear machine Equations (26) and (31) can be used to predict the force output and force ripple of the PM linear machine. The major geometric parameters of the dual Halbach linear machine are illustrated in Figure 14. Rr and Ra are inner radius and outer radius of internal PMs, respectively. Rb and Rs are the inner radius and outer radius of external PMs, respectively. tr is the width of radially magnetized PMs, while tz is the width of axially magnetized PMs. As sum of tr and tz is equal to the pole pitch, tp, only tr and tp will be discussed here. For slotless machines, we assume that the influence of the end effects associated with the finite armature length is negligible. The influence of air gap is not as significant as that in slotted machines (Wang et al., 2001). It can be considered as a constant. Normally, the maximum machine size is limited by particular applications. In this study, the max radius of the linear machine, Rs, is fixed at 40 mm. The force output and force ripple can be seen as functions of the rest five geometric parameters, Rb, Ra, Rr, tp and tr. The technology of nondimensionalization is utilized in this study, i.e. the five-dimensional parameters are replaced with nondimensionalized parameters, kbs ¼ Rb =Rs , kab ¼ Ra =Rb , kra ¼ Rr =Ra , kps ¼ tp =Rs and krp ¼ tr =tp . Nondimensionalization is the partial or full removal of units from a mathematical equation by a suitable substitution of variables. It is generally used in mechanical, economic or sociological areas to evaluate the weight of different inputs, such as the oil gap in the hydrodynamic lubrication system. By using the technique of nondimensionalization, the behavior of the system can be analyzed regardless of the units used to measure the variables.

Figure 14. Design parameters of linear machine with dual Halbach array

B. Inter-dependence of structure parameter For structure design of the linear machine, we must consider the inter-dependence of all five parameters. In other words, the influence between the dimensionless parameters, kbs, kab, kra, kps and krp needs to be analyzed. In this study, if one parameter has little influence on the rest ones, it is denoted as independent parameter. Otherwise, it is denoted dependant parameter. The dependence of the geometric parameters is discussed as follows. (1) Relationship between kbs and other parameters. Figure 15 shows how the fundamental force output varies with respect to kbs, for different values of kab, kra, kps and krp. The fundamental force is defined as the time integral of force output divided by the complete period, i.e.: Z tp ~F ¼ Fdz=2tp : ð39Þ

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2 tp

From Figure 15(b) and (d), it is found that regardless the ratio of kra and krp, the position for the maximum value of fundamental force does not change much, which

(a)

(b)

(c)

(d)

Notes: (a) Force output for eight values of kab; (b) force output for eight values of kra; (c) force output for eight values of kps; (d) force output for eight values of krp

Figure 15. Force output as a function of kbs

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indicates that these two parameters has no significant impact on kbs. However, Figure 15(a) and (c) shows that for different values of kab or kps, the peak value of average force is achieved at different ratios of kbs. Therefore, kab and kps have close relationship with kbs. (2) Relationship between kab and other parameters. Figure 16(a)-(c) presents how the force varies as a function of kab for different values of kra, kps, krp, respectively. From the three figures, it is found that the ratios of kab for the maximum average force does not change much with respect to kra, kps, and krp, which indicates that the three parameters has no close inter-dependence on kab. (3) Relationship between kra and other parameters. The force variation with respect to kra for different values of kps, krp is presented in Figure 17(a) and (b), respectively. Similarly, from Figures 15(b), 16(a), 17(a) and (b), it is found that the ratio kra is not closely related to other four dimensionless parameters. (4) Relationship between kps and other parameters. Figure 18 illustrates the force variation with respect to kps for values of krp. It indicates that for different values of krp, the peak force is achieved at different ratios of kps. Therefore, kps is closely dependent

(b)

(a)

Figure 16. Force output as a function of kab

(c)

Notes: (a) Force output for eight values of kra; (b) force output for eight values of kps; (c) force output for eight values of krp

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(a)

(b)

Notes: (a) Force output for different values of kps; (b) force output for different values of krp

Figure 17. Force output as a function of kra

Figure 18. Force output as a function of kps

on krp. Figure 15(c) presents the dependence of kps and kbs, while Figures 16(b) and 17(a) have shown the independence of kps, kab and kra. Therefore, the value of kps has close relationship with kbs and krp. (5) Relationship summary of all parameters. By summarizing the above analysis, the close relationship exists between parameter pairs of kab and kbs, kps and kbs, krp and kps. And the rest parameter pairs has not close relationship, i.e. kra and kbs, kra and kbs, kra and kab, kps and kab, krp and kab, kps and kra, krp and kra. C. Determination of independent parameters Among the five parameters, only kra is not closely related to the rest four parameters. The force output and force ripple with respect to kra are presented in Figure 19. It is found that when kra arrives at the lowest value, the force output is maximized and the

