Analysis and Design of a Maglev Permanent Magnet Synchronous Linear Motor to Reduce Additional Torque in Dq Current Control

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Article Analysis and Design of a Maglev Permanent Magnet Synchronous Linear Motor to Reduce Additional Torque in dq Current Control

Feng Xing 1 ID , Baoquan Kou 1,*, Lu Zhang 1, Tiecheng Wang 1 and Chaoning Zhang 2 1 School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China; [email protected] (F.X.); [email protected] (L.Z.); [email protected] (T.W.) 2 School of Electrical Engineering, KAIST, Daejeon 34141, Korea; [email protected] * Correspondence: [email protected]; Tel.: +86-451-8640-3771

Received: 21 January 2018; Accepted: 28 February 2018; Published: 5 March 2018

Abstract: The maglev linear motor has three degrees of motion freedom, which are respectively realized by the thrust force in the x-axis, the levitation force in the z-axis and the torque around the y-axis. Both the thrust force and levitation force can be seen as the sum of the forces on the three windings. The resultant thrust force and resultant levitation force are independently controlled by d-axis current and q-axis current respectively. Thus, the commonly used dq transformation control strategy is suitable for realizing the control of the resultant force, either thrust force and levitation force. However, the forces on the three windings also generate additional torque because they do not pass the mover mass center. To realize the maglev system high-precision control, a maglev linear motor with a new structure is proposed in this paper to decrease this torque. First, the electromagnetic model of the motor can be deduced through the Lorenz force formula. Second, the analytic method and ﬁnite element method are used to explore the reason of this additional torque and what factors affect its change trend. Furthermore, a maglev linear motor with a new structure is proposed, with two sets of 90 degrees shifted winding designed on the mover. Under such a structure, the mover position dependent periodic part of the additional torque can be offset. Finally, the theoretical analysis is validated by the simulation result that the additionally generated rotating torque can be offset with little ﬂuctuation in the proposed new-structure maglev linear motor. Moreover, the control system is built in MATLAB/Simulink, which shows that it has small thrust ripple and high-precision performance.

Keywords: maglev; linear motor; additional torque; offset

1. Introduction Linear motors are now an increasingly popular alternative solution for linear positioning applications [1–4]. In general, the linear motor-based positioning systems feature higher performance in terms of accuracy, thrust and acceleration compared with rotating motor based systems [5]. Three common support methods are available for the linear movement, namely mechanical method [6–8], air floatation method [9–11] and magnetic levitation method [12–15]. The mechanical method has the advantage of low cost and has been widely used in simple linear movement applications. However, the friction force is inevitable, which limits the control accuracy. The air flotation method has no physical contact and thus has no friction force as the mechanical method. The disadvantage of this method is that the high-pressure air causes high-frequency vibration. Moreover, this method is subject to disturbance vulnerability and low stiffness. Some ultra-precision equipment needs to work in the vacuum state, which further limits the application of the air flotation method. The maglev permanent magnet synchronous linear motor has the advantages of no mechanical

Energies 2018, 11, 556; doi:10.3390/en11030556 www.mdpi.com/journal/energies Energies 2018, 11, 556 2 of 15 friction and high stiffness and being suitable for working in a vacuum state. Thus, it has been researched by many scholars for wide applications in high-speed, large-acceleration and high-precision linear positioning movement [16–18]. The maglev linear motor has three degrees-of-freedom (DOF). They are realized by the thrust force in the x-axis, the maglev force in the z-axis and the torque around the y-axis respectively. The commonly adopted traditional linear motor control method is based on dq decoupling, which can realize the decoupling control of the thrust and levitation force through the respective iq and id. With the dq transformation, ABC three-phase currents can be obtained from the needed iq and id. The three phase currents under the magnetic field can generate respective position dependent force on the three winding. Both the thrust force and levitation force can be seen as the sum of the forces on the three windings. The resultant thrust force and resultant levitation force are independently controlled by iq current and id current respectively. Thus, the commonly used dq transformation control strategy is suitable for realizing the control of the resultant force, either thrust force and levitation force. However, the forces on the three windings also generate additional torque. For the levitation force, they act on the three different points and thus generate that torque. For the thrust force in the direction of x-axis, that torque still exists because it does not go in line with the center the mover mass, it can also generate torque which is not dependent on the mover position. The traditional linear motor adopting the linear guide and sliding module has only single DOF, thus the imposed torque has little influence on the system due to the mechanical constraint. However, the imposed torque will change the gap of the maglev linear motor at the two ends due to the lack of mechanical constraint. The gap change can make it extremely difficult to realize the proper control of the electromagnetic force. Therefore, how to decrease the imposed torque is the key factor to realize the precise control of the maglev linear motor. The three forces on the three-phase windings are the function of the position in the magnetic field. To decrease the disturbance torque paper [19] has proposed to use two permanent magnet arrays which are placed opposite to each other. When two movers are mounted back-to-back and moved in between the permanent magnet arrays, the torques produced by the movers cancel each other out. This arrangement, however, cannot be used to preload aerostatic bearings or in single side magnetic levitation. A commutation and switching algorithm for multi DOF moving magnet actuators with permanent magnets and integrated active magnetic bearing has been proposed in [20]. Wrench current decoupling method has been applied in the multi DOF control of the linear motor. However, this method requires a full-bridge drive unit to drive one winding, which increases the hardware cost and complexity. Paper [21] has proposed three DOF control strategy for an iron core linear permanent magnet motor, which is based on the dq decomposition. To realize the three DOF control, however, the conventional dq control strategy cannot realize the desired performance and thus more complex decoupling control algorithm needs to be designed. To decrease the disturbance torque caused by the conventional dq control strategy, this paper proposed a new structure of maglev linear motor. The mover has two sets of three-phase windings which are arranged with a phase difference of 90 electrical degrees. When the sets of windings have an initial phase difference of 90 electrical degrees, the disturbance torques in the two sets of windings are in the opposite direction, which can cancel each other out. Traditional dq control strategy can be used to realize the three DOF control without complex decoupling and compensation. Therefore, the traditional drive consisting of three half-bridges can be used to the proposed new-structure maglev linear motor. The proposed new-structure maglev linear motor features advantages such as additional torque offset, simple control strategy and simple hardware requirement. The structure of the paper is as follows. In Section2, the three DOF electromagnetic model of ironless maglev permanent magnet synchronous linear motor is introduced. It includes the coordinate system definition of maglev system, the analytic formula of Halbach magnetic field and the analytic formula of the electromagnetic force and torque. In Section3, the disturbance torque around the center of the mover mass caused in the conventional dq decoupling strategy is explored. In Section4, EnergiesEnergies 20182018, 11, 11, x, 556FOR PEER REVIEW 3 3of of 15 15 the motor structure is optimally designed. In Section 5, the simulation has been done to validate the the new-structure linear motor is proposed to offset the disturbance torque; moreover, the motor disturbance torque offset and the effectiveness of the dq decoupling strategy in the control system. structure is optimally designed. In Section5, the simulation has been done to validate the disturbance Section 6 presents a simple conclusion of this paper. torque offset and the effectiveness of the dq decoupling strategy in the control system. Section6 presents a simple conclusion of this paper. 2. Maglev Permanent Magnet Linear Motor Electromagnetic Model 2. Maglev Permanent Magnet Linear Motor Electromagnetic Model 2.1. Coordinate Definition 2.1.Figure Coordinate 1 show Definition the model of a maglev linear motor which consists of the stator, air gap and the mover.Figure The stator1 show is themade model of ABC of a maglevthree-phase linear wi motorndings which and backplane. consists of The the three-phase stator, air gap windings and the aremover. arranged The stator in the is direction made of ABCof x. three-phaseThe three-phase windings winding and backplane.arrangement The width three-phase is 4τ (τ windings is the pole are pitch).arranged The in stator the direction is made of ofx .magnetic The three-phase conductive winding plate arrangement and permanent width magnets. is 4τ (τ isThe the permanent pole pitch). magnetsThe stator are is arranged made of magneticin the form conductive of Halbach. plate and permanent magnets. The permanent magnets are arranged in the form of Halbach.

