Magnetic Flux Distribution of Linear Machines with

sensors Article

Magnetic Flux Distribution of Linear Machines with Novel Three-Dimensional Hybrid Magnet Arrays Nan Yao, Liang Yan *, Tianyi Wang and Shaoping Wang School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China; [email protected] (N.Y.); [email protected] (T.W.); [email protected] (S.W.) * Correspondence: [email protected]; Tel.: +86-10-8233-9890 Received: 18 September 2017; Accepted: 9 November 2017; Published: 18 November 2017

Abstract: The objective of this paper is to propose a novel tubular linear machine with hybrid permanent magnet arrays and multiple movers, which could be employed for either actuation or sensing technology. The hybrid magnet array produces flux distribution on both sides of windings, and thus helps to increase the signal strength in the windings. The multiple movers are important for airspace technology, because they can improve the system’s redundancy and reliability. The proposed design concept is presented, and the governing equations are obtained based on source free property and Maxwell equations. The magnetic field distribution in the linear machine is thus analytically formulated by using Bessel functions and harmonic expansion of magnetization vector. Numerical simulation is then conducted to validate the analytical solutions of the magnetic flux field. It is proved that the analytical model agrees with the numerical results well. Therefore, it can be utilized for the formulation of signal or force output subsequently, depending on its particular implementation. Keywords: linear machine; hybrid magnet arrays; multiple movers; magnet flux; magnetic vector potential; FEM

1. Introduction Linear machine generates translations directly without rotation-to-transmission conversion mechanisms, and thus achieves compact structure, high efficiency and good dynamic performance. It has wide applications in aerospace industries [1,2], transportation [3,4], high-precision manufacture [5,6], energy harvesting [7,8], robotics [9,10], and medical operations. Linear machine can be used as either generator or actuator to accomplish different works. Magnetic flux density is extremely important for the design of linear machines. High flux density helps to increase force output or signal strength depending on particular tasks. The conventional method is to increase the magnet size and thus the flux density. However, the large size or mass of the system is apparently not preferred for most applications, especially in aerospace technology. Alternatively, various magnet patterns have been proposed by researchers to enhance the magnetic flux density inside the linear machines without changing the system size significantly. Buren et al. [11] presents one linear electromagnetic generator with three axially magnetized PM poles. The relative motion between stator and translator leads to a varying magnetic flux through the armature windings, and the output power proportional to the rate of flux change is induced. Several axially magnetized disc-shaped magnets separated by soft-magnetic spacers are mounted on the translator. The magnetization directions of neighboring magnets are opposite, and the spacers act as flux concentrators and form the magnet poles. The design may lead to high flux leakage, and decrease the force output. An improved version of axial magnetization is proposed by Kim et al. [12] in a tubular linear brushless machine. Cylindrical permanent magnets are placed in an NS-NS-SN-SN fashion with spacers between pairs. The magnets are fixed within a freely sliding brass tube on the

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mover. Coils are configured in three phases to interact with the magnet and cause translation of the mover. This structure and magnetization pattern help to increase flux density near the like pole regions. However, the aluminum tubes used to separate magnets decrease the volume efficiency of the mover. Huang et al. [13] employed convex pole instead of rectangular pole in the axial magnetization pattern to reduce the magnetic saturation and flux leakage of tubular machines. High thrust and less permanent magnet dosage were achieved in the research. However, the magnetization and assembly are relatively challenging. Wang et al. [14] analyzed a tubular linear machine with surface-mounted radially magnetized magnets and came to the conclusion that this pattern can reduce the magnet material and the cost. Special customized fixture may be required for the magnetization. Nirei et al. [15] developed a moving-coil linear machine with typical radial magnetization. There were 16 pieces of permanent magnets mounted along the inner surface of the outer yoke. Due to the same polarization, the multiple magnets are essentially equivalent to one magnet ring with radial magnetization. However, the single-ring pattern unavoidably causes high flux leakage and reduces the force output. Baloch et al. [16] designed a tubular vernier machine adopting dual stator configurations. Multiple PM poles are mounted on the mover along the axial direction. The poles are alternatively magnetized radially, which helps to achieve high force output and large working range compared with the design in [15]. Wang et al. [17] presented a novel linear electromagnetic machine based on the concept of magnetic screw-nut. The radially magnetized permanent magnets are helically disposed on the nut and the screw to produce force and torque simultaneously. However, the two motions are coupled, and cannot be controlled independently. The Halbach array was firstly proposed a few decades ago [18], and implemented into design of linear machines recently [19]. It can enhance the flux density on the one side of PM, and reduce the flux leakage on the other side in a certain degree. Yan et al. [20] proposed a high dynamic performance linear machine with improved Halbach array. A combination of two axial magnets and one radial magnet is utilized in the design of magnet topology. Compared with commonly used quasi-Halbach array, it enhances the self-shielding effect without the utilization of back irons. Therefore, the mover mass is reduced significantly, and thus the dynamic response is increased greatly. Izzeldin [21] analyzed the magnet patterns of three moving-magnet linear machines, including rectangular, trapezoidal and T-shape magnet arrays with quasi-Halbach magnetization. It is shown that the T-shape and trapezoidal magnet arrays achieve better flux linkage than the rectangular one. However, the fabrication and magnetization of magnets are relatively challenging. Yan et al. [22,23] extended conventional magnet array into three-dimensional topology, and proposed the novel dual Halbach array. It helps to improve the axial force output and depress the radial vibration. Magnet patterns have also been employed to improve the performance of flux-switching machines. For example, Zhang et al. [24] proposed a yokeless linear machine topology with double magnets per mover module to enhance thrust density but efficiency of the new machine is marginally higher than that of the conventional yokeless machine. The comparison of magnet patterns is summarized in Table 1. The objective of this study is to propose one novel linear permanent magnet machine with three-layer hybrid magnets and two-layer windings. The employment of the proposed structure offers the following advantages. The combination of conventional Halbach array and alternatively magnetized radial poles helps to enhance the magnetic flux density in the radial direction, and thus increase the force output of the linear machine. In addition, the utilization of multiple movers in the linear machine increases the redundancy and reliability of the system, which is extremely important for the aerospace technology. Furthermore, the winding mass is reduced, and thus the dynamic response could be improved significantly for moving coil designs. The design concept is presented, Laplace’s and Poisson’s equations are obtained from source free property of magnetic field and Maxwell equations. The general solution of magnetic vector potential is represented with Bessel functions. The flux distribution inside the machine is then formulated analytically by utilizing harmonic expansion of magnetization vectors and boundary conditions. Numerical computation is conducted to validate the analytical model and on the proposed design.

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Table 1. Comparison of magnetization patterns of linear machines. Table 1. Comparison of magnetization patterns of linear machines.