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Figure 19. Force output and force ripple as functions of kra

(a)

(b)

Notes: (a) Force variation vs kra; (b) force ripple vs kra

force ripple is minimized. A low ratio of kra represents an increase relative length of internal PMs, and thus high force output. However, large volume of magnets may increase the system cost significantly. In this paper, kra is choose to be 0.35, and thus Rr ¼ 7 mm for assembly convenience. D. Determination of dependent parameters It is known that kbs is closely related to kab and kps, and kps is dependent on krp. The variation of force output and force ripple with respect to krp and kps are illustrated in Figure 20. The force ripple grows rapidly when kps is larger than 3.0, but the force output only rises slightly. Therefore, the value of kps can be set to 3.0 or less. Figures 18 and 20(a) show that when kps ¼ 3.0 and krp ¼ 0.3, the force output approaches to the peak value and the force ripple is less than 5 percent. Therefore, kps ¼ 3.0 and krp ¼ 0.3 are selected for the machine design. To analyze the inter-dependence of kps and kbs, the variations of force output and force ripple with respect to kbs and kps are plotted in Figure 21. From the figure, it is found that when kps ¼ 3.0, kbs ¼ 0.85, the force output is very close to the peak point and the force ripple is less than 5 percent. Similarly, to investigate the dependence of kbs and kab, the variations of force output and force ripple

Figure 20. Force output and force ripple vs kps and krp, at kbs ¼ 0.85, kab ¼ 0.6, kra ¼ 0.35

(a)

Notes: (a) Force output vs kps and krp; (b) force ripple vs kps and krp

(b)

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(a)

Figure 21. Force output and force ripple vs kps and kbs, at kab ¼ 0.6, krp ¼ 0.6, kra ¼ 0.35

(b)

Notes: (a) Force output vs kbs and kps; (b) force ripple vs kbs and kps

with respect to kbs and kab are plotted in Figure 22. According to the figure, when kbs ¼ 0:85 and kab ¼ 0:6, the force output gains the peak value, and the force ripple is less than 5 percent. Therefore, the value of kps is set to 3.0, krp to 0.3, kbs to 0.85, and kab to 0.6 for the design of the PM tubular linear machine. E. Final values of structure parameters From the obtained dimensionless parameters and the maximum radius Rs in the last section, the final values for the structure parameters of the tubular linear machine can be obtained. The result is presented in Table I.

(a)

Figure 22. Force output and force ripple vs kbs and kab, at kra ¼ 0.35, kps ¼ 3, krp ¼ 0.3

(b)

Notes: (a) Force output vs kbs and kab; (b) force ripple vs kbs and kab

Outer radius of external Halbach array Rs Inner radius of external Halbach array Rb Outer radius of internal Halbach array Ra Inner radius of internal Halbach array Rr Magnetic pole pitch tp Axial width of radially magnetized PMs tr

40 34 20 7 120 36

Table I. Final values of structure parameters (mm)