m z Mover s z m y mx s p s y x Air gap

τ q d Stator

FigureFigure 1. 1. CoordinateCoordinate system system of of maglev maglev linear linear motor. motor.

Two coordinate systems are defined in Figure 1. The first one is the stator coordinate system Two coordinate systems are deﬁned in Figure1. The ﬁrst one is the stator coordinate system denoted by the superscript s, which is shown as: denoted by the superscript s, which is shown as: T ssss=   v h x yz i T (1) sv = sx sy sz (1) The other one is the mover coordinate system denoted by the superscript m. It can be expressed as: The other one is the mover coordinate system denoted by the superscript m. It can be expressed as:

h T iT mmmmm=  m m m vv=  xx yzy z (2)(2)

TheThe vector vector pp isis the the position position vector that isis deﬁneddefined asas pointingpointing from from the the stator stator coordinate coordinate system system to tothe the mover mover coordinate coordinate system. system. It It can can be be expressed expressed as: as: T =   p hpppxyz i T (3) p = px py pz (3) The two coordinate systems can be converted into each other with the coordinate transformationThe two coordinate through the systems following can beequations converted [22]: into each other with the coordinate transformation through the following equations [22]: ssm=+ s vRvs m m p (4) v = Rm v + p (4) mmmvRvp=−m () ss v = Rs s( v − p) (5)(5) TheThe orientation orientation transformation transformation matrix matrix is is defined deﬁned as: as:

ssss s R= Rot()()()sx,,,ϕθ Rotyzs Rot s φ Rmm = Rot( x, ϕ)Rot( y, θ)Rot( z, φ) (6)(6) ms==−1T s m RRs s mm−1 Rs T (7) Rs = Rm = Rm (7)  10 0  1 0 0  s   ()ϕϕϕ=− RotRot(sx, ϕx),0cos()sin()=  0 cos(ϕ) − sin(ϕ)  (8)(8)  ϕϕ 00sin()cos() sin(ϕ) cos(ϕ)

Energies 2018, 11, 556 4 of 15

  cos(θ) 0 sin(θ) s   Rot( y, θ) =  0 1 0  (9) − sin(θ) 0 cos(θ)   cos(φ) − sin(φ) 0 s   Rot( z, φ) =  sin(φ) cos(φ) 0  (10) 0 0 1 where φ, θ and ϕ are the rotation angles about the sx-, sy-, and sz-axes, respectively.

2.2. Magnetic Field Analysis The literature review shows that the magnetic field of the Halbach permanent magnet array is predicted by either the harmonic model [23] or the transfer relationship [24], which can be eventually solved as boundary-value problems using the Maxwell equation and the scalar potential. The magnetic field analysis of the linear motor can be done under several hypotheses, such as the neglection of the end effects and the magnetic field saturation, etc. [22,25]. In Figure1, the magnetic flux density B can be decomposed in the x and z directions as follows [26]: √ 2 2µ M π π h π B = − 0 0 1 − exp − h exp − sz + m sin sx (11) x π τ m τ 2 τ √ 2 2µ M π π h π B = 0 0 1 − exp − h exp − sz + m cos sx (12) z π τ m τ 2 τ where µ0 and M0 denote the permeability of the free space and the peak magnetization magnitude of permanent magnets, respectively. hm represents height of a single permanent magnet. To make the above equations more concise, we assume that √ 2 2µ M π πh K = 0 0 1 − exp − h exp − m (13) 1 π τ m 2τ

Based on the above assumption, Equations (11) and (12) can be rewritten as: π π B = −K exp − sz sin sx (14) x 1 τ τ π π B = K exp − sz cos sx (15) z 1 τ τ The above equations can be transformed into those in the mover coordinate system which are shown as: π π B = −K exp − (mz + p ) sin (mx + p ) (16) x 1 τ z τ x π π B = K exp − (mz + p ) cos (mx + p ) (17) z 1 τ z τ x

2.3. Electromagnetic Force and Torque Analysis There are three common analytical ways to solve magnetic force in his research report, namely virtual work method, Maxwell stress method, and the Lorentz force equation method [27]. Among them, Lorentz force equation method is the most straightforward and simple one, which is shown as [28,29]: F = y J × BdV (18) V

T = y r × (J × B)dV (19) V Energies 2018, 11, x FOR PEER REVIEW 5 of 15 Energies 2018, 11, 556 5 of 15 =××() TrJB dV (19) V where B is the magnetic ﬂux density of the magnet array, J is the volume current density in the coil, where B is the magnetic flux density of the magnet array, J is the volume current density in the coil, V is the volume of the coil, r is the vector from the point about which the torque is calculated. V is the volume of the coil, r is the vector from the point about which the torque is calculated. The volume current density is dependent on the current, which is shown as: The volume current density is dependent on the current, which is shown as:

= NiNi J J= (20)(20) wwhccchc where NN isis the the turns turns of of the the winding, winding,wc andwc andhc are hc theare windingthe winding width width and height and height respectively respectively as shown as inshown Figure in2 Figure. 2.

s y m y mx sz sx m M lc A z j A i

m z s z m wc m M j s y x s y x hc

lcw A-A Figure 2. Top views and A-A view of the components of a maglev linear motor: a Halbach array and

a coil. Point Mj isis the the position position in in the the local local coordinate coordinate system.

In the mover local coordinate system, the electromagnetic force and torque on the straight In the mover local coordinate system, the electromagnetic force and torque on the straight segment segment of the coil can be derived as: of the coil can be derived as: m mmmmmm m m m F =FJBvpy=×J × B()( v,dd, p)dx xdy y (21)(21)

m =mmmmm=××× ( × m (m )) m m T TrJBvpyr ()J B(),ddv, p dx xdy y (22)(22) According to the model in Figure2, The model simpliﬁcation method is paper [ 23] is used to According to the model in Figure 2, The model simplification method is paper [23] is used to simplify the winding into 2 ﬁlaments with current i replacing current density J, thus the electromagnetic simplify the winding into 2 filaments with current i replacing current density J, thus the force and torque in the j-th winding (when lcw = τ) can be expressed as: electromagnetic force and torque in the j-th winding (when lcw = τ) can be expressed as:             my + lc m τ my + lc m τ j 2 0 x j + ττj 2 0 x j − Z mm+−2 Z  2 m   ll m  mxx  m   m  m   m  F = N m yy++i c  × Byjj, pd y + m c −i  × B  y , pd y (23) j j j 00j 22j  j j   22          mmmmmmmmy − lc m my − lc   m FBpBp=×jNi20 z j y,d y +−×j 2 i0 yz j ,d y jj j  j    j (23) m −−llc mmm c   yyj 00zzj   22jj                     m x 0 m x + τ m x 0 m x − τ  my + lc  j     j 2   my + lc  j     j 2     j 2        j 2         mT = N R  m  ×   × mB m , pdmy + R  m  ×   × mB m , pdmy j    y   ij   y j ττ   y   −ij   y j    (24)  l      m +−  l     m     my − c   l mm   xx  my − cl         j 2 m y ++c m j mjy 2c  j m r j xxj 0 00z 22j rj  0 z  z 2 j 2 z  j  m=××+×−× m mm m m   mm m  TBpBpjjjNyiyyyiy,d  jj   ,d y      m −−lc m m lc  m (24) y j rz00y j  r   z According to the2  calculationzj results, the electromagnetic2  z force   and torque  j can be derived as:         π mF = −2l NK K i sin mx + p (25) xj c 1 z j τ j x According to the calculation results, the electromagnetic force and torque can be derived as: m π m Fzj = 2lc NK1Kzij cos x j + px (26) τπ mmF =−2sinlNKKi() x + p xj cπ 1 z jτ j x π (25) mT = −2l NK K i cos mx + p x + sin mx + p r (27) yj c 1 z j τ j x j τ j x z

Energies 2018, 11, 556 6 of 15

m where, rz is the equivalent force arm in the z-direction from the coil to the mover mass center. Kz can be expressed as: hc hc Z π Z π K = exp − sz dsz = exp − (mz + p ) dmz (28) z τ τ z 0 0

3. Additional Torque Analysis and Offset

3.1. dq Coordinate Transformation The control of the three-phase AC motor can be seen as that of DC motor through the coordinate transformation. The three-phase currents can be transformed into force current iq and magnetic ﬁeld excitation current id. Thus, the thrust force and the levitation force of linear motor can be independently controlled by the current iq and id respectively. The current transformation from the dq coordinate to the ABC coordinate system is shown as:

   π π  i r cos x − sin x " # A 2 τ τ i   =  π − 2 − π − 2  d  iB   cos τ x 3 sin τ x 3  (29) 3 π 2 π 2 iq iC cos τ x + 3 − sin τ x + 3

Through Equation (29), it can be known that the needed three-phase currents can be calculated from the desired id and iq.

3.2. Torque Analysis Once the coil is ﬁxed on the mover, the relative position between the windings becomes a known constant, and the force and torque on the mass center of the mover can be calculated by substituting the coordinates of each coil into Equations (25)–(27) separately and then add them up. The electromagnetic force generated by three-phase winding is expressed as: π π π mF = −2l NK K sin (mx + p ) i + sin (mx + p ) i + sin (mx + p ) i (30) x c 1 z τ A x A τ B x B τ C x C Substituting the current in Equation (29) and the position of the coil in the mover coordinate system into Equation (30), we can get r ! 2 π 4 2 π 2 2 π 2 mF = 2l NK K i sin (p ) + π + sin (p ) + π + sin (p ) (31) x c 1 z q 3 τ x 3 τ x 3 τ x

Through mathematical analysis it can be known that

π 4 2 π 2 2 π 2 3 sin (p ) + π + sin (p ) + π + sin (p ) ≡ (32) τ x 3 τ x 3 τ x 2

Thus, Equation (31) can be rewritten as: √ m Fx = 6lc NK1Kziq (33)

In a similar way, we can get √ m Fz = 6lcK1Kzid (34) The torque of the three-phase winding on the center of mass is

m π m c π m Ty = −2lc NK1Kz cos τ ( x A + px) x A + sin τ ( x A + px) rz iA π m c π m + cos τ ( xB + px) xB + sin τ ( xB + px) rz iB (35) π m c π m + cos τ ( xC + px) xC + sin τ ( xC + px) rz iC Energies 2018, 11, 556 7 of 15

Substituting the current in Equation (29) into Equation (35), we can get √ m 8 6 π 2 2 π 2 2 Ty = −lc NK1Kzid τ cos (px) − π − cos (px) + π √9 τ 3 τ 3 (36) 4 6 π 2 π 2 +lc NK1Kziq 9 τ sin 2 τ (px) − 3 π − sin 2 τ (px) + 3 π + Fxrz

It can be seen from Equations (33) and (34) that the required Fx and Fz forces can be obtained from the given currents iq and id. From Equation (36) it can be seen that both the d-axis current and the q-axis current can produce a torque which varies with the position, and the thrust force Fx generated by the q-axis current can generate a constant torque related to the arm rz.

3.3. Finite Element Analysis of AdditionalTorque The ﬁnite element simulation software is used to model and simulate the motor structure in Figure1 with the main parameters shown in Table1. In the simulation, two groups of excitation, id = 3.7 A, iq = 0 A and id = 0 A, iq = 3.7 A, are applied respectively. The simulation results are shown in Figure3. When id = 3.7 A, iq = 0 A, the torque Td (blue curve in Figure3) to the centroid of the mover changes from −0.33 N·m to 0.33 N·m, and the period is τ. When id = 0 A, iq = 3.7 A, the torque Tq (red curve in Figure3) to the center of the mover varies from −0.22 N·m to 0.42 N·m, and the period is also τ, but a constant offset is generated. This offset results from the thrust Fx in Equation (36).

Table 1. Motor major parameters.

Parameter Value Unit Pole pitch τ 15 mm Winding size (lc × wc × hc) 100 × 5 × 10 mm Major permanent magnet size (lm1 × wm1 × hm) 100 × 10 × 8 mm Auxiliary permanent magnet size (lm2 × wm2 × hm) 100 × 5 × 8 mm Energies 2018, 11, x FOR PEER REVIEWAir gap δ 1 mm 8 of 15

0.8 Td Tq 0.4 ·m T/N 0

-0.4 0 1 2 3 4 5 6

x/τ

FigureFigure 3. Additional3. Additional torque torque caused caused by byid andid andiq .iq.

4.4. Optimal Optimal and and Design Design of of Motor Motor Structure Structure

4.1.4.1. Structure Structure Design Design InIn order order to to keep keep the the thrust thrust force force and and levitation levitation force force constant constant at a magneticat a magnetic field-free field-free position, position, the forcethe generatedforce generated by each by of each the coils of the varies coils at differentvaries at relativedifferent magnetic relative field magnetic positions, field thus positions, givingrise thus togiving a torque rise relative to a torque to the centroidrelative to of the the mover.centroid According of the mover. to the According torque fluctuations to the torque with fluctuations respect to positionswith respect caused to by positi currentons icausedd and iq ,by the current use of twoid and motor iq, the units use is proposedof two motor to generate units is the proposed opposite to torquegenerate ripple the to opposite cancel each torque other ripple out toto achievecancel each optimal other additional out to achieve torque. optimal Electromagnetic additional force torque. is generatedElectromagnetic by the current force is and generated the magnetic by the field current together, and the and magnetic the current field and together, magnetic and field the variescurrent nearlyand magnetic sinusoidal field (cosine) varies and nearly their periodsinusoidal is both (cosine) twice and the poletheir pitch periodτ. Fromis both Equation twice the (36), pole it canpitch be τ. From Equation (36), it can be known that the period of torque is a pole pitch τ. Figure 3 also shows that the undulating period of the additional torque is a pole pitch τ. Therefore, it is necessary to maintain the relative phase difference 1/2τ of the two motor units to achieve the offset of the additional torque ripple. Figure 4 shows the designed three DOF maglev linear motor, which is mainly composed of the mover and stator. The mover adopts ironless structure and consists of two sets of ABC three-phase winding and the backplane. The stator is composed of with a uniformly arranged Halbach permanent magnet and a yoke plate made from magnetic material. The arrangement of the two sets of windings is as follows. The mover is divided into two parts by its mass center in the thrust x direction, and each part is equipped with a set of three-phase windings to form a complete three-phase motor unit, with the corresponding phase difference between the initial phase of the winding being 4.5τ, as shown in Figure 5.

Figure 4. Basic structure of the proposed maglev linear motor.

Energies 2018, 11, x FOR PEER REVIEW 8 of 15

0.8 Td Tq 0.4 ·m T/N 0

-0.4 0 1 2 3 4 5 6

x/τ

Figure 3. Additional torque caused by id and iq.

4. Optimal and Design of Motor Structure

4.1. Structure Design In order to keep the thrust force and levitation force constant at a magnetic field-free position, the force generated by each of the coils varies at different relative magnetic field positions, thus giving rise to a torque relative to the centroid of the mover. According to the torque fluctuations with respect to positions caused by current id and iq, the use of two motor units is proposed to generate the opposite torque ripple to cancel each other out to achieve optimal additional torque. Electromagnetic force is generated by the current and the magnetic field together, and the current and magneticEnergies 2018 field, 11 ,varies 556 nearly sinusoidal (cosine) and their period is both twice the pole pitch τ. 8 of 15 From Equation (36), it can be known that the period of torque is a pole pitch τ. Figure 3 also shows that the undulating period of the additional torque is a pole pitch τ. Therefore, it is necessary to maintainknown the thatrelative the period phase of difference torque is a 1/2 poleτ pitchof theτ .two Figure motor3 also units shows to that achieve the undulating the offset period of the of the additionaladditional torque torque ripple. is a pole pitch τ. Therefore, it is necessary to maintain the relative phase difference Figure1/2τ of 4 theshows two the motor designed units tothree achieve DOF themaglev offset linear of the motor, additional which torque is mainly ripple. composed of the mover andFigure stator.4 Theshows mover the designedadopts ironless three DOF struct maglevure and linear consists motor, of two which sets is of mainly ABC three-phase composed of the windingmover and and the stator. backplane. The mover The adoptsstator ironlessis comp structureosed of andwith consists a uniformly of two arranged sets of ABC Halbach three-phase permanentwinding magnet and the and backplane. a yoke plate The made stator from is composed magnetic of material. with a uniformly The arrangement arranged of Halbach the two permanent sets of windingsmagnet is and as afollows. yoke plate The made mover from is divided magnetic into material. two parts The arrangementby its mass center of the twoin the sets thrust of windings x direction,is as follows.and each The part mover is equipped is divided with into twoa set parts of bythree-phase its mass center windings in the thrustto formx direction, a complete and each three-phasepart is motor equipped unit, with with a the set corresponding of three-phase ph windingsase difference to form between a complete the initial three-phase phase of motor the unit, windingwith being the corresponding 4.5τ, as shown phase in Figure difference 5. between the initial phase of the winding being 4.5τ, as shown in Figure5.

Energies 2018, 11, x FOR PEER REVIEW 9 of 15

A2 C2 B2 Unit II 4τ

Energies 2018, 11,Figure x FOR Figure PEER4. Basic REVIEW 4. structureBasic structure of the proposed of the proposed maglev maglev linear linearmotor. motor. 9 of 15

A2 C2 B2 Unit II 4τ (4+½)τ Unit I A1 C1 B1

Figure 5. Structure and arrangement of the two sets of windings. (4+½)τ Unit I A1 C1 B1 4.2. Optimization of Halbach Permanent Magnet Array FigureFigure 5. Structure 5. Structure and and arrangement arrangement of the of the two two sets sets of windings. of windings. The size of the main permanent magnet and the auxiliary permanent magnet can affect the 4.2. Optimization of Halbach Permanent Magnet Array distribution of the4.2. magnetic Optimization ofinduction Halbach Permanent intensity Magnet in Array the air gap, thus the structure optimization of the permanent magnetThe Theis size necessarysize of theof the main main permanentfor permanent the magnetdesired magnet and andpe therformance. th auxiliarye auxiliary permanent permanent When magnet themagnet canpole can affect affectpitch the the is fixed, the distributiondistribution of theof the magnetic magnetic induction induction intensity intensity in thein the air air gap, gap, thus thus the the structure structure optimization optimization of theof the variable structurepermanentpermanent dimensions magnet magnet is necessary wism necessary1, w form2, theh form desired of the permanent desired performance. performance. magnet When the When polearray, pitchthe pole which is ﬁxed, pitch the affectisvariable fixed, the the distribution of the magneticstructure inductionvariable dimensions structure intensity dimensionswm1, wm in2, h thewm mof1, w permanentairm2, hgap,m of permanent are magnet shown array, magnet whichin array,Figure affect which the6. affect distribution the distribution of the magneticof the magnetic induction induction intensity inintensity the air gap,in the are air shown gap, are in shown Figure6 in. Figure 6.

Figure 6. Selection of the permanent magnet array variable. Figure 6. Selection of the permanent magnet array variable. Figure 6. Selection of the permanent magnet array variable. The wm1, wm2 and hm in Figure 6 are set as variables to optimize the force coefficient k of the

linearThe wmotor.m1, w mThe2 and widthhm inof Figurethe main6 are permanent set as variables magnet towm optimize1 is in the the range force of 1–14 coefﬁcient mm, thek ofwidth the of the auxiliary permanent magnet wm2 is in the range of 14–1 mm, and the thickness of the permanent The wm1, linearwm2 and motor. h Them in width Figure of the 6 main are permanent set as magnetvariableswm1 is to in theoptimize range of 1–14 the mm, force the width coefficient of k of the themagnet auxiliary hm permanent is in the range magnet of 1–25wm2 mm.is in theBesides, range the of 14–1relation mm, of and wm1 the + w thicknessm2 = τ should of the be permanent satisfied. The linear motor. Thefinite width element of simulationthe main software permanent is used and magnet the force w coefficientm1 is in thek of differentrange ofsizes 1–14 of structures mm, the width of the auxiliary permanentwith in different magnet variable w mvalues2 is in is calculated.the range The of simulation 14–1 mm, results and are shown the thickness in Figure 7. of the permanent magnet hm is in the range of 1–25 mm. Besides, the relation of wm1 + wm2 = τ should be satisfied. The finite element simulation software is used15 and the force coefficient k of different sizes of structures with in different variable values is calculated. The simulation results are shown in Figure 7. 10 k 5

0 15 15 10 30 20 5 10 wm1 /mm 0 hm /mm 10 0 Figure 7. Force coefficient curve in terms of permanent magnet size. k 5 It can be seen from Figure 7 that the main permanent magnet width wm1 and the auxiliary permanent magnet width wm2 have influence on the force coefficient k. The force coefficient is small 0 in the initial stage during15 the change of wm1. The force coefficient keeps increasing with the increase of wm1 until a certain critical 10point. After this critical point, the force coefficient30 begins to decrease 20 when wm1 continues to increase. In Figure5 7, the maximum value of force coefficient k appears when w /mm 10 h /mm wm1 is in the range of 8–9 mm, som1 the simulation0 0 is further carriedm out in the interval of 8–9 mm with steps of 0.1 mm. The optimal force coefficient is achieved when the width of the permanent magnet is wmFigure1 = 8.7 mm 7. andForce wm2 coefficient= 6.3 mm. The curve relationship in terms between of permanent the force coefficient magnet k and size. the height of the permanent magnet hm with the above optimized width of the permanent magnet is shown in Figure 8. It can be seen from Figure 7 that the main permanent magnet width wm1 and the auxiliary permanent magnet width wm2 have influence on the force coefficient k. The force coefficient is small in the initial stage during the change of wm1. The force coefficient keeps increasing with the increase of wm1 until a certain critical point. After this critical point, the force coefficient begins to decrease when wm1 continues to increase. In Figure 7, the maximum value of force coefficient k appears when wm1 is in the range of 8–9 mm, so the simulation is further carried out in the interval of 8–9 mm with steps of 0.1 mm. The optimal force coefficient is achieved when the width of the permanent magnet is wm1 = 8.7 mm and wm2 = 6.3 mm. The relationship between the force coefficient k and the height of the permanent magnet hm with the above optimized width of the permanent magnet is shown in Figure 8.

Energies 2018, 11, x FOR PEER REVIEW 9 of 15

A2 C2 B2 Unit II 4τ

(4+½)τ Unit I A1 C1 B1

Figure 5. Structure and arrangement of the two sets of windings.

4.2. Optimization of Halbach Permanent Magnet Array The size of the main permanent magnet and the auxiliary permanent magnet can affect the distribution of the magnetic induction intensity in the air gap, thus the structure optimization of the permanent magnet is necessary for the desired performance. When the pole pitch is fixed, the variable structure dimensions wm1, wm2, hm of permanent magnet array, which affect the distribution of the magnetic induction intensity in the air gap, are shown in Figure 6.

Figure 6. Selection of the permanent magnet array variable.

The wm1, wm2 and hm in Figure 6 are set as variables to optimize the force coefficient k of the Energies 2018, 11, 556 9 of 15 linear motor. The width of the main permanent magnet wm1 is in the range of 1–14 mm, the width of the auxiliary permanent magnet wm2 is in the range of 14–1 mm, and the thickness of the permanent magnet hm isis in in the the range range of of 1–25 1–25 mm. mm. Besides, Besides, the the relation relation of ofwmw1 m+ 1w+m2w =m τ2 should= τ should be satisfied. be satisfied. The Thefinite finite element element simulation simulation software software is used is used and and the the force force coefficient coefficient k kofof different different sizes sizes of of structures structures with in different variable values isis calculated.calculated. The simulation resultsresults areare shownshown inin FigureFigure7 7..

10 k 5

0 15 10 30 20 5 w /mm 10 h /mm m1 0 0 m

Figure 7. ForceForce coefficient coefﬁcient curve in terms of permanent magnet size.

It can be seen from Figure 7 that the main permanent magnet width wm1 and the auxiliary It can be seen from Figure7 that the main permanent magnet width wm1 and the auxiliary permanent magnet width wm2 have influence on the force coefficient k. The force coefficient is small permanent magnet width wm2 have influence on the force coefficient k. The force coefficient is small in the initial stage during the change of wm1. The force coefficient keeps increasing with the increase in the initial stage during the change of wm1. The force coefficient keeps increasing with the increase of wm1 until a certain critical point. After this critical point, the force coefficient begins to decrease of wm1 until a certain critical point. After this critical point, the force coefficient begins to decrease when wm1 continues to increase. In Figure 7, the maximum value of force coefficient k appears when when wm1 continues to increase. In Figure7, the maximum value of force coefficient k appears when wm1 is in the range of 8–9 mm, so the simulation is further carried out in the interval of 8–9 mm with wm1 is in the range of 8–9 mm, so the simulation is further carried out in the interval of 8–9 mm with steps of 0.1 mm. The optimal force coefficient is achieved when the width of the permanent magnet steps of 0.1 mm. The optimal force coefficient is achieved when the width of the permanent magnet is is wm1 = 8.7 mm and wm2 = 6.3 mm. The relationship between the force coefficient k and the height of wm1 = 8.7 mm and wm2 = 6.3 mm. The relationship between the force coefficient k and the height of the theEnergies permanent 2018, 11, x magnetFOR PEER h REVIEWm with the above optimized width of the permanent magnet is shown10 of in 15 permanent magnet hm with the above optimized width of the permanent magnet is shown in Figure8. Figure 8.

8 k

2 0 5 10 15 20 25 h / mm m Figure 8. Force coefﬁcientcoefficient k curve with respectrespect toto thethe thicknessthickness ofof permanentpermanent magnet.magnet.

ItIt cancan be be seen seen from from Figure Figure8 that 8 thethat influence the influence permanent permanent magnet thicknessmagnet thickness on the force on coefficient the force kcoefficientand can be k roughlyand can divided be roughly into three divided ranges. into In three the range ranges. of 1–10In the mm range for the of permanent 1–10 mm magnetfor the thickness,permanent the magnet change thickness, of force coefficient the changek increases of force almost coefficient linearly k increases with the increasealmost linearly of the permanent with the magnetincrease thickness. of the permanent In the range magnet of 1–25 thickness. mm, the In force the coefficient range of 1–25 increases mm, slowlythe force with coefficient the increase increases of the thicknessslowly with of the the permanent increase of magnet. the thickness When the of thicknessthe permanent is larger magnet. than 20 When mm, and the thethickness force coefficient is larger isthan nearly 20 mm, constant and withthe force the increase coefficient of theis nearly thickness constant of the with permanent the increase magnet. of Thethe thickness selection of thethe thicknesspermanent of magnet. the permanent The selection magnet of should the thickness depend of on the the permanent type of motor magnet and should specific depend requirements. on the Astype the of motor motor is moving-coiland specific type requirements. maglev permanent As the magnetmotor synchronousis moving-coil linear type motor, maglev the statorpermanent mass doesmagnet not synchronous affect the ratio linear of the motor, thrust the to the stator moving mass parts, does the not thickness affect the of ratio the permanent of the thrust magnet to the is selectedmoving asparts, 20 mm. the thickness of the permanent magnet is selected as 20 mm.

4.3. Optimization of Coil Thickness After the optimization of the permanent magnet size, the thickness of the coil in the z-direction is selected from 1–20 mm, and the simulation result is shown in Figure 9.

k 10

0 0 2 4 6 8 10 12 14 16 18 20 h /mm c Figure 9. Force coefficient k curve with respect to the thickness of coil.

Figure 9 shows the relationship between a motor unit thrust coefficient and the thickness of the coil. It is seen from Figure 9 that in the initial stage, the force coefficient k increases significantly with the increase of the coil thickness. When the coil thickness is beyond 12 mm, the increase of the coil thickness has little influence on the coil thrust output, which is caused by the attenuation of the magnetic field, thus the coil thickness hc is selected to be 12 mm.

5. Simulation Result and Analysis The proposed motor structure has been explored using finite element simulation software. In the simulation, the open-loop current is given to the motor based on the relative position between the magnetic field and the motor mover to validate the offset of the additional torques. Moreover, MATLAB/Simulink has been used to build the motor control system to explore its control accuracy in the three DOF and its robustness.

Energies 2018, 11, x FOR PEER REVIEW 10 of 15

8 k

2 0 5 10 15 20 25 h / mm m Figure 8. Force coefficient k curve with respect to the thickness of permanent magnet.

It can be seen from Figure 8 that the influence permanent magnet thickness on the force coefficient k and can be roughly divided into three ranges. In the range of 1–10 mm for the permanent magnet thickness, the change of force coefficient k increases almost linearly with the increase of the permanent magnet thickness. In the range of 1–25 mm, the force coefficient increases slowly with the increase of the thickness of the permanent magnet. When the thickness is larger than 20 mm, and the force coefficient is nearly constant with the increase of the thickness of the permanent magnet. The selection of the thickness of the permanent magnet should depend on the type of motor and specific requirements. As the motor is moving-coil type maglev permanent magnet synchronous linear motor, the stator mass does not affect the ratio of the thrust to the Energiesmoving2018 parts,, 11, 556the thickness of the permanent magnet is selected as 20 mm. 10 of 15

4.3. Optimization of Coil Thickness 4.3. Optimization of Coil Thickness After the optimization of the permanent magnet size, the thickness of the coil in the z-direction After the optimization of the permanent magnet size, the thickness of the coil in the z-direction is is selected from 1–20 mm, and the simulation result is shown in Figure 9. selected from 1–20 mm, and the simulation result is shown in Figure9.

k 10

0 0 2 4 6 8 10 12 14 16 18 20 h /mm c

Figure 9. Force coefﬁcientcoefficient k curve with respect toto thethe thicknessthickness ofof coil.coil.

Figure 9 shows the relationship between a motor unit thrust coefficient and the thickness of the Figure9 shows the relationship between a motor unit thrust coefficient and the thickness of the coil. It is seen from Figure 9 that in the initial stage, the force coefficient k increases significantly coil. It is seen from Figure9 that in the initial stage, the force coefficient k increases significantly with the increase of the coil thickness. When the coil thickness is beyond 12 mm, the increase of the with the increase of the coil thickness. When the coil thickness is beyond 12 mm, the increase of the coil thickness has little influence on the coil thrust output, which is caused by the attenuation of the coil thickness has little influence on the coil thrust output, which is caused by the attenuation of the magnetic field, thus the coil thickness hc is selected to be 12 mm. magnetic field, thus the coil thickness hc is selected to be 12 mm. 5. Simulation Result and Analysis 5. Simulation Result and Analysis The proposed motor structure has been explored using finite element simulation software. In The proposed motor structure has been explored using finite element simulation software. the simulation, the open-loop current is given to the motor based on the relative position between In the simulation, the open-loop current is given to the motor based on the relative position the magnetic field and the motor mover to validate the offset of the additional torques. Moreover, between the magnetic field and the motor mover to validate the offset of the additional torques. MATLAB/Simulink has been used to build the motor control system to explore its control accuracy Moreover, MATLAB/Simulink has been used to build the motor control system to explore its control in the three DOF and its robustness. accuracy in the three DOF and its robustness.

5.1. Finite Element Simulation The proposed maglev linear motor is simulated in the ﬁnite element software based on the motor parameters given in Table2.

Table 2. Motor major parameters.

Parameter Value Unit Pole pitch τ 15 mm Air gap δ 1 mm Winding size (lc × wc × hc) 100 × 5 × 12 mm Major permanent magnet size (lm1 × wm1 × hm) 100 × 8.7 × 20 mm Auxiliary permanent magnet size (lm2 × wm2 × hm) 100 × 6.3 × 20 mm Mover size (l × w × h) 150 × 120 × 20 mm Mover mass 2.1 kg Number of turns N 100 -

The amplitude of three phases current is 2 A. The motor mover can move in the range of 0–6τ (0–90 mm). The simulation result is shown in Figures 10 and 11. Energies 2018, 11, x FOR PEER REVIEW 11 of 15

Energies 2018, 11, x FOR PEER REVIEW 11 of 15 5.1. Finite Element Simulation 5.1. FiniteThe Elementproposed Simulation maglev linear motor is simulated in the finite element software based on the motorThe parameters proposed givenmaglev in linearTable 2.moto r is simulated in the finite element software based on the motor parameters given in Table 2. Table 2. Motor major parameters.

ParameterTable 2. Motor major parameters. Value Unit ParameterPole pitch τ Value15 Unitmm PoleAir pitch gap τδ 151 mmmm Winding size (lc × wc × hc) 100 × 5 × 12 mm Air gap δ 1 mm Major permanent magnet size (lm1 × wm1 × hm) 100 × 8.7 × 20 mm Winding size (lc × wc × hc) 100 × 5 × 12 mm Auxiliary permanent magnet size (lm2 × wm2 × hm) 100 × 6.3 × 20 mm Major permanent magnet size (lm1 × wm1 × hm) 100 × 8.7 × 20 mm Mover size (l × w × h) 150 × 120 × 20 mm Auxiliary permanent magnet size (lm2 × wm2 × hm) 100 × 6.3 × 20 mm MoverMover size (l mass× w × h) 150 × 120 2.1 × 20 mm kg Number of turns N 100 - Mover mass 2.1 kg Number of turns N 100 - The amplitude of three phases current is 2 A. The motor mover can move in the range of 0–6τ (0–90The mm). amplitude The simulation of three resultphases is current shown isin 2Figures A. The 10 motor and 11.mover can move in the range of 0–6τ Energies(0–90 mm).2018, 11The, 556 simulation result is shown in Figures 10 and 11. 11 of 15 1.5 TIyI TIIyII 1.5 1 TIyI TIIyII 1 0.5

0.5 0

0 -0.5

-0.5 -1 0 1 2 3 4 5 6 -1 Displacement/τ 0 1 2 3 4 5 6 Displacement/τ Figure 10. Additional torque generated by the unit I and the unit II. Figure 10. Additional torque generated by the unit I and the unit II.

40 0.5

40 0.5 0.48 20 Fx Fz 0.48 20 Fx ·m 0.46 /N

0 Fz y T ·m 0.46 0.44 /N

0 y

-20 T 0.44 0.42

-20 0.42 -40 0.4 0 1 2 3 4 5 6 0 1 2 3 4 5 6 -40 0.4 0 1 2 3 4 5 6 0 1 2 3 4 5 6

(a) (b) (b) FigureFigure 11. 11.Electromagnetic Electromagnetic(a) force force and and torque: torque: (a) ( electromagnetica) electromagnetic thrust thrust force forceFx and Fx and levitation levitation force forceFz;

(bF)z; torque (b) torqueTy. Ty. Figure 11. Electromagnetic force and torque: (a) electromagnetic thrust force Fx and levitation force

Fz; (b) torque Ty. FigureFigure 10 10 shows shows the the electromagnetic electromagnetic torques torques of of the the two two winding winding units. units. The The blue blue line line represents represents thethe electromagnetic Figureelectromagnetic 10 shows torque torquethe electromagnetic ofof windingwinding unitunit torques II whilewhile of thethe redtwo line linewinding represents represents units. that thatThe forblue for winding winding line represents unit unit II. II. It theItcan can electromagnetic be be seen seen from from Figure torque Figure 10 of10 thatwinding that the the electromagnetic unit electromagnetic I while the torquered torque line is represents is dependent dependent that on on forthe the winding mover mover position unit position II. It in caninx-axis,x -axis,be seen moreover moreover from Figure they they are10 are thatin in opposite oppositethe electromagnetic directions directions.. Figuretorque Figure is11 dependentshowsshows thethe on simulationsimulation the mover resultsresults position ofof the inthe proposedx-axis,proposed moreover new-structure new-structure they are maglev maglev in opposite linear linear motor motordirections with with.d -axisFigured-axis and and11q -axisshowsq-axis currents currentsthe simulation fed. fed. It It can can results be be seen seen of from fromthe proposedFigure 11 anew-structure that the thrust maglev force andlinear levitation motor with force d-axis generated and q-axis bylinear currents motor fed. areIt can the be sum seen of from two winding units. Fx and Fz are kept constant when q-axis and d-axis currents are kept constant. Figure 11b shows that the mover position dependent periodic part of the additional torque has been offset with a constant part left and it ﬂuctuates around 0.455 N·m. The torque of 0.455 N·m reﬂects the last part in Equation (36). This constant part is because the thrust force Fx does not go in line with the center of the mass. Its height can be affected by the mover load shape and its mass. Nevertheless, it can be further offset by adjusting the levitation forces in the two sets of winding units, which is explored in next section.

5.2. Position Closed Loop Veriﬁcation

The control system is shown in Figure 12. xr, zr and θr are the given references in the three DOF. They are compared with the practical values, after which the proportional-integral-derivative (PID) controller is used to regulate the needed force. Currents id1, id2, iq1 and iq2 are then calculated from the regulated signals through the force distribution unit. The dq currents can be then further transformed into three-phase currents through the coordinate transformation. Finally, the real three-phase currents can be given to the maglev linear motor through the current source inverter. The controller parameters are shown in Table3. Energies 2018, 11, x FOR PEER REVIEW 12 of 15

Figure 11a that the thrust force and levitation force generated by linear motor are the sum of two winding units. Fx and Fz are kept constant when q-axis and d-axis currents are kept constant. Figure 11b shows that the mover position dependent periodic part of the additional torque has been offset with a constant part left and it fluctuates around 0.455 N·m. The torque of 0.455 N·m reflects the last part in Equation (36). This constant part is because the thrust force Fx does not go in line with the center of the mass. Its height can be affected by the mover load shape and its mass. Nevertheless, it can be further offset by adjusting the levitation forces in the two sets of winding units, which is explored in next section.

5.2. Position Closed Loop Verification

Table 3. Proportional-integral-derivative (PID) controller parameters.

Parameter Controller I Controller II Controller III Kp 98 98 223 Ki 0.1 0.1 0.07 Kd 1.5 1.5 3.6

The forces and torque in the three DOF are shown as: =+ FFFxx12 x  =+ FFFzz12 z (37)  =−() TLFFyzz21 Energies 2018, 11, 556 12 of 15 where, L is the equivalent force arm between the winding unit and the mover center of mass.

Figure 12. Control system structure.

The control systemTable 3.forProportional-integral-derivative the proposed maglev linear (PID) motor controller is built parameters. in MATLAB/Simulink. The simulation time is set to be 0.2 s. At 0 s, the given reference in z-direction is 1 mm and at 0.05 s the given reference in x-directionParameter step rises Controller to 0.1 m. I TheController rotation angle II θ Controller is kept to IIIbe 0. The simulation result is shown in FigureK p13. 98 98 223 Ki 0.1 0.1 0.07 K 1.5 1.5 3.6 d

The forces and torque in the three DOF are shown as:  F = F + F  x x1 x2 Fz = Fz1 + Fz2 (37)   Ty = L(Fz2 − Fz1) where, L is the equivalent force arm between the winding unit and the mover center of mass. The control system for the proposed maglev linear motor is built in MATLAB/Simulink. The simulation time is set to be 0.2 s. At 0 s, the given reference in z-direction is 1 mm and at

0.05Energies s the given2018, 11, referencex FOR PEER REVIEW in x-direction step rises to 0.1 m. The rotation angle θ is kept13 of to15 be 0. The simulation result is shown in Figure 13.

0.1 00 /N /m .505 x 0 x F

0 00 0 0.04 0.08 0.12 0.16 0.2 0 0.04 0.08 0.12 0.16 0.2 -3 Time/s 4 xx10 10 10 00 /N /m 5 z z F

0 0 0 0.04 0.08 0.12 0.16 0.2 0 0.04 0.08 0.12 0.16 0.2 -6 Time/s 6 x x1010 5 0 ·m /N /rad 0 y 0 θ T

-5 0 0 0.04 0.08 0.12 0.16 0.2 0 0.04 0.08 0.12 0.16 0.2 Time/s

(a) (b)

FigureFigure 13. Three13. Three degrees-of-freedom degrees-of-freedom (DOF)(DOF) positionposition close-loop close-loop simulation simulation result: result: (a) position (a) position and and angle;angle; (b) electromagnetic (b) electromagnetic force force and and torque. torque.

It has been seen from Figure 13a that the control system for the proposed maglev linear motor based on dq decoupling strategy has good positioning accuracy in the three DOF. When the linear motor moves in the direction of x-axis, the rotation angle θ fluctuates between −2.89 μrad and 2.63 μrad with a very small torque change. 300 N is the largest thrust that can be generated by the motor. The simulation result shows that the mover position dependent periodic part can be offset with the two sets of winding units, moreover the constant part, not dependent on the mover position, can be further offset by adjusting the levitation forces in the two sets of winding units through position close-loop control.

6. Conclusions This paper has proposed a new-structure maglev linear motor that has thee DOF. The main innovation of this motor is to offset the mover position dependent periodic part of the additional torque caused in the typical dq decoupling control method through designing two winding units with the relative phase difference that can make the torques caused by d-axis current and q-axis current have the opposite direction and thus can offset each other. The conventional dq decoupling control strategy and three-phase motor driver can be used for the control system adopting this new-structure maglev linear motor. The finite element simulation has been done to prove that the proposed motor can offset the mover position dependent periodic part of additional. Moreover, the control system is designed to control it with the conventional dq decoupling control strategy. It has been found that the constant part, not dependent on the mover position, can be further offset by adjusting the levitation forces in the two sets of winding units through position close-loop control. The thrust and torque ripples are very small, which guarantees high positioning precision.

Author Contributions: Design of the research scheme and research supervision: Feng Xing, Baoquan Kou and Tiecheng Wang. Performing of the research and writing of the manuscript: Lu Zhang, Chaoning Zhang.

Conflicts of Interest: The authors declare no conflict of interest.

Energies 2018, 11, 556 13 of 15

It has been seen from Figure 13a that the control system for the proposed maglev linear motor based on dq decoupling strategy has good positioning accuracy in the three DOF. When the linear motor moves in the direction of x-axis, the rotation angle θ ﬂuctuates between −2.89 µrad and 2.63 µrad with a very small torque change. 300 N is the largest thrust that can be generated by the motor. The simulation result shows that the mover position dependent periodic part can be offset with the two sets of winding units, moreover the constant part, not dependent on the mover position, can be further offset by adjusting the levitation forces in the two sets of winding units through position close-loop control.

6. Conclusions This paper has proposed a new-structure maglev linear motor that has thee DOF. The main innovation of this motor is to offset the mover position dependent periodic part of the additional torque caused in the typical dq decoupling control method through designing two winding units with the relative phase difference that can make the torques caused by d-axis current and q-axis current have the opposite direction and thus can offset each other. The conventional dq decoupling control strategy and three-phase motor driver can be used for the control system adopting this new-structure maglev linear motor. The ﬁnite element simulation has been done to prove that the proposed motor can offset the mover position dependent periodic part of additional. Moreover, the control system is designed to control it with the conventional dq decoupling control strategy. It has been found that the constant part, not dependent on the mover position, can be further offset by adjusting the levitation forces in the two sets of winding units through position close-loop control. The thrust and torque ripples are very small, which guarantees high positioning precision.

Acknowledgments: This work was supported in part by the State Key Program of National Natural Science of China under Grant (51537002), in part by the National Natural Science Foundation of China (51607044), in part by the China Postdoctoral Science Foundation Funded Project (2015M80262), in part by “the Fundamental Research Funds for the Central Universities” (Grant No. HIT. NSRIF. 2017010), and in part by the National Science and Technology Major Project (2009ZX02207-001). Author Contributions: Design of the research scheme and research supervision: Feng Xing, Baoquan Kou and Tiecheng Wang. Performing of the research and writing of the manuscript: Lu Zhang, Chaoning Zhang. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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