MagnetPatterns Patterns Magnet

Axial Axial magnetization magnetization

Radial magnetization Radial

magnetization

Halbach array and its extension Halbach array and

its extension

2. Concept Design

Particular ParticularDesigns Designs Three magnets with axial Three magnets with magnetization axial magnetization NS-NS-SN-SNfashion fashion NS-NS-SN-SN withspacers spacers with

Features Features Simple structure, easy control; high Simple structure, easy control; high fluxflux leakage, leakage, force output lowlow force output High flux density nearthe thelike likepole poleregions, regions, High flux density near improve force output;low lowvolume volumeefficiency efficiency improve force output; Low magnetic saturation,low lowflux fluxleakage; leakage; Low magnetic saturation, Convexmagnet magnetpole pole Convex difficult fabrication and assembly difficult fabrication and assembly Surface-mounted Less magnet Surface-mounted radial Less magnet material, material,high highdynamics; dynamics;lower lowerflux radial magnetization density, special magnetization flux density, specialmagnetization magnetizationfixture fixture Single ring magnet with No special requirement of magnetization fixture; Single ring magnet with No special requirement of magnetization multiple sectors high flux leakage, low force output multiple sectors fixture; leakage, force output Multiple magnets in Highhigh forceflux output, largelow working range; Multiple magnets in axial High force output, large working range; axial direction magnetization and assembly challenge direction magnetization and torque assembly challenge Produce force and simultaneously; Magnetic screw-nut coupled in two directions Produce force motions and torque simultaneously; Magnetic screw-nut motions inside twoof directions Highcoupled flux density on one PM; flux leakage Conventional Halbach array Conventional Halbach High fluxalthough density on side of reduced. PM; flux exists it isone relatively Improved compounded Goodexists shielding effect,itlow mover mass, high array leakage although is relatively reduced. Halbach array dynamic response; difficult magnet assembly Improved compounded Good shielding effect, low mover mass, high T-shaped and High magnetic fluxdifficult linkage;magnet Non-regular shape Halbach array dynamic response; assembly trapezoidal magnet of magnets, difficult for fabrication T-shaped and trapezoidal High magnetic flux linkage; Non-regular High axial force output, low radial vibration; shapeenhancement of magnets, is difficult for fabrication Dual magnet Halbach array Force not significant, especially High axial force output, low radial for large size machine. vibration; Dual Halbach array Force enhancement is not significant, especially for large size machine.

2.The Concept Designstructure of the proposed tubular linear machine is illustrated in Figure 1a. schematic

It consists ofschematic two layersstructure of windings three layers of linear magnet poles. is The windings mounted The of theand proposed tubular machine illustrated inare Figure 1a. It on the consists mover, and the magnet poles are on the stator. The relative motion between the stator and of two layers of windings and three layers of magnet poles. The windings are mountedmover on generates voltages onmagnet the windings the motion speed. It is worth pointing out that the mover, and the poles areproportional on the stator. to The relative motion between the stator and mover voltages on the could windings proportional to the motion speed. It voltage is worth signal, pointing that the generates two layers of windings be connected together to enhance the orout separated the two layers of windings could be connected together to enhance the voltage signal, or separated to detect motions of individual rigid bodies. The redundancy property improves the reliability of detectmachines. motions ofThe individual rigid bodies. The redundancy propertythe improves the reliabilityThe of the the to linear multiple-winding structure also benefits heat dissipation. same linear machines. The multiple-winding structure also benefits the heat dissipation. The structure can be utilized for design of actuators in aircrafts. Magnetization pattern of thesame hybrid structure canisbeillustrated utilized for actuators in aircrafts. Magnetization pattern of theare hybrid magnet arrays in design Figureof1b. One layer of radially magnetized PM poles placed magnet arrays is illustrated in Figure 1b. One layer of radially magnetized PM poles are placed in in between the two layers of Halbach arrays. All radially magnetized PM poles in the three layers between the two layers of Halbach arrays. All radially magnetized PM poles in the three layers have have the same polarization while the axial PMs of two Halbach layers are magnetized with opposite the same polarization while the axial PMs of two Halbach layers are magnetized with opposite directions. The proposed hybrid arrangement can increase the flux density in the radial direction directions. The proposed hybrid arrangement can increase the flux density in the radial direction greatly. Therefore, thethe axial force dueto tothe thecross crossproduct product flux density greatly. Therefore, axial forcegeneration generationisisimproved improved due of of flux density andand current input. current input.

(a)

(b)

Figure 1. Tubular linear machine with hybrid magnet arrays: (a) Schematic machine structure; (b)

Figure 1. Tubular linear machine with hybrid magnet arrays: (a) Schematic machine structure; Magnetization and flux. (b) Magnetization and flux.

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3. Governing Equations In this section, the flux field in the proposed linear machine is formulated analytically. The obtained mathematical model could be employed for subsequent design optimization, modeling of thrust or output current, and high precision motion control. 3.1. Assumptions (a) (b) (c) (d)

The machine has a periodic magnetic structure along axial direction z. The axial length is infinite, and thus the end effects can be ignored. The distribution of magnetic field is axially symmetric. The magnetic permeability of back iron is infinite.

3.2. Characterization of Materials and Governing Equations To formulate the magnetic field mathematically, the space in linear machine under study is divided into two regions based on their magnetic characteristics. The air space is denoted as Region I. The PM pole filled with rare-earth magnetic material is denoted as Region II (Figure 2). According to different material properties, it is easy to obtain equations relating magnetic field intensity H(A/m) to flux density B(T) for these two regions B = µ0 H,

B = µ0 µr H + µ0 M,

(1)

  where µ0 is the permeability of vacuum with a value of 4π × 10−7 N/A2 , dimensionless quantity µr is the relative permeability of permanent magnets, and M = Brem /µ0 (A/m) is the residual magnetization vector. B is equal to the curl of magnetic vector potential, i.e.,

∇ × A = B.

(2)

The Coulomb gauge is used as a constraint to uniquely determine the divergence of a vector. As a result, we have Laplace’s Equation, or governing equation for region I

∇2 A = 0

(3)

∇2 A = −µ0 ∇ × M.

(4)

and Poisson’s Equation, for region II

4. Formulation of Magnetic Field 4.1. General Solution to Magnetic Potential in Region I Based on the symmetric distribution of flux density in tubular linear machines, the Laplace’s equation, i.e., Equation (3) in cylindrical coordinates can be expanded and simplified as ∂ ∂r



1 ∂(rAθ ) r ∂r



+

∂2 A θ = 0. ∂z2

(5)

Since Aθ is only a function of independent variables, r and z, it can be represented with separation principle of variables as A θ = R (r ) Z ( z ). (6) Substituting Equation (6) into (5) gives 1 ∂2 R (r ) 1 ∂R(r ) 1 1 ∂2 Z ( z ) + − 2+ = 0. 2 R(r ) ∂r R(r )r ∂r Z (z) ∂z2 r

(7)

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Aθ is determined uniquely based on the structure of tubular linear machines. The magnetic field in tubular linear machines is periodically and symmetrically distributed along z axial, thus, the solution to magnetic potential in Region I is Aθ =

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∑ ∞n=1 [an I1 (mn r) + bn K1 (mn r)] sin(mn z),

(8) 5 of 14 (2n−1)π

where I1 and K1 are the modified Bessel functions of the first and second kind [25], mn = , n is (2𝑛−1)𝜋 τ where 𝐼1 and 𝐾1 are the modified Bessel functions of the first and second kind [25], 𝑚𝑛 = p ,n a positive integer. 𝜏𝑝 is a positive integer. 4.2. General Solution to Magnetic Potential in Region II

4.2. General Magnetic in Region II For theSolution specific to structure ofPotential tubular linear machine under study in this paper, Poisson’s Equation in cylindrical coordinates can of betubular expanded as machine under study in this paper, Poisson’s Equation For the specific structure linear

in cylindrical coordinates canbe expanded  as 2   ∂ Aθ ∂ 1 ∂(rAθ ) ∂Mr ∂Mz , − 𝜕 1 𝜕(𝑟𝐴 )+ 𝜕2 𝐴 = − µ0 𝜕𝑀 𝜕𝑀 ∂r r (∂r 𝜃 ) +∂z22𝜃 = −𝜇0 ( 𝑟∂z− 𝑧),∂r 𝜕𝑟 𝑟

𝜕𝑟

𝜕𝑧

𝜕𝑧

(9) (9)

𝜕𝑟

the radial radialand andaxial axial components of magnetization vector M, respectively. where M 𝑀𝑟r and M 𝑀𝑧z are the components of magnetization vector 𝐌, respectively. The The homogeneous solution to Poisson’s Equation is exactly the same as the general solution to homogeneous solution to Poisson’s Equation is exactly the same as the general solution to Laplace’s Laplace’s equation. deriving the particular solution of Equation (9) will determine the equation. Therefore, Therefore, deriving the particular solution of Equation (9) will determine the general general solution of Poisson’s equation. To formulate the particular solution, weto need to substitute the solution of Poisson’s equation. To formulate the particular solution, we need substitute the right right side of Equation (9) with its harmonic expansion. As shown in Figure 1, the two layers of Halbach side of Equation (9) with its harmonic expansion. As shown in Figure 1, the two layers of Halbach composed of axial and radial magnets whereas the arrays in the proposed novel linear machine are composed middle layer only has radially magnetized magnets.

(a)

(b)

Figure 2. Components Figure 2. Components ofof magnetization magnetization vector vector of of Halbach Halbach layers: layers: (a) (a) Radial Radial component; component; (b) Axial component. (b) Axial component.

As illustrated in Figure 2a, 𝑀𝑟 is a non-continuous periodic even function with a period of 2𝜏𝑝 . As illustrated in Figure 2a, Mr is a non-continuous periodic even function with a period of 2τp . Its harmonic expansion is Its harmonic expansion is 4𝐵

(2𝑛−1)𝜋𝜏

𝑟𝑒𝑚 𝑟  𝑛 𝑧), 𝑀𝑟 = ∑∞ ( ) cos(𝑚 (10) 𝑛=1 (2𝑛−1)𝜋𝜇 sin  2𝜏𝑝− 1) πτr 4Brem 0 (2n ∞ Mr = ∑ n=1 sin cos(mn z), (10) − 1)of πµradial 2τp and 𝐵 0 where 𝜏𝑝 is the pole pitch, 𝜏𝑟 is the(2n width magnets, 𝑟𝑒𝑚 is the remanence. As shown in Figure 2b, 𝑀𝑧 is a non-continuous periodic odd function with a period of 2𝜏𝑝 . Its harmonic where τp is the pole pitch, τr is the width of radial magnets, and Brem is the remanence. As shown in expansion is is a non-continuous periodic odd function with a period of 2τ . Its harmonic expansion is Figure 2b, M

z

p

4𝐵𝑟𝑒𝑚

(2𝑛−1)𝜋𝜏𝑟

 𝑛 𝑧), 𝑀𝑧 = − ∑∞ cos ( ) sin(𝑚 (11) 𝑛=14B (2𝑛−1)𝜋𝜇 2𝜏𝑝− 1) πτr (2n rem 0 Mz = − ∑ ∞ sin(mn z), (11) cos n =1 2τp (2n − 1)πµ0 where 𝜏𝑧 is the width of axial magnets. The pattern of the middle PM layer is exactly the same as the radial magnets of where τz magnetization is the width of axial magnets. Halbach layers (Figure 3). Therefore, the radial PM component of magnetization vector can bemagnets expressed The magnetization pattern of the middle layer is exactly the same as the radial of with Equation (10). Substituting Equations (10) and (11) into (9) yields the following form of Poisson’s Halbach layers (Figure 3). Therefore, the radial component of magnetization vector can be expressed equation for Halbach arrays and radial PM array, i.e., 𝜕

(

1 𝜕(𝑟𝐴𝜃 )

= ∑∞ 𝑛=1

4𝐵𝑟𝑒𝑚

𝐴 = ∑∞ {[𝑎 𝐼 (𝑚 𝑟) + 𝑏 𝐾 (𝑚 𝑟)]sin(𝑚 𝑧) + 𝑠(𝑟, 𝑧)}.

(13)

𝜕𝑧 2

𝜏𝑝

sin (

(2𝑛−1)𝜋𝜏𝑟

(12)

𝜕𝑟

)+

𝜕2 𝐴𝜃

) sin(𝑚𝑛 𝑧).

𝜕𝑟 𝑟

2𝜏𝑝

Thus, the general solution to Poisson’s equation is

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with Equation (10). Substituting Equations (10) and (11) into (9) yields the following form of Poisson’s equation for Halbach arrays and radial PM array, i.e., ∂ ∂r



1 ∂(rAθ ) r ∂r



∂2 A θ + = ∂z2

∑

∞ 4Brem n =1 τp

(2n − 1)πτr sin 2τp 

 sin(mn z).

(12)

Thus, the general solution to Poisson’s equation is Aθ =

∑ ∞n=1 {[an I1 (mn r) + bn K1 (mn r)] sin(mn z) + s(r, z)}.

(13)

The homogeneous part of Equation (13) is the same as Equation (8) and the particular part s(r, z) can be derived by using separation of variables as Sensors 2017, 17, 2662

s(r, z) = L1 (mn r )

6 of 14

πPn sin(mn z), 2mn 2

(14)

𝜋𝑃

𝑠(𝑟, 𝑧)functions = 𝐿1 (𝑚𝑛 𝑟)[26].𝑛2Therefore, sin(𝑚𝑛 𝑧), the general solution to Poisson’s (14) where L1 (mn r ) are the modified Struve 2𝑚𝑛 equation where 𝐿 is (𝑚 𝑟)are the modified Struve functions [26]. Therefore, the general solution to Poisson’s 1

equation is

𝑛

  πPn sin m z Aθ = ∑ ∞ a I m r + b K m r sin m z + L m r . [ ( ) ( )] ( ) ( ) ( ) n 1 n n 1 n n n n 1 n =1 𝜋𝑃𝑛 2 2m n sin(𝑚 𝑧)}. 𝐴𝜃 = ∑∞ 𝑛 𝑛=1 {[𝑎𝑛 𝐼1 (𝑚𝑛 𝑟) + 𝑏𝑛 𝐾1 (𝑚𝑛 𝑟)]sin(𝑚𝑛 𝑧) + 𝐿1 (𝑚𝑛 𝑟) 2 2𝑚𝑛

(15) (15)

Figure Figure 3. 3. Component Component of of magnetization magnetization vector vector of of middle middle layer. layer.

4.3. Analytical Model of Flux Density 4.3. Analytical Model of Flux Density From Equations (1), (8) and (15), the general solution of flux density is obtained From Equations (1), (8) and (15), the general solution of flux density is obtained 𝜕𝐴

𝑖 𝑖 𝑖 𝐵𝑟1 = − 𝜃 = − ∑∞ 𝑛 𝑧) 𝑛=1 𝑚𝑛 [𝑎 h1𝑛 𝐼1 (𝑚𝑛 𝑟) + 𝑏1𝑛 𝐾1 (𝑚𝑛 𝑟)]cos(𝑚 i ∂A𝜕𝑧θ i ∞ i i Br1 = − = − ∑∞ m a I m r + b K m r cos m ( ) ( ) ( n n n z) n=1 n 1n 1 1n 1 ∂z 𝜕𝐴 𝐴 𝜃 𝜃 𝑖 𝑖 𝑖 𝐵𝑧1 = + = ∑ 𝑚𝑛 [𝑎1𝑛 𝐼0 (𝑚𝑛 𝑟) − 𝑏1𝑛 𝐾0 (𝑚𝑛 𝑟)]sin(𝑚𝑛 𝑧) ∞ 𝑟 𝜕𝑟   ∂A A i I ( m r ) − bi K ( m r ) sin( m z ) i = θ + θ 𝑛=1 = m a Bz1 ∑ n 1n 0 n n n 1n 0 r ∂r n =1 𝑖 = 1,2; i = 1, 2; ∞   𝜋𝑃𝑛  ∞ 𝑗j 𝑗 πP j𝑗 n ] cos(𝑚𝑛 𝑧))} ∑m 𝑚𝑛 {[𝑎a2𝑛 𝐼I1 (𝑚 𝑟) + b𝑏j2𝑛K 𝐾1((𝑚 𝑟) + L𝐿1((𝑚 𝑟) 𝑛 𝑛 𝑛 2 B𝐵r2𝑟2==−− ∑ m r + cos m z m r + m r ( )) ( ) ) ) n n n n n 1 2𝑚𝑛2 2n 1 2n 1 2m n n𝑛=1 =1

(16) (16) (17) (17)

(18) (18)

∞ nh i o ∞ 𝜋𝑃n𝑛 πP 𝑗j 𝑗j 𝑗j B mn𝑛z𝑧))} = m𝑛n {[𝑎a2𝑛 r )− −𝑏b2𝑛 K00(𝑚 r) + + 𝐿L00(𝑚 r ) 2m ( )) (m𝑛n𝑟) (m𝑛n𝑟) (m𝑛n𝑟) ∑𝑚 𝐵𝑧2 sin(𝑚 2 ]sin z2 = ∑ 2n𝐼0I0(𝑚 2n𝐾 (19) 2𝑚n𝑛 2 n =1 (19) 𝑛=1 j = 1, 2, 3. 𝑗 = 1,2,3. i and Bi represent the radial and axial magnetic field in the air region, respectively, while B j and Br1 z1𝑖 r2 𝑗 𝑖 𝐵𝑟1 j and 𝐵𝑧1 represent the radial and axial magnetic field in the air region, respectively, while 𝐵𝑟2 Bz2 are𝑗 in magnet regions. The upper script, i = 1, 2, represents two sections of air region, while j = 1, 2, and 𝐵𝑧2 are in magnet regions. The upper script, i = 1, 2, represents two sections of air region, while 3 represents internal, middle and external PMs respectively. j = 1, 2, 3 represents internal, middle and external PMs respectively.

4.4. Boundary Conditions 1 2 1 There are 10 unknown coefficients in the analytical model of flux density, i.e., 𝑎1𝑛 ，𝑎1𝑛 ，𝑏1𝑛 ， 3 and 𝑏2𝑛 . As shown in Figure 4, based on the continuity of flux density

3 2 1 1 2 2 𝑏1𝑛 ，𝑎2𝑛 ，𝑏2𝑛 ，𝑎2𝑛 ，𝑏2𝑛 ，𝑎2𝑛

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4.4. Boundary Conditions 1 , b2 , There are 10 unknown coefficients in the analytical model of flux density, i.e., a11n , a21n , b1n 1n 1 2 2 3 3 b2n , a2n , b2n , a2n and b2n . As shown in Figure 4, based on the continuity of flux density and Ampere circuital theorem, the perpendicular component of flux density is continuous in two adjacent media and the tangential component of field intensity is continuous at the boundary of two media when the surface current is zero [27,28]. Ten boundary conditions are employed to determine all the unknowns, i.e., 1 1 1 1 1 3 = 0; Hz2 = Hz1 ; Br2 = Br1 ; Hz2 = 0; Hz2 r = R6 r = R2 r = R2 r = R2 r = R2 r = R1 Sensors 2017, 17, 2662 7 of 14 2 2662 1 2 1 2 2 2 2 Sensors 2017,H17, 7 (20) of 14 z2 r = R3 = Hz1 r = R3 ; Br2 r = R3 = Br1 r = R3 ; Hz2 r = R4 = Hz1 r = R4 ; Br2 r = R4 = Br1 r = R4 ; where 3 2 3 2 Hz2 = Hz1 ; Br2 = Br1 , r = R5 r = R5 r = R5 r = R5 where 𝑖 𝑖 𝑖 ∑∞ 𝑚𝑛 [𝑎1𝑛 𝐼 (𝑚 𝑟) − 𝑏1𝑛 𝐾0 (𝑚𝑛 𝑟)]sin(𝑚𝑛 𝑧) 𝐵𝑧1 𝑖 𝑖 0 (𝑚𝑛 𝑖 𝑖 = ∑𝑛=1 ∞ where 𝐻𝑧1 = 𝑚 [𝑎 𝐼 𝑟) − 𝑏 𝐾0 (𝑚𝑛 𝑟)]sin(𝑚𝑛 𝑧) 𝐵 𝑛 0 𝑛 𝑛=1 1𝑛 1𝑛 𝑧1 𝑖 ∞ i i i 𝜇 𝜇 mn [ a1n I0 (mn r )−0b1n K0 (mn r )] sin(mn z) 𝐻𝑧1 = i 0 =Bz1 ∑ Hz1 𝜇0= µ0 = n=1 𝜇µ00 𝑗 nh 𝑗 i 𝜋𝑃𝑛 ] sin(𝑚 o 𝑗 ∑∞ 𝑟) − 𝑏2𝑛 (𝑚 𝑟) L+(m 𝐿0 (𝑚 j ∞ 2𝑛 𝑛n𝑟) 2𝑚 𝑛 𝑧))} 𝑛=1 𝑚𝑛∑{[𝑎 𝜋𝑃(𝑛m 2 )) 𝑗 n𝐼0 (𝑚 𝑗b j 𝐾 j 𝑀𝑧2 ∞ r ) πP K00(𝑚 sin m a2n𝑛I0𝑟) (mn𝑛r )+ (mn−r )− 𝑗 0 𝐿 n (𝑚 𝑗 𝑛 ]n z n = 1 2 2n ∑ (𝑚 j 𝑚 {[𝑎 𝐼 𝑏 𝐾 𝑟) + 𝑟) sin(𝑚 𝑧))} 2m z2 − 𝑀 0 𝑛 0 𝑛 0 𝑛n 2𝑚 2 𝑛M 𝑛=1= 𝑛 2𝑛 2𝑛 𝐻𝑧2 = H − 𝑗 𝑛 (21) µr − 𝜇𝑧2 𝜇µ00 µ𝜇r𝑟 𝐻𝑧2 = z2 𝑟 (21) (21) 𝜇0 𝜇𝑟 𝜇𝑟   ∞ ∞ (2n−1)πτr 4Brem 1 3 Mz2 = − M3 z2 = −∞ ∑ 4𝐵 cos (2𝑛 −2τ1)𝜋𝜏 𝑟 sin( m n z ) −1)πµ (2n𝑟𝑒𝑚 1 p

a12n ,

𝑀𝑧2 = −𝑀𝑧2 = − ∑n=1 4𝐵𝑟𝑒𝑚 0cos ((2𝑛 − 1)𝜋𝜏𝑟 ) sin(𝑚𝑛 𝑧) 3 1 2𝜏𝑝 𝑀𝑧2 = −𝑀𝑧2 = − 𝑛=1 ∑ (2𝑛 − 1)𝜋𝜇0 cos ( ) sin(𝑚𝑛 𝑧) (2𝑛 −M1)𝜋𝜇 2𝜏𝑝 2 =0 0 𝑛=1

z2

2 𝑀𝑧2 =0 2 𝑀𝑧2 =0

n n B1 B1

ΔS ΔS

μ1 > μ 2 μ1 > μ 2

n n

H1 H1

A A

μ1 μ1 μ2 μ2

B B

D D

H2 H2

μ1 > μ 2 μ1 > μ 2

B2 B2

μ1 μ μ12 μ2

C C

Figure intensity. Figure 4. 4. Boundary Boundary conditions conditions of of flux flux density density and and field field intensity. Figure 4. Boundary conditions of flux density and field intensity.

The major structural parameters in the boundary conditions are presented in Figure 5. The major structural parameters parameters in in the the boundary boundary conditions conditions are are presented presented in in Figure Figure5.5.

r r

R7

τr τr

R2 R1 R2 R1

τz τz

R5 R4 R 5 R 3 R4 R3

R6 R 7 R6

z z

τp τp

Figure 5. Major structural parameters of the proposed linear machine. Figure 5. 5. Major parameters of of the the proposed proposed linear linear machine. machine. Figure Major structural structural parameters

Substituting Equations (16)–(18), (19) and (21) into Equation (20) will derive ten expanded Substituting Equations (16)–(18), (19) and (21) into Equation (20) will derive ten expanded equations describing ten boundary conditions respectively. They are shown as follows: equations describing ten boundary conditions respectively. They are shown as follows: 𝜋𝑃 4𝐵 (2𝑛−1)𝜋𝜏 1 1 𝑎2𝑛 𝐼 (𝑚 𝑅 ) − 𝑏2𝑛 𝐾0 (𝑚𝑛 𝑅1 ) = −𝐿0 (𝑚𝑛 𝑅1 ) 𝜋𝑃𝑛𝑛2 − 4𝐵𝑟𝑒𝑚 cos ((2𝑛−1)𝜋𝜏𝑟𝑟 ) , (22.a) 𝑟𝑒𝑚 𝑛 1 0 (𝑚𝑛 1 ) 1 𝑎2𝑛 𝐼0 𝑛 𝑅1 − 𝑏2𝑛 𝐾0 (𝑚𝑛 𝑅1 ) = −𝐿0 (𝑚𝑛 𝑅1 ) 2𝑚𝑛 2 − (2𝑛−1)𝜋𝑚 cos ( 2𝜏𝑝 ) , (22.a) 2𝑚 (2𝑛−1)𝜋𝑚 2𝜏 𝑛

𝜋𝑃

4𝐵

𝑛

𝑝

(2𝑛−1)𝜋𝜏

3 3 𝑎2𝑛 𝐼 (𝑚 𝑅 ) − 𝑏2𝑛 𝐾0 (𝑚𝑛 𝑅6 ) = −𝐿0 (𝑚𝑛 𝑅6 ) 𝜋𝑃𝑛𝑛2 + 4𝐵𝑟𝑒𝑚 cos ((2𝑛−1)𝜋𝜏𝑟𝑟 ) , 𝑟𝑒𝑚 𝑛 3 0 (𝑚𝑛 6 ) 3 𝑎2𝑛 𝐼0 𝑛 𝑅6 − 𝑏2𝑛 𝐾0 (𝑚𝑛 𝑅6 ) = −𝐿0 (𝑚𝑛 𝑅6 ) 2𝑚𝑛 + (2𝑛−1)𝜋𝑚 cos ( 2𝜏𝑝 ) ,

(22.b) (22.b)

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Substituting Equations (16)–(18), (19) and (21) into Equation (20) will derive ten expanded equations describing ten boundary conditions respectively. They are shown as follows: 1 a12n I0 (mn R1 ) − b2n K0 ( m n R 1 )

  4Brem πPn (2n − 1)πτr − cos , = − L0 ( m n R1 ) 2τp (2n − 1)πmn 2mn 2

(22a)

3 K0 ( m n R 6 ) a32n I0 (mn R6 ) − b2n

  πPn 4Brem (2n − 1)πτr = − L0 ( m n R6 ) + cos , 2τp (2n − 1)πmn 2mn 2

(22b)

1 K ( m R ) − a1 I ( m R ) + b1 K ( m R ) µr a11n I0 (mn R2 ) − µr b1n n 2 n 2 n 2 0 2n 0 2n 0   (2n−1)πτr πPn 4Brem = L0 (mn R2 ) 2m , 2 + (2n −1) πm cos 2τp

(22c)

n

n

1 1 K1 ( m n R 2 ) = − L 1 ( m n R 2 ) K1 (mn R2 ) − a11n I1 (mn R2 ) − b1n a12n I1 (mn R2 ) + b2n

πPn , 2mn 2

1 2 µr a11n I0 (mn R3 ) − µr b1n K0 (mn R3 ) − a22n I0 (mn R3 ) + b2n K0 ( m n R 3 ) = L 0 ( m n R 3 )

πPn , 2mn 2

1 2 K1 ( m n R 3 ) = − L 1 ( m n R 3 ) K1 (mn R3 ) − a11n I1 (mn R3 ) − b1n a22n I1 (mn R3 ) + b2n 2 2 µr a21n I0 (mn R4 ) − µr b1n K0 (mn R4 ) − a22n I0 (mn R4 ) + b2n K0 ( m n R 4 ) = L 0 ( m n R 4 )

(22d) (22e)

πPn , 2mn 2

πPn , 2mn 2

2 2 K1 (mn R4 ) − a21n I1 (mn R4 ) − b1n K1 ( m n R 4 ) = − L 1 ( m n R 4 ) a22n I1 (mn R4 ) + b2n

(22f) (22g)

πPn , 2mn 2

2 K ( m R ) − a3 I ( m R ) + b3 K ( m R ) = µr a21n I0 (mn R5 ) − µr b1n n 5 n 5 n 5 0 2n 0 2n 0   (2n−1)πτr 4Brem πPn , L0 (mn R5 ) 2m 2 − (2n −1) πm cos 2τp

(22h)

(22i)

n

n

3 2 a32n I1 (mn R5 ) + b2n K1 (mn R5 ) − a21n I1 (mn R5 ) − b1n K1 ( m n R 5 ) = − L 1 ( m n R 5 )

πPn , 2mn 2

(22j)

The matrix form of Equations (22a) until (22j) is AX = F                  

where

0 0 A31 A41 A51 A61 0 0 0 0

0 0 A32 A42 A52 A62 0 0 0 0

0 0 0 0 0 0 A73 A83 A93 A103

0 0 0 0 0 0 A74 A84 A94 A104

A15 0 A35 A45 0 0 0 0 0 0

A16 0 A36 A46 0 0 0 0 0 0

0 0 0 0 A57 A67 A77 A87 0 0

0 0 0 0 A58 A68 A78 A88 0 0

0 A29 0 0 0 0 0 0 A99 A109

0 A210 0 0 0 0 0 0 A910 A1010

                 

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10





                =                

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10

         ,        

(23)

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πPn A15 = I0 (mn R1 ); A16 = −K0 (mn R1 ); F1 = − L0 (mn R1 ) 2m 2 − n

πPn A29 = I0 (mn R6 ); A210 = −K0 (mn R6 ); F2 = − L0 (mn R6 ) 2m 2 + n

4Brem (2n−1)πmn 4Brem (2n−1)πmn

cos



cos

(2n−1)πτr 2τp





(2n−1)πτr 2τp

; 

;

A31 = µr I0 (mn R2 ); A32 = −µr K0 (mn R2 ); A35 = − I0 (mn R2 ); A36 = K0 (mn R2 );   (2n−1)πτr 4Brem πPn F3 = L0 (mn R2 ) 2m ; A45 = I1 (mn R2 ); A46 = K1 (mn R2 ); 2 + (2n −1) πm cos 2τp n

n

πPn A41 = − I1 (mn R2 ); A42 = −K1 (mn R2 ); F4 = − L1 (mn R2 ) 2m 2 ; A51 = µr I0 ( m n R3 ); n

πPn A52 = −µr K0 (mn R3 ); A57 = − I0 (mn R3 ); A58 = K0 (mn R3 ); F5 = L0 (mn R3 ) 2m 2; n

A67 = I1 (mn R3 ); A68 = K1 (mn R3 ); A61 = − I1 (mn R3 ); A62 = −K1 (mn R3 ); πPn F6 = − L1 (mn R3 ) 2m 2 ; A73 = µr I0 ( m n R4 ); A74 = − µr K0 ( m n R4 ); A77 = − I0 ( m n R4 ); n

πPn A78 = K0 (mn R4 ); F7 = L0 (mn R4 ) 2m 2 ; A87 = I1 ( m n R4 ); A88 = K1 ( m n R4 ); A83 = − I1 ( m n R4 ); n

πPn A84 = −K1 (mn R4 ); F8 = − L1 (mn R4 ) 2m 2 ; A93 = µr I0 ( m n R5 ); A94 = − µr K0 ( m n R5 ); n   (2n−1)πτr πPn 4Brem ; A99 = − I0 (mn R5 ); A910 = K0 (mn R5 ); F9 = L0 (mn R5 ) 2m 2 − (2n −1) πmn cos 2τp n

A109 = I1 (mn R5 ); A1010 = K1 (mn R5 ); A103 = − I1 (mn R5 ); A104 = −K1 (mn R5 ); πPn F10 = − L1 (mn R5 ) 2m 2. n

5. Numerical Simulation and Analysis 5.1. Numerical Model The numerical approach is an effective way to analyze the magnetic flux field distribution of electromagnetic actuators. The values of major parameters for numerical computation are listed in Table 2. These parameter values are obtained by maximizing the force output. With the values in Table 2, the maximum force output is 112 N. The details of design optimization based on force model and motion control implementation with force feedback will be covered in another paper. Table 2. Major parameters of the linear machine for numerical analysis. Parameter Items Thickness of back iron, R1 Outer radius of inner Halbach layer, R2 Inner radius of middle PM layer, R3 Outer radius of middle PM layer, R4 Inner radius of outer Halbach layer, R5 Outer radius of outer Halbach layer, R6 Outer radius of outer back iron, R7 Width of radial magnets, τr Width of axial magnets, τz Pole pitch, τp

Value

Relative permeability of PMs, µr

0.005(m) 0.010(m) 0.017(m) 0.020(m) 0.028(m) 0.032(m) 0.035(m) 0.016(m) 0.008(m) 0.024(m)   4π × 10−7 N/A2   1.07 N/A2

Remenance, Brem

1.23(T)

Permeability of vacuum, µ0

The structure of the proposed linear machine is axially symmetric, so Magnetostatic 2D solver is conducted on Maxwell to model and analyze the flux field. In this study, the whole model is divided into 179,958 mesh elements to derive accurate solution, with the RMS edge length ranging from 0.00012 mm to 0.00051 mm for different parts of solving regions. The relationship between the number of iterative solving rounds and the energy error percentage is shown in Figure 6. The energy error

Remenance, 𝐵𝑟𝑒𝑚

1.23(T)

The structure of the proposed linear machine is axially symmetric, so Magnetostatic 2D solver is conducted on Maxwell to model and analyze the flux field. In this study, the whole model is divided into 179,958 mesh elements to derive accurate solution, with the RMS edge length ranging from Sensors 2017, 17, 2662 10 of 15 0.00012 mm to 0.00051 mm for different parts of solving regions. The relationship between the number of iterative solving rounds and the energy error percentage is shown in Figure 6. The energy error percentage is used to represent the accuracy of numerical solution in Maxwell. canfound be found percentage is used to represent the accuracy of numerical solution in Maxwell. It canIt be that that 10 rounds of iterative solving, the energy is below 5%, and it is acceptable for analysis afterafter 10 rounds of iterative solving, the energy errorerror is below 5%, and it is acceptable for analysis and and validation of electromagnetic machines. validation of electromagnetic machines.

Iterative solving rounds

Figure Figure 6. 6. Relation Relation between between iterative iterative solving solving rounds rounds and and energy energy error error percentage. percentage.

Flux Flux distribution distribution corresponding corresponding to to the the initial initial and and maximum maximum working working positions positions of of coils coils without without and with load is shown in Figure 7a–c respectively. It can be found that the magnetic flux is generated and with load is shown in Figure 7a–c respectively. It can be found that the magnetic flux is generated by by the the permanent permanent magnet, magnet, goes goes across across the the air air gap gap filled filled with with windings, windings, and and returns returns through through the the back back irons, forming a close loop. As indicated in Figure 7b,c, flux distribution varies for different irons, forming a close loop. As indicated in Figure 7b,c, flux distribution varies for different coil coil positions, not affected affected significantly significantly by by power power supply supply of of coils. coils. positions, but but is is not Sensors 2017, 17, 2662

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(a)

(b)

(c)

Figure 7. Magnetic flux distribution: (a) Initial position without power supply; (b) Initial position with Figure 7. Magnetic flux distribution: (a) Initial position without power supply; (b) Initial position with power supply; supply; (c) (c) Maximum Maximum working working position position with with power power supply. supply. power

B/(T)

Figure 8 illustrated the BH curve to analyze magnetic saturation in the machine. It is found that Figure 8 illustrated the BH curve to analyze magnetic saturation in the machine. It is found that there is no significant saturation in the system. Only slight saturation exists at the inner back iron due there is no significant saturation in the system. Only slight saturation exists at the inner back iron to the relatively small volume. Further design optimization might be conducted to reduce the due to the relatively small volume. Further design optimization might be conducted to reduce the saturation if necessary. saturation if necessary. To precisely observe the variation of magnetic flux field, Figure 9a,b show the flux distribution with respect to axial and radial positions in outer and inner air gaps, respectively. Only the radial flux component is presented, because only this component can produce voltage signal or axial thrust in this machine. Figure 9a shows that the radial flux density does not change significantly in the radial DT4 B-H Curve direction. However, it is slightly larger at positions close to the inner magnet layer. Similarly, Figure 9b 2.5 indicates that the flux density close to the outer layer of Halbach array is relatively larger. In the axial 2 direction, the flux density for both air gaps varies in trigonometric form, which is consistent with the magnetization pattern of PM poles in this direction. 1.5

1 0.5 0

power supply; (c) Maximum working position with power supply.

Figure 8 illustrated the BH curve to analyze magnetic saturation in the machine. It is found that there is no significant saturation in the system. Only slight saturation exists at the inner back iron due to the2017, relatively Sensors 17, 2662 small volume. Further design optimization might be conducted to reduce 11 ofthe 15 saturation if necessary.

DT4 B-H Curve 2.5

B/(T)

2 1.5 1 0.5 0 0

0.5

1

1.5

2 H/(A/m)

2.5

3

3.5

4 x 10

4

Figure 8. Flux density in the linear machine and BH curve of back iron. Figure 8. Flux density in the linear machine and BH curve of back iron. SensorsTo 2017, 17, 2662 observe precisely Sensors 2017, 17, 2662

11 of 14 the variation of magnetic flux field, Figure 9a,b show the flux distribution 11 of 14 with respect to axial and radial positions in outer and inner air gaps, respectively. Only the radial flux component is presented, because only this component can produce voltage signal or axial thrust in this machine. Figure 9a shows that the radial flux density does not change significantly in the radial direction. However, it is slightly larger at positions close to the inner magnet layer. Similarly, Figure 9b indicates that the flux density close to the outer layer of Halbach array is relatively larger. In the axial direction, the flux density for both air gaps varies in trigonometric form, which is consistent with the magnetization pattern of PM poles in this direction.

(a) (a)

(b) (b)

Figure 9. 9. Magnetic flux flux density with with respect to to axial and and radial directions: directions: (a) Inner winding winding region; Figure Figure 9. Magnetic Magnetic flux density density with respect respect to axial axial and radial radial directions: (a) Inner Inner winding region; (b) Outer winding region. (b) (b) Outer Outerwinding windingregion. region.

5.2. Validation of Analytical Models 5.2. Validation Validation of of Analytical Analytical Models Models 5.2. The analytical model is a powerful tool for design optimization and control implementation of The analytical model is implementation of The is aa powerful powerfultool toolfor fordesign designoptimization optimizationand andcontrol control implementation electromagnetic actuators. In the numerical approach is efficient and reliable way to electromagnetic actuators. In contrast, contrast, thethe numerical approach is an an efficient and reliable way to of electromagnetic actuators. In contrast, numerical approach is an efficient and reliable way validate analytical results. To the result and model precisely, four validate analytical results. To compare compare the numerical numerical resultresult and analytical analytical model precisely, four to validate analytical results. To compare the numerical and analytical model precisely, positions indicated by in air region are for as in 10. positions indicated by lines lines in the the are utilized utilized for simulation, simulation, as shown shown in Figure Figure 10. 10. four positions indicated by lines inair theregion air region are utilized for simulation, as shown in Figure

rr Line4 Line4 Line3 Line3 Line2 Line2 Line1 Line1 zz Figure 10. Positions for comparison of numerical result and analytical models. Figure 10. 10. Positions Positions for for comparison comparison of of numerical numerical result result and and analytical analytical models. models. Figure

The The comparison comparison result result of of the the analytical analytical model model and and the the numerical numerical computation computation is is presented presented in in Figure 11. Figure 11a,b represent the variation of flux component in radial and axial directions, Figure 11. Figure 11a,b represent the variation of flux component in radial and axial directions, respectively, respectively, for for Line Line 1. 1. Similarly, Similarly, the the flux flux variation variation for for Line Line 2–4 2–4 is is presented presented in in Figure Figure 11c–h. 11c–h. It It is is found that the analytical model fits with the numerical result well. It could be employed for the study found that the analytical model fits with the numerical result well. It could be employed for the study on on modeling modeling of of force force or or current current output, output, and and subsequent subsequent design design optimization. optimization. Figure Figure 11a,c 11a,c indicate indicate

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The comparison result of the analytical model and the numerical computation is presented in Figure 11. Figure 11a,b represent the variation of flux component in radial and axial directions, respectively, for Line 1. Similarly, the flux variation for Line 2–4 is presented in Figure 11c–h. It is found that the analytical model fits with the numerical result well. It could be employed for the study on modeling of force or current output, and subsequent design optimization. Figure 11a,c indicate that the flux density close to the internal layer of Halbach array is relatively larger than that near the middle magnet layer. Similarly, Figure 11e,g indicate that the flux density close to the external layer of Halbach array is relatively larger than that near the middle layer. This result is consistent with the comparison in Figure 9. Figure 11 also shows that both radial and axial components of magnetic flux density change alternatively with positive and negative signs, which is caused by the alternatively Sensors 2017, 17, 2662 12 of 14 magnetized PM poles in the axial direction.

(a)

(b)

(c)

(d)

(e)

(f) Figure 11. Cont.

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(e)

(f)

(g)

(h)

Figure 11. Comparison of analytical model and numerical result: (a) Radial flux density on line 1; (b) Figure 11. Comparison of analytical model and numerical result: (a) Radial flux density on line 1; Axial flux density on line 1. (c) Radial flux density on line 2; (d) Axial flux density on line 2. (e) Radial (b) Axial flux density on line 1. (c) Radial flux density on line 2; (d) Axial flux density on line 2. flux density on line 3; (f) Axial flux density on line 3. (g) Radial flux density on line 4; (h) Axial flux (e) Radial flux density on line 3; (f) Axial flux density on line 3. (g) Radial flux density on line 4; density on line 4. (h) Axial flux density on line 4.

6. Discussion The objective of this study is to propose one novel tubular linear machine with hybrid magnet arrays and multiple layers of windings. It could be implemented for actuation or motion sensing depending on particular tasks. The employment of hybrid magnet array with Halbach and alternatively magnetized radial poles helps to enhance the magnetic flux density in the radial direction, and thus increase the current signal strength or force output. Furthermore, the utilization of multiple windings offers the redundancy property, and thus increases the system reliability significantly. Based on the proposed topological design, the magnetic vector potential is formulated analytically from governing equations. The flux distribution is then obtained from the curl of the potential. Numerical computation is conducted to validate the analytical model. It shows that the analytical model agrees with the numerical result well. In addition, the variation of magnetic flux density in the linear machine is consistent with the magnet patterns. The proposed structure topology and developed analytical model could be employed for subsequent study on current signal or force generation, and real-time motion control. Acknowledgments: The authors acknowledge the financial support from the National Natural Science Foundation of China (NSFC) under the Grant No. 51575026, the National Key Basic Research Program of China (973 Program, 2014CB046406), NSFC51235002, the Fundamental Research Funds for the Central Universities, and the Science and Technology on Aircraft Control Laboratory. Author Contributions: N. Yao and L. Yan did major contribution to this paper; they proposed the novel concept of the linear machine, and did substantial works to prove it; T. Wang did some numerical simulation; S. Wang gave important suggestion on mathematical modeling. Conflicts of Interest: The authors declare no conflict of interest.

References 1.

2. 3.

Kou, B.; Huang, X.; Wu, H.; Li, L. Thrust and thermal characteristics of electromagnetic launcher based on permanent magnet linear synchronous motors. In Proceedings of the 2008 14th Symposium on Electromagnetic Launch Technology, Victoria, BC, Canada, 10–13 June 2008; pp. 1–6. Musolino, A.; Rizzo, R.; Tripodi, E. The double-sided tubular linear induction motor and its possible use in the electromagnetic aircraft launch system. IEEE Trans. Plasma Sci. 2013, 41, 1193–1200. [CrossRef] Hellinger, R.; Mnich, P. Linear motor-powered transportation: History, present status, and future outlook. Proc. IEEE 2009, 97, 1892–1900. [CrossRef]

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4.

5.

6. 7. 8.

9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20.

21.

22.

23. 24. 25. 26.

14 of 15

Lu, Z.; Zhou, H.; Liu, P.; Liu, G.; Chen, L. Design and analysis of five-phase fault-tolerant radially magnetized tubular permanent magnet motor for active vehicle suspensions. In Proceedings of the 2016 19th International Conference on Electrical Machines and Systems (ICEMS), Chiba, Japan, 13–16 November 2016; pp. 1–4. Commins, P.A.; Moscrop, J.W.; Cook, C.D. Novel tooth design for a tubular linear motor for machine tool axis. In Proceedings of the 2011 IEEE International Conference on Mechatronics (ICM), Istanbul, Turkey, 13–15 Apirl 2011; pp. 660–665. Shukor, A.Z.; Fujimoto, Y. Direct-drive position control of a spiral motor as a monoarticular actuator. IEEE Trans. Ind. Electron. 2014, 61, 1063–1071. [CrossRef] Tang, X.; Lin, T.; Zuo, L. Design and optimization of a tubular linear electromagnetic vibration energy harvester. IEEE/ASME Trans. Mechatron. 2014, 19, 615–622. [CrossRef] Asama, J.; Burkhardt, M.R.; Davoodi, F.; Burdick, J.W. Investigation of energy harvesting circuit using a capacitor-sourced buck converter for a tubular linear generator of a moball: A spherical wind-driven exploration robot. In Proceedings of the 2015 IEEE Energy Conversion Congress and Exposition (ECCE), Montreal, QC, Canada, 20–24 September 2015; pp. 3167–3171. Slamani, M.; Nubiola, A.; Bonev, I.A. Effect of servo systems on the contouring errors in industrial robots. Trans. Can. Soc. Mech. Eng. 2012, 36, 83–96. Kwak, H.-J.; Park, G.-T. Study on the mobility of service robots. Int. J. Eng. Technol. Innov. 2012, 2, 97–112. Von Büren, T.; Tröster, G. Design and optimization of a linear vibration-driven electromagnetic micro-power generator. Sens. Actuators A Phys. 2007, 135, 765–775. [CrossRef] Kim, W.-J.; Murphy, B.C. Development of a novel direct-drive tubular linear brushless permanent-magnet motor. Int. J. Control Autom. Syst. 2004, 2, 279–288. Huang, X.; Tan, Q.; Wang, Q.; Li, J. Optimization for the Pole Structure of Slot-Less Tubular Permanent Magnet Synchronous Linear Motor and Segmented Detent Force Compensation. IEEE Trans. Appl. Supercond. 2016, 26, 0611405. [CrossRef] Wang, J.; Jewell, G.; Howe, D. Design optimisation and comparison of tubular permanent magnet machine topologies. IEE Proc. Electr. Power Appl. 2001, 148, 456–464. [CrossRef] Nirei, M.; Tang, Y.; Mizuno, T.; Yamamoto, H.; Shibuya, K.; Yamada, H. Iron loss analysis of moving-coil-type linear DC motor. Sens. Actuators A Phys. 2000, 81, 305–308. [CrossRef] Baloch, N.; Khaliq, S.; Kwon, B.-I. A High Force Density HTS Tubular Vernier Machine. IEEE Trans. Magn. 2017, 53, 8111205. [CrossRef] Wang, J.; Atallah, K.; Wang, W. Analysis of a magnetic screw for high force density linear electromagnetic actuators. IEEE Trans. Magn. 2011, 47, 4477–4480. [CrossRef] Halbach, K. Design of permanent multipole magnets with oriented rare earth cobalt material. Nuclear Instrum. Methods 1980, 169, 1–10. [CrossRef] Wang, J.; Jewell, G.W.; Howe, D. A general framework for the analysis and design of tubular linear permanent magnet machines. IEEE Trans. Magn. 1999, 35, 1986–2000. [CrossRef] Wang, T.; Jiao, Z.; Yan, L.; Chen, C.-Y.; Chen, I.-M. Design and analysis of an improved Halbach tubular linear motor with non-ferromagnetic mover tube for direct-driven EHA. In Proceedings of the 2014 IEEE Chinese Guidance, Navigation and Control Conference (CGNCC), Yantai, China, 8–10 August 2014; pp. 797–802. Abdalla, I.I.; Ibrahim, T.; Nor, N.M. Design and modeling of a single-phase linear permanent magnet motor for household refrigerator applications. In Proceedings of the 2013 IEEE Business Engineering and Industrial Applications Colloquium (BEIAC), Langkawi, Malaysia, 7–9 Apirl 2013; pp. 634–639. Zhang, L.; Yan, L.; Yao, N.; Wang, T.; Jiao, Z.; Chen, I.-M. Force formulation of a three-phase tubular linear machine with dual Halbach array. In Proceedings of the 2012 10th IEEE International Conference on Industrial Informatics (INDIN), Beijing, China, 25–27 July 2012; pp. 647–652. Yan, L.; Zhang, L.; Jiao, Z.; Hu, H.; Chen, C.-Y.; Chen, I.-M. A tubular linear machine with dual Halbach array. Eng. Comput. 2014, 31, 177–200. [CrossRef] Zhang, B.; Cheng, M.; Wang, J.; Zhu, S. Optimization and analysis of a yokeless linear flux-switching permanent magnet machine with high thrust density. IEEE Trans. Magn. 2015, 51, 8204804. [CrossRef] Wikipedia. Available online: https://en.wikipedia.org/wiki/Bessel_function (accessed on 9 November 2017). Wolframe MathWorld. Available online: http://mathworld.com/ModifiedStruveFunction.html (accessed on 9 November 2017).

Sensors 2017, 17, 2662

27. 28.

15 of 15

Ibala, A.; Masmoudi, A. Accounting for the armature magnetic reaction and saturation effects in the reluctance model of a new concept of claw-pole alternator. IEEE Trans. Magn. 2010, 46, 3955–3961. [CrossRef] Gaussens, B.; Hoang, E.; de la Barriere, O.; Saint-Michel, J.; Manfe, P.; Lecrivain, M.; Gabsi, M. Analytical armature reaction field prediction in field-excited flux-switching machines using an exact relative permeance function. IEEE Trans. Magn. 2013, 49, 628–641. [CrossRef] © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).