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Conclusions A novel tubular linear machine with dual Halbach arrays is proposed in this paper. Based on the flux field models from magnetic vector potential, the force output for linear machines with single-phase, double-phase and three-phase winding patterns is formulated analytically. By using the derived mathematic models, the force output for different winding phases is simulated. Furthermore, numerical results are obtained to validate the derived force models. According to force analysis, the three-phase winding pattern can achieve relatively high force output and low force ripple. The analytical models also provide a useful tool for assessing the influence of structure parameters of linear machines on their output performance. Based on the parameter analysis, one set of values are selected for the structure design of the tubular linear machine with dual Halbach array. A research prototype will be fabricated in the future for further study. References Buren, T.V. and Troster, G. (2007), “Design and optimization of a linear vibration-driven electromagnetic micro-power generator”, Sensors and Actuators, A: Physical, Vol. 135 No. 2, pp. 765-775. Chang, H.W., Lee, H.W., Lin, C.T. and Wen, Z.Q. (2012), “Development of blue laser direct-write lithography system”, International Journal of Engineering and Technology Innovation, Vol. 2 No. 1, pp. 63-71. Chayopitak, N. and Taylor, D. (2005), “Thermal analysis of linear variable reluctance motor for manufacturing automation applications”, Proceeding of IEEE International Conference on Electric Machines and Drives, San Antonio, Texas, May, pp. 866-873. Chris, A.M. and Marc, A. (2011), “Kinematic analysis of a translational 3-dof tensegrity mechanism”, Transactions of the Canadian Society for Mechanical Engineering, Vol. 35 No. 4, pp. 573-584. Costamagna, E., Barba, P.D. and Palka, R. (2012), “Field models of high-temperature superconductor devices for magnetic levitation”, Engineering Computations, Vol. 29 No. 6, pp. 605-616. Demenko, A., Mendrela, E. and Szelag, W. (2006), “Finite element analysis of saturation effects in a tubular linear permanent magnet machine”, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 25 No. 1, pp. 44-54. Hong, J., Kang, D., Joo, S. and Hahn, S. (2001), “Variation of force density in BLDC linear motor on the width of PM and coil”, Proceedings of IEEE International Symposium on Industrial Electronics, Pusan, Korea, June, pp. 979-982. Karunanidhi, S. and Singaperumal, M. (2010), “Design, analysis and simulation of magnetostrictive actuator and its application to high dynamic servo valve”, Sensors and Actuators, A: Physical, Vol. 157 No. 2, pp. 185-197. Kim, W.J. and Murphy, B.C. (2004), “Development of a novel direct-drive tubular linear brushless permanent-magnet motor”, International Journal of Control, Automation, and Systems, Vol. 2 No. 3, pp. 279-288. Kwak, H.J. and Park, G.T. (2012), “Study on the mobility of service robots”, International Journal of Engineering and Technology Innovation, Vol. 2 No. 2, pp. 13-28. Lin, C.H. and Lin, C.P. (2011), “Integral backstepping control for a PMLSM using adaptive RNNUO”, International Journal of Engineering and Technology Innovation, Vol. 1 No. 1, pp. 53-64.

Manolas, D.A., Borchers, I. and Tsahalis, D.T. (2000), “Simultaneous optimization of the sensor and actuator positions for an active noise and/or vibration control system using genetic algorithms, applied in a Dornier aircraft”, Engineering Computations, Vol. 17 No. 5, pp. 620-630. Mashimo, T. and Toyama, S. (2008), “Micro rotary-linear ultrasonic motor for endovascular diagnosis and surgery”, Proceeding of IEEE International Conference on Robotics and Automation, Pasadena, CA, May, pp. 3600-3605. Mazeika, D., Vasiljev, P., Kulvietis, G. and Vaiciuliene, S. (2009), “New linear piezoelectric actuator based on traveling wave”, Journal of Vibroengineering, Vol. 11 No. 1, pp. 68-77. Mohamed, S., Albert, N. and Ilian, A.B. (2011), “Effect of servo systems on the contouring errors in industrial robots”, Transactions of the Canadian Society for Mechanical Engineering, Vol. 36 No. 1, pp. 83-96. Nirei, M., Tang, Y., Mizuno, T., Yamamoto, H., Shibuya, K. and Yamada, H. (2000), “Iron loss analysis of moving-coil-type linear DC motor”, Sensors and Actuators, A: Physical, Vol. 81 Nos 1/3, pp. 305-308. Palka, R., Barba, P.D. and Costamagna, E. (2012), “Field models of high-temperature superconductor devices for magnetic levitation”, Engineering Computations, Vol. 29 No. 6, pp. 1-15. Qidwai, S.M. and DeGiorgi, V.G. (2010), “Numerical analysis of the effect of poling-loading misalignment and mechanical boundary conditions on piezoelectric actuation”, Engineering Computations, Vol. 27 No. 8, pp. 909-929. Ragulskis, T. (2008), “Constant magnet motors”, Journal of Vibroengineering, Vol. 10 No. 3, pp. 269-271. Shigeki, T., Shota, K. and Akifumi, Y. (2009), “Development of piezoelectric actuators with rotational and translational motions”, Journal of Vibroengineering, Vol. 11 No. 3, pp. 374-378. Shkolnikov, V., Ramunas, J. and Santiago, J.G. (2010), “A self-priming, rollerfree, miniature, peristaltic pump operable with a single, reciprocating actuator”, Sensors and Actuators, A: Physical, Vol. 160 No. 1, pp. 141-146. Shutov, M.V., Sandoz, E.E., Howard, D.L., Hsia, T.C., Smith, R.L. and Collins, S.D. (2005), “A microfabricated electromagnetic linear synchronous motor”, Sensors and Actuators, A: Physical, Vol. 121 No. 2, pp. 566-575. Wang, J., Howe, D. and Jewell, G.W. (2002), “An improved axially magnetized tubular permanent magnet machine topology”, Proceedings of IET International Conference on Power Electronics, Machines and Drives, Bath, UK, June, pp. 303-308. Wang, J., Jewell, G.W. and Howe, D. (2001), “Design optimisation and comparison of tubular permanent magnet machine topologies”, IEE Proceedings of Electric Power Applications, UK, September, Vol. 148 No. 5, pp. 456-464. Wang, J., Wang, W., Jewell, G.W. and Howe, D. (1998), “Design and experimental characterisation of a linear reciprocating generator”, IEE Proceedings of Electric Power Applications, UK, November, Vol. 145 No. 6, pp. 509-518. Yajima, H., Wakiwaka, H., Minegishi, K., Fujiwara, N. and Tamura, K. (2000), “Design of linear DC motor for high-speed positioning”, Sensors and Actuators, A: Physical, Vol. 81 No. 1, pp. 281-284. Zhao, M., Huang, Z.D. and Chen, L.P. (2008), “Multidisciplinary design optimization of tool head for heavy duty CNC vertical turning mill”, Engineering Computations, Vol. 25 No. 7, pp. 658-676.

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Further reading Zhu, Z., Howe, D., Bolte, E. and Ackermann, B. (1993), “Instantaneous magnetic field distribution in brushless permanent magnet DC motors. I. Opencircuit field”, IEEE Transactions on Magnetics, Vol. 29 No. 1, pp. 124-135. About the authors Liang Yan received the Bachelor degree from North China Institute of Technology, Shanxi, China in 1995, Master degree from Beijing Institute of Technology, Beijing, China, in 1998, and the PhD degree from Nanyang Technological University, Singapore, in 2006. He was with Beijing Institute of Technology as a Lecturer from 1998 to 2002, Nanyang Technological University as a Research Fellow from 2006 to 2009, and Beijing University of Aeronautics and Astronautics as an Associate Professor currently. His research interests include robotics, intelligent actuators and sensors, navigation system and expert system design. Professor Yan was the recipient of the National Defense Science and Technology Award, China in 2002. He was the Program Chairman of 2012 IEEE 10th International Conference on Industrial Informatics, Publication Chairman of 2008 IEEE International Conference on Cybernetics and Intelligent Systems, and 2009 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, etc. Liang Yan is the corresponding author and can be contacted at: [email protected] Lei Zhang obtained her Bachelor degree in 2011 from North China Electric Power University. She is now with Beihang University as a Master student. She is the recipient of Mechanics Competition of Hebei Province in 2010. Her research interests include electromagnetic actuators, sensor, and robotics. Zongxia Jiao received the BS and PhD degrees from Zhejiang University, China, in 1985 and 1991, respectively. He was with Beihang University (BUAA) as a Postdoc from 1991 to 1993, and a Professor since 1994. He was a Visiting Professor at the Institute for Aircraft System Engineering, Technische University Hamburg-Harburg, Germany in 2000. He is currently the College Dean of the School of Automation Science and Electrical Engineering, BUAA. His research interest includes actuators, sensors, fluid power and transmission. He was the recipient of Changjiang Scholar Professor in 2006, and Distinguished Young Scholar in 2008. He serves 2011 International Conference on Fluid Power and Transmission, 2012 IEEE International Conference on Industrial Informatics as general chairs. He has published more than 100 papers in international journals and refereed conferences. Dr Hongjie Hu is based at the School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics. Chin-Yin Chen received his PhD degree from the Department of Mechanical and Electro-mechanical Engineering at the National Sun Yat-Sen University in 2008. He is currently an Assistant Researcher with the Taiwan Ocean Research Institute, Kaohsiung, Taiwan. His research interests include mechatronics design, integrated structure/control design, and robotics. I-Ming Chen received the BS degree from the National Taiwan University, Taipei, Taiwan, in 1986, and the MS and the PhD degrees from the California Institute of Technology, Pasadena, USA, in 1989 and 1994, respectively. Since 1995, he has been with the School of Mechanical and Aerospace Engineering of Nanyang Technological University (NTU), Singapore. He is the Director of Intelligent Systems Center in NTU. He was JSPS Visiting Scholar in Kyoto University, Japan in 1999, Visiting Scholar in the Department of Mechanical Engineering of MIT in 2004, and currently Fellow of Singapore-MIT Alliance under Manufacturing Systems and Technology (MST) Program, and Fellow of IEEE/ASME. His research interests are in wearable sensors, human-robot interaction, reconfigurable automation, parallel kinematics machines (PKM), and smart material-based actuators. To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints