Design of a New 1D Halbach Magnet Array with Good Sinusoidal Magnetic Field by Analyzing the Curved Surface

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Article Design of a New 1D Halbach Magnet Array with Good Sinusoidal Magnetic Field by Analyzing the Curved Surface

Guangdou Liu, Shiqin Hou, Xingping Xu and Wensheng Xiao *

College of Mechanical and Electronic Engineering, China University of Petroleum, Qingdao 266580, China; [email protected] (G.L.); [email protected] (S.H.); [email protected] (X.X.) * Correspondence: [email protected]; Tel.: +86-189-5420-7088

Abstract: In the linear and planar motors, the 1D Halbach magnet array is extensively used. The sinusoidal property of the magnetic field deteriorates by analyzing the magnetic field at a small air gap. Therefore, a new 1D Halbach magnet array is proposed, in which the permanent magnet with a curved surface is applied. Based on the superposition of principle and Fourier series, the magnetic flux density distribution is derived. The optimized curved surface is obtained and fitted by a polynomial. The sinusoidal magnetic field is verified by comparing it with the magnetic flux density of the finite element model. Through the analysis of different dimensions of the permanent magnet array, the optimization result has good applicability. The force ripple can be significantly reduced by the new magnet array. The effect on the mass and air gap is investigated compared with a conventional magnet array with rectangular permanent magnets. In conclusion, the new magnet array design has the scalability to be extended to various sizes of motor and is especially suitable for small air gap applications. Keywords: Halbach magnet array; curved surface; good sinusoidal magnetic field; linear motor;

Citation: Liu, G.; Hou, S.; Xu, X.; planar motor Xiao, W. Design of a New 1D Halbach Magnet Array with Good Sinusoidal Magnetic Field by Analyzing the Curved Surface. Sensors 2021, 21, 1. Introduction 2522. https://doi.org/10.3390/ Linear and planar motors can directly convert electric energy into linear motion s21072522 mechanical energy without any transmission device of intermediate conversion mecha- nism [1,2]. They have the advantages of compact structure, high transmission stiffness, Academic Editor: Cristina Gómez-Polo fast dynamic response, and high positioning accuracy due to the absence of mechanical conversion parts, and are extensively used in various fields, such as logistics delivery sys- Received: 5 March 2021 tems, computer numerical control (CNC) machine systems, lithography, and magnetically Accepted: 26 March 2021 Published: 4 April 2021 levitated train systems. The magnetic field on one side of the 1D Halbach permanent magnet array is significantly enhanced due to the arrangement of permanent magnets and

Publisher’s Note: MDPI stays neutral has good sinusoidal characteristics. Therefore, the 1D Halbach magnet array is widely with regard to jurisdictional claims in applied in linear, planar, and other motors [3–8]. published maps and institutional affil- Many other researchers have studied linear or planar motors by applying the 1D iations. Halbach magnet array. Won-Jong Kim et al. [9,10] proposed the planar magnetic levitation device, which is composed of four permanent magnet linear motors. The 1D Halbach magnet array with rectangular magnets is used and the magnetic field is represented by Fourier series expansion and the complex theory. M Lee et al. [11] replaced the rectangular magnets with trapezoidal magnets in the double-sided linear motor to produce more force. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. The magnetic flux density distribution was obtained by adding the magnetic flux density This article is an open access article of each permanent magnet. Chen Jun-Wei et al. [12] proposed a new Halbach magnetic distributed under the terms and array by applying the permanent magnet with a sinusoidal edge. The larger flux density conditions of the Creative Commons and smaller harmonic distortion can be obtained by the magnet array in a linear motor. Attribution (CC BY) license (https:// The magnetic flux density was derived by generalized blending function mapping and creativecommons.org/licenses/by/ superposition. Irfan-Ur-Rab Usman et al. [13] developed a 6 degree of freedom (DOF) 4.0/). planar levitating synchronous motor. It consists of four linear motors with 1D magnet

Sensors 2021, 21, 2522. https://doi.org/10.3390/s21072522 https://www.mdpi.com/journal/sensors Sensors 2021, 21, x FOR PEER REVIEW 2 of 18 Sensors 2021, 21, 2522 2 of 17

and superposition. Irfan-Ur-Rab Usman et al. [13] developed a 6 degree of freedom (DOF) arrays.planar The levitating coil is madesynchronous into a printedmotor. It circuit consists board. of four The linear force motors fluctuation with can1D bemagnet reduced byarrays. designing The coil proper is made separation into a printed and spacing circuit board. magnet The arrays. force fluctuation Rui Chen can [14 ]be analyzed reduced the ◦ M-Magnetby designing array proper with separation a magnetization and spacing axis inmagnet a 45 arrays.direction Rui relative Chen [14] to itsanalyzed side surfaces, the insteadM-Magnet of 0◦ arrayor 90 with◦ magnetization a magnetization pieces axis usedin a 45° in adirection conventional relative Halbach to its side array. surfaces, Then, a novelinstead hybrid of 0° array or 90° based magnetization on the M-Magnet pieces us arrayed in a is conventional presented which Halbach can array. attenuate Then, the a 6th forcenovel ripple. hybrid A. array Boduroglu based on et the al. [M-Magnet15] proposed array a is new, presented skewed which magnet can arrangementattenuate the 6th for the linearforce motor. ripple. ByA. Boduroglu comparing et the al. conventional[15] proposed magneta new, skewed arrangement, magnet arrangement the permanent for magnetthe withlinear asymmetric motor. By comparing V shape arrangement the conventional can reducemagnet thearrangement, force ripple the andpermanent has little magnet effect on thewith average asymmetric force. V shape arrangement can reduce the force ripple and has little effect on theAccording average force. to the above analysis, in order to improve the performance of the permanent magnetAccording array, a variety to the above of studies analysis, have in been order carried to improve out, such the performance as high force, of small the perma- harmonic distortion,nent magnet and array, low forcea variety ripple. of studies In the linearhave been and carried planar out, motor, such the as airhigh gap force, is very small small harmonic distortion, and low force ripple. In the linear and planar motor, the air gap is between the permanent magnets and coils in order to get higher thrust. However, the very small between the permanent magnets and coils in order to get higher thrust. How- sinusoidal property of the magnetic field deteriorates when the air gap is small. This can be ever, the sinusoidal property of the magnetic field deteriorates when the air gap is small. seen in Figure1, obtained by the conventional 1D Halbach magnet array with rectangular This can be seen in Figure 1, obtained by the conventional 1D Halbach magnet array with magnets when the air gap is 1 mm. In the real-time control, the first harmonic of the rectangular magnets when the air gap is 1 mm. In the real-time control, the first harmonic magneticof the magnetic flux density flux density is usually is usually used toused calculate to calculate the force,the force, which which will will cause cause the the force rippleforcedue ripple to due the errorto the witherror thewith actual the actual magnetic magnetic flux flux density. density.

B (T)

(a) (b)

FigureFigure 1. (a )1. The (a) firstThe harmonicfirst harmonic and FEMand FEM of magnetic of magnetic flux density;flux density; (b) the (b conventional) the conventional magnet magnet array array with rectangularwith rectangular magnets. magnets. It is found that the magnetic flux density has poor sinusoidal property and relatively large errorIt is found compared that the with magnetic the first flux harmonic. density has As poor we sinusoidal know, one property permanent and relatively magnet can belarge regarded error compared as a superposition with the first of countlessharmonic. As small we know, pieces one according permanent tothe magnet superposition can be regarded as a superposition of countless small pieces according to the superposition prin- principle, so does the magnetic flux density of the permanent magnet. Therefore, the ciple, so does the magnetic flux density of the permanent magnet. Therefore, the main main cause of poor waveform is that the rectangular permanent magnet is used, such as cause of poor waveform is that the rectangular permanent magnet is used, such as in [1– in [1–10,13,14]. Though the trapezoidal magnet [11], the sinusoidal-edged magnet [12], and 10,13,14]. Though the trapezoidal magnet [11], the sinusoidal-edged magnet [12], and the the skewed magnet [15] are applied in the motors, the top and bottom surfaces of permanent skewed magnet [15] are applied in the motors, the top and bottom surfaces of permanent magnetsmagnets are are flat flat and and thethe sinusoidalsinusoidal property property of of the the magnetic magnetic field field also also deteriorates deteriorates in the in the smallsmall air air gap. gap. Therefore, Therefore, thethe paper is is analyz analyzeded from from a anew new perspective. perspective. The The surface surface shape shape ofof the the permanent permanent magnet magnet isis designed to to be be curv curveded instead instead of offlat flat so sothat that the thewaveform waveform of of magneticmagnetic flux flux density density cancan bebe improved. InIn order order to to obtain obtain thethe magneticmagnetic flux flux density density with with good good sinusoidal sinusoidal characteristics, characteristics, the the permanentpermanent magnet magnet isis divideddivided into small small pieces pieces and and the the heights heights of ofthe the small small pieces pieces are are designed. The design of a permanent magnet is a constrained nonlinear multivariable optimization problem. The appropriate optimization algorithm [16–20] should be chosen to realize the design of the new magnet array. In terms of numerical effect and stability, the sequential quadratic programming (SQP) algorithm is considered to be one of the most effective methods to solve nonlinear constrained optimization. Therefore, the heights of Sensors 2021, 21, x FOR PEER REVIEW 3 of 18

designed. The design of a permanent magnet is a constrained nonlinear multivariable op- Sensors 2021, 21, 2522 timization problem. The appropriate optimization algorithm [16–20] should be chosen to 3 of 17 realize the design of the new magnet array. In terms of numerical effect and stability, the sequential quadratic programming (SQP) algorithm is considered to be one of the most effective methods to solve nonlinear constrained optimization. Therefore, the heights of the small pieces are optimized by SQP. The shape of the curved surface is obtained by the small pieces are optimized by SQP. The shape of the curved surface is obtained by optimization results and is fitted by a least squares method [21,22]. Consequently, a new optimization results and is fitted by a least squares method [21,22]. Consequently, a new 1D Halbach magnet array with a curved surface is proposed. 1D Halbach magnet array with a curved surface is proposed. 2. Modeling of the New Magnet Array 2. Modeling of the New Magnet Array TheThe magnetic magnetic flux flux density density is modeled is modeled in this in this section. section. The new The newmagnet magnet array arrayis pro- is proposed based based on on the the conventional conventional magnet magnet array array with with rectangular rectangular magnets. magnets. The permanent The permanent magnet is designed to obtain a goodgood sinusoidalsinusoidal magneticmagnetic field.field. Therefore, the analysis steps aresteps as are follows. as follows. Firstly, Firstly, the permanentthe permanent magnet magnet is divided is divided into into small small pieces. pieces. Then Then the the height ofheight each of small each piecesmall ispiece independently is independently set. Finally, set. Finally, the magnetic the magnetic flux densityflux density is derived is de- based onrived the based Fourier on the series. Fourier series. FigureFigure 2 shows the cross-section of of the the magnet magnet array array and and the the MMx projectionx projection distribu- distribution fortion small for small pieces pieces of permanent of permanent magnets magnets in in the thex magnetizationx magnetization direction. direction.The The magnetizationmagnet- directionization direction is denoted is denoted by the by arrow. the arrow. The The direction direction of theof the arrow arrow is is from from the theS Spole pole toto the N pole.the N Eachpole. Each small small piece piece of permanent of permanent magnet magnet can can be be seen seen as as one one rectangular rectangular magnet magnet and is indicatedand is indicated by the by blue the bar.blue bar.

x p t (a) Mx 1 d (t+p)/2 x (t-p)/2 -1 (b) Figure 2. 2. TheThe magnet magnet array array and and magnetization magnetization distributi distribution:on: (a) cross-section (a) cross-section of the magnet of the magnetarray; array; (b) Mx projection distribution. (b) Mx projection distribution.

OnOn the the basis basis of of the the periodic periodic and and symmetry symmetry characteristics characteristics of the of magnet the magnet array, array,all the all the permanent magnets magnets are are divided divided into into the the same same number number of small of small pieces. pieces. The side The length side length of of each piece piece is is the the same. same. For For each each permanent permanent ma magnet,gnet, the axis the axisof symmetry of symmetry is in the is in middle the middle and both sides sides have have a asymmetrical symmetrical distribution. distribution. Therefore, Therefore, there there are four are small four small pieces pieces with with the same heights in one period, and these pieces can be regarded as one group shown in the same heights in one period, and these pieces can be regarded as one group shown in Figure 2. If the number of all small pieces of one permanent magnet is 2n, where n is an Figure2. If the number of all small pieces of one permanent magnet is 2 n, where n is an integer, the number of the small pieces of half of one permanent magnet is n. From this, integer, the number of the small pieces of half of one permanent magnet is n. From this, the the permanent magnets magnetized in the x-direction of the magnet array are composed permanentof n groups of magnets pieces, magnetizedand so are the in permanent the x-direction magnets of themagnetized magnet in array the z are-direction. composed of n groups of pieces, and so are the permanent magnets magnetized in the z-direction. The Mx of magnet array is modeled based on the Fourier series and expressed as The Mx of magnet array is modeled based on the Fourier series and expressed as ∞ n = () (ω ) M x Maikkx∞ n ,cos , (1) ki==11 Mx = M ∑ ∑ a(i, k) cos(kωx), (1) k=1 i=1

where M = Br/µ0, ω = π/t, µ0 is the permeability of vacuum, t is the pole pitch, k is the harmonic numbers, a(i,k) is projection distribution coefﬁcient, Br is the remanence of the permanent magnet,

a(i, k) = 8/kπ sin(kωd/2) sin(kπ/2) sin(kω(p + (2i − 1)d)/2), (2) p d = , (3) 2n Sensors 2021, 21, x FOR PEER REVIEW 4 of 18

where M=Br/μ0, ω=π/t, μ0 is the permeability of vacuum, t is the pole pitch, k is the harmonic numbers, a(i,k) is projection distribution coefficient, Br is the remanence of the permanent magnet,

aik(),=+−8 sin() kωπω d 2 sin() k 2 sin() k() p () 2 i 1 d 2 kπ , (2)

Sensors 2021, 21, 2522 4 of 17 p d = , (3) 2n where d is the side length of the small piece, and p is the side length of the permanent wheremagnet.d is the side length of the small piece, and p is the side length of the permanent magnet. The magnetizationThe magnetization vector vector of of thethe magnet array array consists consists of M ofx andMx Mandz components.Mz components. The The Mz componentcomponent also also can can be bemodeled modeled by using by using the Fourier the Fourier series.series. Figure Figure3 shows3 showsthe Mz the Mz projection distribution distribution of ofthe the permanent permanent magnets magnets in the in z magnetization the z magnetization direction. direction.

(a)

Mz 1 (t-p)/2 (t+p)/2 x d -1 (b) Figure 3. 3. TheThe magnet magnet array array and andmagnetization magnetization distributi distribution:on: (a) cross-section (a) cross-section of the magnet of the array; magnet array; (b) Mz projection distribution. (b) Mz projection distribution.

The expression of Mz is The expression of Mz is ∞ n = ()(ω ) M z Mbkikx∞ n ,sin , (4) == Mz = Mki11∑ ∑ b(k, i) sin(kωx), (4) = = where b(i,k) is the projection distribution coefficient,k 1 i 1

where b(i,k) is thebik() projection,sin2sin2cos212=−8 distribution() kωπω d() coefﬁcient, k() k() p − () i − d kπ . (5) The magnetizationb(i, k) = −vector8/kπ sinof the(kω magnetd/2) sin array(kπ is/2 obtained) cos(kω and(p − expressed(2i − 1) das) /2). (5)

 M ∞ n aik(),cos ( kω x ) The magnetization vectorMM== ofx the magnet array is obtained and expressed as  ()(ω ), (6) M z ki==11bik,sin k x → ∞ n Mx a(i, k) cos(kωx) There are three regionsM = of the magnetic= M field in space. From top to bottom,, they are (6) Mz ∑ ∑ b(i, k) sin(kωx) air, permanent magnet array, and air. The magnetick=1 i=1 scalar potential method is applied to solve the magnetic flux density because there is no conduction current. There are three regions of the magnetic field in space. From top to bottom, they are air, The boundary conditions of the interface can be derived from Maxwell’s equations. permanent magnet array, and air. The magnetic scalar potential method is applied to solve So the magnetic field problem comes down to solve the Poisson equation of magnetic sca- thelar potential magnetic [9]. flux The density magnetic because flux density there is noobtained conduction by using current. the variable separation method.The For boundary the region conditions below the magnet of the interfacearray, it is can expressed be derived as from Maxwell’s equations. So the magnetic field problem comes down to solve the Poisson equation of magnetic ∞ ω  Bkxλ cos() scalar potential [9]. The magnetic==−x fluxμω densityz is obtained by using the variable separation BKek 0  , (7) Bkx= sin ()ω method. For the region below thez magnetk 1 array, it is expressed as

→ ∞ Bx λz cos(kωx) B = = −µ0ω Ke k , (7) Bz ∑ sin(kωx) k=1

where ht(i) and hb(i) are the position of the top and bottom surfaces of each piece of the permanent magnet, respectively,

n Br −λht(i) −λh (i) K(µr = 1) = ∑ e − e b (b(i, k) − a(i, k)), (8) 2µ0λ i=1

λ = kω, (9)

The relative permeability for permanent magnet µr is assumed as 1.0. Because high- quality sintered NdFeB permanent magnets (µr = 1.03~1.05) will be used, the error due to this assumption can be neglected. Sensors 2021, 21, 2522 5 of 17

3. Design of New Magnet Array 3.1. Optimization The main purpose is to obtain a good sinusoidal magnetic field, which is to make the actual magnetic flux density consistent with the first harmonic of the magnetic flux density. As a result, the actual magnetic field has good sinusoidal characteristics, and the expression can be simplified as that of the first harmonic. In the real-time control of electrical machines, the first harmonic of the magnetic flux density is usually used to calculate the force. The new magnet array is designed based on the conventional magnet array, and the shape of the surface is optimized. Therefore, the first harmonic of magnetic flux density of the conventional magnet array is chosen in the optimization. The optimization is realized by reducing the higher harmonics. The shape of the curved surface is obtained by optimizing the height of small pieces. Generally, the main performance of the motors is reflected by the horizontal thrust. It is produced by the z component of the magnetic flux density. So the minimization of the higher harmonics of the z component is chosen to be the objective. For the conventional magnet array, the first harmonic of the magnetic flux density can be obtained when n takes 1 and expressed as

→ Bx1 ωz cos(ωx) B1 = = −µ0ωK1e , (10) Bz1 sin(ωx)

where Bx1 and Bz1 are the first harmonic of x and z component of magnetic flux density, respectively, K1 is the coefficient, mh is the height of the permanent magnet, mt and mb are the position of the top and bottom surfaces of the rectangular permanent magnet, respectively,

B 1 r −ωmt −ωmb K1 = e − e ∑(b(i, k) − a(i, k)), (11) 2µ0ω i=1

mh = mt − mb. (12)

In order to determine the harmonic numbers for optimizing, we take the Br, t, p, and mh parameters of the permanent magnet as 1.2 T, 20 mm, 10 mm, and 10 mm for analysis, respectively. The maximum of the magnetic flux density at 2 mm below the x-axis is about 1.677 × 10−5 T when k = 25, which is much less than the geomagnetic field (about 6 × 10−5 T). The harmonic components can be ignored in optimization if the harmonic numbers, k, are more than 25. Therefore, the approximate expression of the magnetic flux density can be represented by a certain number of harmonics. The z component is given by

m m λz Bz = −µ0ω ∑ Ke k sin(kωx), (13) k=1

where m = 25. The higher harmonic components of the new magnet array can be evaluated by

25 Bzh = Bz − Bz1. (14)

According to the periodicity of the magnetic ﬁeld, the region of half period is chosen. The region is placed 2 mm below the x-axis and divided into 41 points. The objective function is given according to Equations (10) and (14)

41 41

f (ht(i), hp(i)) = ∑ Bzh(xl, zl)/ ∑ Bz1(xl, zl) × 100%, (15) m=1 m=1

where xl = 0.025(m − 1)t and zl = −0.002 are the coordinate values. Sensors 2021, 21, 2522 6 of 17

From the objective function, Equation (15), the reduction of higher harmonics is a constrained nonlinear multivariable optimization problem. The sequential quadratic programming (SQP) algorithm has the advantages of good convergence, high computational efficiency, and strong boundary searchability, which is chosen to realize the optimization. In order to avoid the singular points and obtain better optimization results, the number of pieces of half of one permanent magnet is set to five after analysis. For the whole permanent magnet, the total number of small pieces is 10. It is enough to exhibit well the shape of the curved surface of the permanent magnet. The optimization parameters and variables are shown in Tables1 and2, respectively. Considering the assembly of the permanent magnets, the positions of the top surface of each piece (ht(1), ht(2), ... , ht(5)) remain the same, and the optimization variables are the positions of the bottom surface of each piece of half of one permanent magnet (hp(1), hp(2), ... , hp(5)). The ‘constraints’ in Table2 define the range of optimization variables. Since the constraints of the five optimization variables are the same, they are represented by hp(1)~hp(5). Considering the optimization parameters, the upper bound of optimization variables does not exceed the position of the top surface of the small pieces, i.e., 10 mm. The lower bound of optimization variables does not exceed the position of the objective region, i.e., −2 mm.

Table 1. Optimization parameters.

Parameters Symbol Value Unit pole pitch t 20 mm the side length of the magnets p 10 mm remanence of the permanent magnets Br 1.2 T the number of pieces of half of one permanent magnet n 5 - the positions of the top surface of each piece ht(1)~ht(5) 10 mm

Table 2. Optimization variables.

Optimization variables Constraints Unit

hp(1)~hp(5) [−2,10] mm

The results can be seen in Figure4, the minimization of the objective function is obtained. In Figure4, the ‘current point’ denotes the value of the optimization variables Sensors 2021, 21, x FOR PEER REVIEWand is the best point the solver found in its run. The ‘current function value’ is the7 value of 18 of

the objective function at the current point.

FigureFigure 4. The optimization iteration iteration and and variables. variables.

The small pieces of one permanent magnet based on the optimization variables are shown in Figure 5. In the new magnet array, the optimized curved surfaces of permanent magnets in the x and z magnetization directions are the same. The symmetry axis of the curved surface is in the middle and both sides have a symmetrical distribution. hp(1)~hp(5) are the optimization variables and also the positions of bottom surfaces of small pieces of half of one permanent magnet. So the small pieces of one permanent magnet can be obtained according to symmetry. Symmetry axis

x hp(1) hp(2) hp(3) hp(4) hp(5) Figure 5. The small magnet pieces, based on the optimization variables of one permanent magnet.

We take the permanent magnet whose x coordinate of the center is at zero as an example. The shape of the optimized curved surface of the permanent magnet is shown in Figure 6. The data used to construct the curved surface shape are obtained according to the optimization results. It can be seen that the shape is similar to a sine or power function. The least squares polynomial fitting is used to fit the data. By comparing the results of a polynomial of degrees 2, 3, and 4, the fitting of the polynomial of degree 4 is more accurate. The polynomial of degree 4 is chosen. For better expression and without affecting the accuracy, the first- and third-order terms with very small coefficients are removed, and the sum of squares due to error (SSE) is 1.376×10−10 m2. The computed data of the polynomial is also shown in Figure 6 and the expression is as follows ()=++42 f xaxaxa123 (16)

where a1=3.763×105, a2=34.28, and a3=-3.699×10-4 are the coefficients, and the value range of x is [−p/2, p/2].

Sensors 2021, 21, x FOR PEER REVIEW 7 of 18

Sensors 2021, 21, 2522 7 of 17

Figure 4. The optimization iteration and variables. The small pieces of one permanent magnet based on the optimization variables are Theshown small in Figurepieces5 of. In one the permanent new magnet magnet array, based the optimized on the optimiza curved surfacestion variables of permanent are shownmagnets in Figure in the5. Inx theand newz magnetization magnet array, directions the optimized are the curved same. surfaces The symmetry of permanent axis of the magnetscurved in the surface x and is z in magnetization the middle and directions both sides are have thea same. symmetrical The symmetry distribution. axis ofhp (1)~the hp(5) curvedare surface the optimization is in the middle variables and both and sides also thehave positions a symmetrical of bottom distribution. surfaces ofhp(1)~ smallhp(5) pieces are theof optimization half of one permanent variables and magnet. also the So posi the smalltions of pieces bottom of onesurfaces permanent of small magnet pieces of can be half ofobtained one permanent according magnet. to symmetry. So the small pieces of one permanent magnet can be

Symmetry axis

x hp(1) hp(2) hp(3) hp(4) hp(5)

FigureFigure 5. The 5. smallThe smallmagnet magnet pieces, pieces, based basedon the on opti themization optimization variables variables of one of permanent one permanent magnet. magnet.

We takeWe the take permanent the permanent magnet magnet whose whose x coordinatex coordinate of the center of the is center at zero is as at an zero ex- as an ample.example. The shape The of shape the optimized of the optimized curved curved surface surface of the ofpermanent the permanent magnet magnet is shown is shown in in FigureFigure 6. The6. Thedata data used used to construct to construct the the curved curved surface surface shape shape are are obtained obtained according according to to the the optimizationoptimization results. results. It can It can be beseen seen that that the theshape shape is similar is similar to a sine to a sineor power or power function. function. The leastThe leastsquares squares polynomial polynomial fitting fitting is used is usedto fit tothe fit data. the data.By comparing By comparing the results the results of a of a polynomialpolynomial of degrees of degrees 2, 3, 2,and 3, and4, the 4, thefitting fitting of the of thepolynomial polynomial of degree of degree 4 is 4 ismore more accu- accurate. rate. The polynomial polynomial of of degree degree 4 4is ischosen. chosen. For For better better expression expression and andwithout without affecting affecting the the accuracy,accuracy, the first- the first-and third-order and third-order terms terms with withvery verysmall small coefficients coefficients are removed, are removed, and and −10 2 the sumthe of sum squares of squares due to due error to (SSE) error is (SSE) 1.376×10 is 1.376−10 m2×. The10 computedm . The data computed of the polyno- data of the mial polynomialis also shown is in also Figure shown 6 and in Figure the expression6 and the expressionis as follows is as follows ()=++424 2 f fxaxaxa(x) =123a1x + a2x + a3 (16) (16)

Sensors 2021, 21, x FOR PEER REVIEW 5 × 5 -4 − × −4 8 of 18 wherewhere a1=3.763×10a1 = 3.763, a2=34.28,10 , anda2 = a3 34.28,=-3.699×10 and a3 are= the3.699 coefficients,10 are and the the coefﬁcients, value range and of the x is [−valuep/2, p/2]. range of x is [−p/2, p/2].

0.8 Optimization results 0.6 f(x)

0.4

0.2 z (mm)

−0.2

−0.4 -5−4−3−2−1012345 x (mm) FigureFigure 6.6.The The polynomialpolynomial ﬁttingfitting datadataf f((xx)) andand thethe optimizationoptimization resultsresults forfor oneone permanentpermanent magnet.magnet.

According to the optimization and polynomial fitting, the analytical model of the magnetic flux density is simple and expressed as Equation (10).

3.2. Verification with FEM The new magnet array with a curved surface is obtained by applying the optimization results, and the axonometric drawing is shown in Figure 7. In order to verify the optimization, the magnetic flux density with different z coordinates are compared with the finite element model (FEM). The FEM is built and analyzed by the Ansoft Maxwell. With precision-driven adaptive subdivision technology and a powerful post-processor, Ansoft Maxwell is an excellent high-performance electromagnetic design software in the industry.

Figure 7. The axonometric drawing of the new magnet array with a curved surface.

The magnetic flux density is obtained by the FEM and analytical model. The x and z components are shown in Figure 8. It is found that the magnetic flux density has a good sinusoidal waveform. The analytical model results fit very well with the FEM results. The total harmonic distortion (THD) [23] is introduced to evaluate the magnetic flux density. The root mean square (RMS) values of the error between the analytical model and FEM and THD of magnetic flux density are shown in Table 3.

Sensors 2021, 21, x FOR PEER REVIEW 8 of 18

0.8 Optimization results 0.6 f(x)

0.4

0.2 z (mm)

−0.2

−0.4 -5−4−3−2−1012345 x (mm) Sensors 2021, 21, 2522 8 of 17 Figure 6. The polynomial fitting data f(x) and the optimization results for one permanent magnet.

According to the optimization and polynomial fitting, the analytical model of the magneticAccording flux density to the is optimizationsimple and expressed and polynomial as Equation fitting, (10). the analytical model of the magnetic flux density is simple and expressed as Equation (10). 3.2. Verification with FEM 3.2. VerificationThe new magnet with FEM array with a curved surface is obtained by applying the optimization results,The new and magnet the axonometric array with drawing a curved is surface shown isin obtained Figure 7. by In applyingorder to verify the optimization the op- results,timization, and the the magnetic axonometric flux density drawing with is different shown inz coordinates Figure7. In are order compared to verify with the the optimization,finite element the model magnetic (FEM). flux The density FEM is with built different and analyzedz coordinates by the Ansoft are compared Maxwell. With with the finiteprecision-driven element model adaptive (FEM). subdivision The FEM techno is builtlogy and and analyzed a powerful by thepost-processor, Ansoft Maxwell. Ansoft With Maxwell is an excellent high-performance electromagnetic design software in the indus- precision-driven adaptive subdivision technology and a powerful post-processor, Ansoft try. Maxwell is an excellent high-performance electromagnetic design software in the industry.

Figure 7. TheThe axonometric axonometric drawing drawing of of the the new new ma magnetgnet array array with with a curved a curved surface. surface.

The magnetic magnetic flux flux density density is is obtained obtained by by the the FEM FEM and and analytical analytical model. model. The Thex andx andz z components are are shown shown in in Figure Figure 8.8 .It It is is foun foundd that that the the magnetic magnetic flux flux density density has has a good a good sinusoidal waveform. waveform. The The analytical analytical model model results results fit fitvery very well well with with the theFEM FEM results. results. The The total harmonic distortion (THD) [23] is introduced to evaluate the magnetic flux density. Sensors 2021, 21, x FOR PEER REVIEW total harmonic distortion (THD) [23] is introduced to evaluate the magnetic flux density. 9 of 18 TheThe root mean square square (RMS) (RMS) values values of of the the error error between between the the analytical analytical model model and andFEM FEM and THD of of magnetic magnetic flux flux density density are are shown shown in inTable Table 3.3 .

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 FEM (z=−1 mm) 0 FEM (z=−1 mm) Analytical model (z=−1 mm) Analytical model (z=−1 mm) FEM (z=−2 mm) −0.2 −0.2 FEM (z=−2 mm) Analytical model (z=−2 mm) Analytical model (z=−2 mm) FEM (z=−3 mm) FEM (z=−3 mm) −0.4 Analytical model (z=−3 mm) −0.4 Analytical model (z=−3 mm) FEM (z=−4 mm) FEM (z=−4 mm) Analytical model (z=−4 mm) Analytical model (z=−4 mm) −0.6 FEM (z=−5 mm) −0.6 FEM (z=−5 mm) Analytical model (z=−5 mm) Analytical model (z=−5 mm) −0.8 −0.8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 x (m) x (m) (a) (b)

Figure 8. TheFigure magnetic 8. The flux magnetic density ﬂux for density different for differentz coordinates:z coordinates: (a) the x (a component;) the x component; (b) the (b )z thecomponent.z component.

TableTable 3. The 3. RMSThe RMS values values of the of the error error between between the the analytical analytical modelmodel and and FEM FEM and and THD THD of magneticof magnetic ﬂux density.flux density.

x Ratio of Peak Value x z Ratio of Peak Value z z (mm)RMS( RMS∆B ) ( ∆ Bx) (T)Ratio of peak value THD(THD (BBx)) (%) RMSRMS( (∆B∆zB) (T)) Ratio of peak valueTHD (BTHD(z) (%) B ) z (mm) of Bx of FEM (%) of Bz of FEM (%) (T) of Bx of FEM (%) (%) (T) of Bz of FEM (%) (%) −1 0.0159 2.19 2.69 0.0154 2.14 2.69 −1 −20.0159 0.00942.19 1.50 2.69 1.43 0.0154 0.0075 1.202.14 1.43 2.69 −2 −30.0094 0.00751.50 1.40 1.43 0.76 0.0075 0.0058 1.081.20 0.76 1.43 −3 −40.0075 0.00631.40 1.38 0.76 0.41 0.0058 0.0059 1.281.08 0.41 0.76 −5 0.0060 1.52 0.22 0.0051 1.29 0.22 −4 0.0063 1.38 0.41 0.0059 1.28 0.41 −5 0.0060 1.52 0.22 0.0051 1.29 0.22

The RMS(ΔBx), RMS(ΔBz), THD(Bx) and THD(Bz) values for different z coordinates are very small. The maximum RMS(ΔBx) and RMS(ΔBz) values occurred when z is -1 mm, which are 2.19% and 2.14% of the peak value of Bx and Bz of FEM, respectively. The THD(Bx) and THD(Bz) values are the same according to Equation (7). The maximum THD(Bx) and THD(Bz) values occurred when z is -1 mm, which are both 2.69%. As a result, a new permanent magnet array with a good sinusoidal magnetic field and a simple analytical model of magnetic flux density is obtained.

4. Discussion

4.1. The Applicability of Optimization Results 4.1.1. Different Lengths of Permanent Magnet In this section, the applicability of optimization results will be discussed when the dimensions of the permanent magnet have changed. The optimization results are obtained with the specific parameters of the permanent magnet array. The fit expression must be improved for different dimensions of the permanent magnet, and the new expression is described as p ()=++42 1 fx00() axaxa 10 20 3 , (17) p

where p1 is the side length of new permanent magnets which are magnetized in z and x directions, x1 is the new x coordinate, and the value range is [-p1/2, p1/2], = x01xpp 1 (18)

We take p1 as 5 mm, 20 mm, 30 mm, and 40 mm, respectively, as an example to verify the new fit expression. Figure 9 shows the results of optimization and the computed data of the new fit expression. It can be seen that the new fit expression has good consistency

Sensors 2021, 21, 2522 9 of 17

The RMS(∆Bx), RMS(∆Bz), THD(Bx) and THD(Bz) values for different z coordinates are very small. The maximum RMS(∆Bx) and RMS(∆Bz) values occurred when z is −1 mm, which are 2.19% and 2.14% of the peak value of Bx and Bz of FEM, respectively. The THD(Bx) and THD(Bz) values are the same according to Equation (7). The maximum THD(Bx) and THD(Bz) values occurred when z is −1 mm, which are both 2.69%. As a result, a new permanent magnet array with a good sinusoidal magnetic ﬁeld and a simple analytical model of magnetic ﬂux density is obtained.

4. Discussion 4.1. The Applicability of Optimization Results 4.1.1. Different Lengths of Permanent Magnet In this section, the applicability of optimization results will be discussed when the dimensions of the permanent magnet have changed. The optimization results are obtained with the speciﬁc parameters of the permanent magnet array. The ﬁt expression must be improved for different dimensions of the permanent magnet, and the new expression is described as p f (x ) = a x 4 + a x 2 + a 1 , (17) 0 0 1 0 2 0 3 p

where p1 is the side length of new permanent magnets which are magnetized in z and x directions, x1 is the new x coordinate, and the value range is [−p1/2, p1/2],

x0 = x1 p/p1 (18)

Sensors 2021, 21, x FOR PEER REVIEW 10 of 18 We take p1 as 5 mm, 20 mm, 30 mm, and 40 mm, respectively, as an example to verify the new fit expression. Figure9 shows the results of optimization and the computed data of the new fit expression. It can be seen that the new fit expression has good consistency withwith thethe optimizationoptimization results.results. Therefore,Therefore, thethe newnew fitfit expressionexpression has has good good applicability applicability to to thethe permanentpermanent magnetmagnet with with different different side side lengths. lengths. z (mm)

Figure 9. The optimization results and fitting data f0(x0) for one permanent magnet with different Figure 9. The optimization results and ﬁtting data f 0(x0) for one permanent magnet with different p1 values.p1 values.

4.1.2.4.1.2. DifferentDifferent HeightsHeights ofof PermanentPermanent MagnetMagnet TheThe applicabilityapplicability of optimization optimization results results is is also also analyzed analyzed when when the the height height of the of per- the permanentmanent magnet magnet is ischanged. changed. The The magnetic magnetic fl ﬂuxux density density of of the the analytical model andand FEMFEM areare comparedcompared when ht((ii))( (ii=1,2,…,5)= 1, 2, ... takes, 5) takes 5 mm 5 and mm 15 and mm, 15 respectively, mm, respectively, which whichare shown are shownin Figure in Figures10 and Figure10 and 11.11. From Figures 10 and 11, the magnetic flux density also shows a good sinusoidal waveform. The analytical model results fit very well with the FEM results. The maximum RMS(ΔBx) and RMS(ΔBz) values are 0.0151 T and 0.0139 T, and 3.03% and 2.78% of the peak value of Bx and Bz of FEM when ht(i) (i=1,2,…,5) takes 5 mm, respectively. The maximum RMS(ΔBx) and RMS(ΔBz) values are 0.0183 T and 0.0168 T, and 2.22% and 2.06% of the peak value of Bx and Bz of FEM when ht(i) (i=1,2,…,5) takes 15 mm, respectively. The maximum THD(Bx) and THD(Bz) values also occur when z takes -1 mm, which are 3.53% and 2.37% of the peak value when ht(i) (i=1,2,…,5) takes 5 mm and 15 mm respectively. Therefore, the new fit expression has good applicability to the permanent magnet with different heights.

) =5 mm ) i ( (i)=5 mm) t t h (

) T ( Bx Bz (T) (h (T) Bz

(a) (b)

Figure 10. The magnetic flux density in different z coordinates when ht(i) (i=1,2,…,5) takes 5 mm: (a) the x component; (b) the z component.

Sensors 2021, 21, x FOR PEER REVIEW 10 of 18

with the optimization results. Therefore, the new fit expression has good applicability to the permanent magnet with different side lengths. z (mm)

Figure 9. The optimization results and fitting data f0(x0) for one permanent magnet with different p1 values.

4.1.2. Different Heights of Permanent Magnet The applicability of optimization results is also analyzed when the height of the permanent magnet is changed. The magnetic flux density of the analytical model and FEM are compared when ht(i) (i=1,2,…,5) takes 5 mm and 15 mm, respectively, which are shown in Figure 10 and Figure 11. From Figures 10 and 11, the magnetic flux density also shows a good sinusoidal waveform. The analytical model results fit very well with the FEM results. The maximum RMS(ΔBx) and RMS(ΔBz) values are 0.0151 T and 0.0139 T, and 3.03% and 2.78% of the peak value of Bx and Bz of FEM when ht(i) (i=1,2,…,5) takes 5 mm, respectively. The maximum RMS(ΔBx) and RMS(ΔBz) values are 0.0183 T and 0.0168 T, and 2.22% and 2.06% of the peak value of Bx and Bz of FEM when ht(i) (i=1,2,…,5) takes 15 mm, respectively. The maximum THD(Bx) and THD(Bz) values also occur when z takes -1 mm, which are 3.53% and 2.37% of the peak value when ht(i) (i=1,2,…,5) takes 5 mm and 15 mm respectively. Sensors 2021, 21, 2522 10 of 17 Therefore, the new fit expression has good applicability to the permanent magnet with different heights.

) =5 mm ) i ( (i)=5 mm) t t h (

) T ( Bx Bz (T) (h (T) Bz

(a) (b) Sensors 2021, 21, x FOR PEER REVIEW 11 of 18 FigureFigure 10. The 10. magneticThe magnetic flux ﬂux density density in indifferent different zz coordinatescoordinateswhen whenh th(it()(i)i (=i=1,2,…,5) 1, 2, ... , 5) takes takes 5 5 mm: mm: ( a) thethe xxcomponent; component; (b) the z component.(b) the z component.

) 5 mm 5 1 = ) i ( (i)=15 mm) t t h (

) T ( Bz Bx (T) (h

(a) (b)

FigureFigure 11. The 11. magneticThe magnetic flux ﬂux density density in in different different zz coordinatescoordinates when whenht h(it)((i)i (=i=1,2,…,5) 1, 2, ... , 5) takes takes 15 15 mm: (aa)) thethex xcomponent; component; (b) the z (component.b) the z component.

TheFrom applicability Figures 10 of and optimization 11, the magnetic results flux is verified density alsoby analyzing shows a good the new sinusoidal magnet ar- waveform. The analytical model results fit very well with the FEM results. The maximum rays with different lengths and heights of permanent magnets. Therefore, the new magnet RMS(∆B ) and RMS(∆B ) values are 0.0151 T and 0.0139 T, and 3.03% and 2.78% of the array can bex flexibly designedz according to the feature. As long as the shape of the bottom peak value of Bx and Bz of FEM when ht(i)(i = 1, 2, ... , 5) takes 5 mm, respectively. The curved surface is consistent with the optimization results, the desired magnetic field maximum RMS(∆Bx) and RMS(∆Bz) values are 0.0183 T and 0.0168 T, and 2.22% and 2.06% strengthof the peakcan valuebe obtained of Bx and byB zsimplyof FEM changing when ht(i)( thei = 1,size 2, ... of ,the 5) takes permanent 15 mm, respectively.magnet and the magneticThe maximum field will THD( stillB xhave) and good THD( Bsinusoidalz) values also characteristics. occur when z takes −1 mm, which are 3.53% and 2.37% of the peak value when ht(i)(i = 1, 2, ... , 5) takes 5 mm and 15 mm 4.2.respectively. The Effect on Therefore, the Mass the new fit expression has good applicability to the permanent magnet with different heights. In the new magnet array, the cross-section of the permanent magnet is no longer a The applicability of optimization results is verified by analyzing the new magnet rectanglearrays withwhen different the curved lengths surface andheights is applied. of permanent It is necessary magnets. to analyze Therefore, the theeffect new on the mass.magnet The arraytotal mass can be can flexibly be obtained designed by according summing to up the the feature. mass As of longall permanent as the shape magnets. of It isthe assumed bottom curved that there surface is no is consistent difference with between the optimization permanent results, magnets, the desired so the magnetic effect on the massfield can strength be indicated can be obtainedby analyzing by simply one permanent changing the magnet. size of the By permanentanalyzing magnetthe cross-section and of thethe magneticpermanent field magnet, will still havethe change good sinusoidal of mass characteristics.can be obtained by integrating equation f0(x0). The mass decreases when the result is positive, otherwise, it increases. The integral is expressed as

p1 p 2 2 () =+42 + 1 Ifxdxapapa=p () 3 20 240 . f0 − 1 00 0 1 2 3 (19) 2 240p The area ratio of the cross-section between the new permanent magnet and the rectangular permanent magnet is analyzed. The area of the cross-section of the rectangular permanent magnet can be described as × 2 Achh==pkpkp111, (20)

where kh×p1 is the height of the cross-section and kh is the coefficient. The ratio is obtained by I 5 f0 ×=42 + + Rapapam = 100%() 312 20 240 3. (21) Ach12 pk

It is found that the ratio is independent of the parameter p1. The parameter p is set to 10 mm in the design of the surface. The ratio is a fixed value when kh is given. Taking kh =1 as an example, the ratio value is -0.372%. The increase is very little and has almost no effect on the mass.

4.3. The Effect on the Air Gap

Sensors 2021, 21, 2522 11 of 17

4.2. The Effect on the Mass In the new magnet array, the cross-section of the permanent magnet is no longer a rectangle when the curved surface is applied. It is necessary to analyze the effect on the mass. The total mass can be obtained by summing up the mass of all permanent magnets. It is assumed that there is no difference between permanent magnets, so the effect on the mass can be indicated by analyzing one permanent magnet. By analyzing the cross-section of the permanent magnet, the change of mass can be obtained by integrating equation f 0(x0). The mass decreases when the result is positive, otherwise, it increases. The integral is expressed as

Z p1 2 2 4 2 p1 If = f0(x0)dx0 = 3a1 p + 20a2 p + 240a3 . (19) 0 p1 − 2 240p The area ratio of the cross-section between the new permanent magnet and the rectangular permanent magnet is analyzed. The area of the cross-section of the rectangular permanent magnet can be described as

2 Ac = p1 × kh p1 = kh p1 , (20)

where kh × p1 is the height of the cross-section and kh is the coefﬁcient. The ratio is obtained by

I f0 4 2 5 Rm = × 100% = 3a1 p + 20a2 p + 240a3 . (21) Ac 12pkh

It is found that the ratio is independent of the parameter p1. The parameter p is set to 10 mm in the design of the surface. The ratio is a ﬁxed value when kh is given. Taking kh = 1 as an example, the ratio value is −0.372%. The increase is very little and has almost no effect on the mass.

4.3. The Effect on the Air Gap Due to the curved surface of permanent magnets, the bottom of the new magnet array is not flat. A part of the bottom curved surface is below the x-axis, according to the optimization results. Compared with the conventional magnet array, the actual air gap between the magnet and the coil is a little smaller. The magnetic field strength will be changed when the air gap remains the same as the conventional magnet array. The influence of the curved surface on the air gap is investigated further. The lowest point of the curved surface is obtained when x0 is 0 mm in f 0(x0). To keep the same air gap, the new z coordinate value, at which the magnetic flux density is calculated, is a p z = z + f (x = 0) = z + 3 1 , (22) 0 0 0 0 p

where z0 is the original coordinate value, such as −1 mm, or −2 mm, and so on. The magnetic ﬂux density is proportional to the expression of eωz according to Equa- tion (10). The ratio of the magnetic ﬂux density calculated at the new z coordinate value to the original one is expressed as

ω z ω z Rz = e 1 /e 1 0 × 100%, (23)

where ω1 is redeﬁned for different side lengths of the permanent magnet, π ω1 = . (24) 2p1

The ratio is also a ﬁxed value and independent of the parameter p1, which equals π/2/p×a e 3. The value of the ratio is 94.36% for different lengths of the permanent magnet. Sensors 2021, 21, x FOR PEER REVIEW 12 of 18

Due to the curved surface of permanent magnets, the bottom of the new magnet array is not flat. A part of the bottom curved surface is below the x-axis, according to the optimization results. Compared with the conventional magnet array, the actual air gap between the magnet and the coil is a little smaller. The magnetic field strength will be changed when the air gap remains the same as the conventional magnet array. The influence of the curved surface on the air gap is investigated further. The lowest point of the curved surface is obtained when x0 is 0 mm in f0(x0). To keep the same air gap, the new z coordinate value, at which the magnetic flux density is calculated, is ap zz=+ fx() ==+0 z 31, (22) 000 0 p

where z0 is the original coordinate value, such as -1 mm, or -2 mm, and so on. The magnetic flux density is proportional to the expression of eωz according to Equa- tion (10). The ratio of the magnetic flux density calculated at the new z coordinate value to the original one is expressed as

ω ω =×1z 10z Reez 100% , (23)

where ω1 is redefined for different side lengths of the permanent magnet, π ω = 1 . (24) 2p1 Sensors 2021, 21, 2522 12 of 17 The ratio is also a fixed value and independent of the parameter p1, which equals × eπ/2/p a3. The value of the ratio is 94.36% for different lengths of the permanent magnet. Considering the air gap, the magnetic flux density is slightly decreased. In other words, Considering the air gap, the magnetic flux density is slightly decreased. In other words, the the new magnet array has a good sinusoidal magnetic field in a very small air gap, sacri- new magnet array has a good sinusoidal magnetic field in a very small air gap, sacrificing ficing very little magnetic flux density, which is especially suitable for precision position- very little magnetic flux density, which is especially suitable for precision positioning ing apparatus with strict requirements for a small air gap. apparatus with strict requirements for a small air gap.

4.4.4.4. The The Effect Effect on on the the Force Force InIn order order to to investigate investigate the the effect effect on on the the fo forcerce of of the the new new magnet magnet array, array, the the force force and and forceforce ripple ripple of of the the new new magnet magnet array array and and the the conventional conventional magnet magnet array array are are compared. compared. In In thethe analytical analytical model model for for real-time real-time control, control, the the expressions expressions of of the the magnetic magnetic flux ﬂux density density of of thethe two two magnet magnet arrays arrays are are the the same same and and obtained obtained by by Equation Equation (10). (10). WeWe take take the the linear linear motor motor with with a amoving moving magnet magnet array array as as an an example. example. Figure Figure 12 12 shows shows thethe axonometric axonometric diagram diagram of of the the motor motor with with the the new new magnet magnet array array and and coils. coils. The The motor motor withwith the the conventional conventional magnet magnet array array is is similar similar and and the the diagram diagram is is not not given. given.

Sensors 2021, 21, x FOR PEER REVIEW 13 of 18

FigureFigure 12. 12. AnAn axonometric axonometric diagram diagram of of the the motor motor with the new magnet arrayarray andandthe the three-phase three-phase coil. coil. For the convenience of analysis, Figure 13 shows the cross-section of the motor with For the convenience of analysis, Figure 13 shows the cross-section of the motor with the new magnet array and the conventional magnet array. In order to remove the position the new magnet array and the conventional magnet array. In order to remove the position dependency and generate the position-independent force on the array, the dq0 transforma- dependency and generate the position-independent force on the array, the dq0 transfor- tion and the three-phase coil are applied. mation and the three-phase coil are applied. z

x p h p c t a d c dc cw Coil ACoil B Coil C (a)

x p t Coil ACoil B Coil C (b)

Figure 13. A cross-section of the motor with three-phase coils: (a) the new magnet array; (b) the Figure 13. A cross-section of the motor with three-phase coils: (a) the new magnet array; (b) the conventional magnet array. conventional magnet array.

By using the magnetic flux density of Equation (10), the Lorentz force on the magnet array, generated by one coil, is calculated by solving the volume integral and expressed as ()ω Fxcxωωcos co =−NIK() ecctb e  F ()ω , (25) Fxczsin co where ωω()+−() 8mK ccdw cc wd K = l 1 sin sin F ()− ω2 (26) cchw c d 44 =− ccchtb (27)

where N is the number of coil turns, I is the current in the coil, KF is the coefficient, ct and cb are the position of the top and bottom surfaces of the coil, respectively. xco is the coordinate of the coil center in the x-direction, ml is the length of the magnet array in the y- direction, cd is the width of the air core of the coil, cw is the width of the coil, and ch is the height of the coil. For the three-phase coil group, the force can be expressed as

 Fx ωω  ==()cctb − FNKeei FFT Fz ()ωωω () () (28) ωωcosxxxabc cos cos  =−NK() ecctb e i F ()ωωω () () sinxxxabc sin sin where  = []T iiiiabc (29)

Sensors 2021, 21, 2522 13 of 17

By using the magnetic ﬂux density of Equation (10), the Lorentz force on the magnet array, generated by one coil, is calculated by solving the volume integral and expressed as

( ) Fcx ωct ωcb cos ωxco = NIKF(e − e ) , (25) Fcz sin(ωxco)

where 8m K ω(c + c ) ω(c − c ) = l 1 d w w d KF 2 sin sin (26) ch(cw − cd)ω 4 4

ch = ct − cb (27)

where N is the number of coil turns, I is the current in the coil, KF is the coefﬁcient, ct and cb are the position of the top and bottom surfaces of the coil, respectively. xco is the coordinate of the coil center in the x-direction, ml is the length of the magnet array in the y-direction, cd is the width of the air core of the coil, cw is the width of the coil, and ch is the height of the coil. For the three-phase coil group, the force can be expressed as

→ → Fx F = = NK (eωct − eωcb )T i F F F z (28) ( ) ( ) ( ) → ωct ωc cos ωxa cos ωxb cos ωxc = NKF(e − e b ) i sin(ωxa) sin(ωxb) sin(ωxc)

where → T i = ia ib ic (29)

xb = xa + dc (30)

xc = xa + 2dc (31) 2t d = (32) c 3

where TF is the matrix, xa, xb, and xc are the coordinates of the coil center of the three-phase coil in the x-direction, and dc is the distance between two coil centers. The dq0 transformation matrix is used to remove the position dependency and expressed as

 2 2  cos(ωx) cos ωx + 3 π cos ωx − 3 π 2 2 2 Tdq0 =  − sin(ωx) − sin ωx + 3 π − sin ωx − 3 π  (33) 3 1 1 1 2 2 2 By applying the dq0 transformation matrix, the force of the three-phase coil group is replaced as

 i  → d 3 Fx ωct ωcb −1 ωct ωcb id F = = NKF(e − e )TFT  iq  = NKF(e − e ) (34) F dq0 2 −i z 0 q

So the current of the coil is expressed as     → ia id −1 i =  ib  = Tdq0 iq  (35) ic 0

From Equation (34), the levitation force and horizontal thrust are produced by the d axis and q axis current, respectively. Therefore, the position-independent force on the array generated by the three-phase coil is obtained. Sensors 2021, 21, 2522 14 of 17

The force and force ripple of the two magnet arrays are analyzed and compared. The dimensions of the magnet array shown in Table1 are used, and the other dimensions of the motor are given in Table4.

Table 4. The dimensions of the motor.

Parameters Symbol Value Unit

air gap ap 1 mm the length of the magnet array in the y-direction ml 40 mm the height of the coil ch 4 mm the width of the coil cw 12 mm the width of the air core of the coil cd 2 mm the distance between two coil centers dc 13.33 mm the number of coil turns N 100 -

Sensors 2021, 21, x FOR PEER REVIEW 15 of 18

The force of the analytical model can be directly calculated by Equation (34) when id and iq are given. In order to better analyze the effect on the force of the two magnet arrays, the forces of the two magnet arrays calculated by the FEM and analytical model are shown in Figure 14 when the magnet array moves in one period. The id and iq take -5 A and 0 A in Figure 14 when the magnet array moves in one period. The id and iq take −5 A and 0respectively. A respectively.

Figure 14. The force calculated by the FEM, the analytical model of the new magnet array, and the Figure 14. The force calculated by the FEM, the analytical model of the new magnet array, and the conventional magnet array in x, y, and z directions. conventional magnet array in x, y, and z directions.

It is found that the Fx ripple and Fz ripple of the new magnet array are much smaller x z thanIt the is conventionalfound that the magnet F ripple array. and The F ripple disturbing of the force, new magnetFy, is produced array are by much the end smaller ofthan the the coil conventional due to the end magnet effect ofarray. magnetic The disturbing flux density force, according Fy, is produced to the Lorentz by the force end of formula.the coil due The toFy theripple end for effect both of magnet magnetic arrays flux is smalldensity and according essentially to zero.the Lorentz The Fx of force the for- newmula. magnet The F arrayy ripple is slightlyfor both smaller magnet than arrays theforce is small of the and conventional essentiallymagnet zero. The array Fx of which the new canmagnet be inferred array is from slightly the above smaller analysis than ofthe the force effect of on the the conventional air gap. magnet array which can beThe inferred new magnet from array the above is built analysis based on of the the fitting effect curve, on the and air the gap. magnetic flux density of theThe new new magnet magnet array array is expressed is built inbased Equation on the (10), fitting based curve, on the and analysis the magnetic in Section flux3. It den- cansity beof seenthe new from magnet Figure array7 that is the expressed magnetic in flux Equation density (10), in different based onz thecoordinates analysis hasin Section a very3. It can small be error seen between from Figure the analytical 7 that the model magnetic and FEM,flux density but the accumulativein different z errorcoordinates will be has produceda very small in the error force between calculation the analytical due to the model volume and integral FEM, by but using the theaccumulative analytical model. error will Therefore, the correction coefficient, K , is introduced to establish the analytical model of be produced in the force calculationc due to the volume integral by using the analytical force for the new magnet array. The force of the new magnet array is modified as follows model. Therefore, the correction coefficient, Kc, is introduced to establish the analytical model of force for the new magnet array. The force of the new magnet array is modified as follows

 Fxnew 3 ωωid FNKKee==()cctb − new c F −i (36) Fznew 2 q where F = xFEM Kc (37) Fx

where Fx is the x component of the force obtained by Equation (34),⎯FxFEM is the average value of the x component of the force obtained by the FEM in one period. The force of the analytical model and RMS values of the error between the analytical model and FEM for the two magnet arrays are shown in Table 5.

Table 5. The analytical force and RMS values for the two magnet arrays.

RMS(∆Fx) RMS(∆Fy) RMS(∆Fz) Magnet array Fx (N) Fy (N) Fz (N) (N) (N) (N)

Sensors 2021, 21, 2522 15 of 17

→ 3 Fxnew ωct ωcb id F new = = NKcKF(e − e ) (36) Fznew 2 −iq where FxFEM Kc = (37) Fx

where Fx is the x component of the force obtained by Equation (34), FxFEM is the average value of the x component of the force obtained by the FEM in one period. The force of the analytical model and RMS values of the error between the analytical model and FEM for the two magnet arrays are shown in Table5.

Sensors 2021, 21, x FOR PEER REVIEW 16 of 18 Table 5. The analytical force and RMS values for the two magnet arrays.

Magnet Array Fx (N) RMS(∆Fx) (N) Fy (N) RMS (∆Fy) (N) Fz (N) RMS(∆Fz) (N) The conventional Theconventional magnet array 16.5901 0.228016.5901 0.2280 0 0 0.0244 0.0244 0 0 0.2611 0.2611 The newmagnet array magnet15.8390 array 0.0365 0 0.0138 0 0.0829 The new magnet 15.8390 0.0365 0 0.0138 0 0.0829 array From Table5, the Fx value of the new magnet array is 95.47% of the conventional From Table 5, the Fx value of the new magnet array is 95.47% of the conventional magnet array. The RMS(∆Fy) for the two magnet arrays are all very small. The RMS(∆Fx) magnet array. The RMS(∆Fy) for the two magnet arrays are all very small. The RMS(∆Fx) and RMS(∆F ) values of the new magnet array are reduced a lot, which are 16.01% and and RMS(∆Fz)z values of the new magnet array are reduced a lot, which are 16.01% and 31.75% of the conventional magnet magnet array. array. Therefore, Therefore, the the new new magnet magnet array array can can signifi- significantly reducecantly reduce the force the ripple force withoutripple without sacrificing sacrif tooicing much too forcemuch in force comparison in comparison to the conventionalto the magnetconventional array. magnet array. InIn order order to to better better observe observe the the advantages advantages of of the the new new magnet magnet array array in inthe the small small air air gap andgap and verify verify the the applicability applicability of of the the correction correction coefficient, coefficient, thethe airair gap,gap, ap,, is is changed changed from 1from mm 1 to mm 0.5 to mm. 0.5 Themm. force The force of the of new the magnetnew magnet array array calculated calculated by theby FEMthe FEM and and analytical modelanalytical based model on based the new on the air new gap air is gap shown is sh inown Figure in Figure 15. The15. The force force of of the the analytical analytical model andmodel RMS and values RMS values of the of error the error between between the analyticalthe analytical model model and and FEM FEM for for the the new new magnet arraymagnet based array on based the on new the air new gap air is gap shown is shown in Table in Table6. 6.

Figure 15. 15. TheThe force force of of the the new new magnet magnet array array calculated calculated by the by FEM the FEMand analytical and analytical model, model, based based on theon the new new air air gap. gap.

Table 6. The analyticalTable 6. force The analytical and RMS force values and for RMS the values newmagnet for the new array, magnet based array, on the based new on air the gap. new air gap.

RMS(∆Fx) RMS(∆Fy) RMS(∆Fz) Magnet Array FxMagnet(N) array RMS( ∆FxF)x (N) Fy (N)Fy (N) RMS( ∆Fy) (N) FFzz (N)(N) RMS(∆Fz) (N) (N) (N) (N) The new magnet array 17.1322 0.0527 0 0.0116 0 0.0857 The new magnet 17.1322 0.0527 0 0.0116 0 0.0857 array

It is found that the new magnet array still has a small force ripple with the small air gap from Figure 15. The correction coefficient is verified by analyzing the RMS values of the error between the analytical model and FEM from Table 6. The RMS(∆Fy) of the new magnet array is also very small. Compared with the conventional magnet array, with the air gap of 1 mm, the force of the new magnet array is higher and the RMS(∆Fx) and RMS(∆Fz) values are also relatively low, which are 23.11% and 32.82% of the conventional magnet array.

Sensors 2021, 21, 2522 16 of 17

It is found that the new magnet array still has a small force ripple with the small air gap from Figure 15. The correction coefficient is verified by analyzing the RMS values of the error between the analytical model and FEM from Table6. The RMS( ∆Fy) of the new magnet array is also very small. Compared with the conventional magnet array, with the air gap of 1 mm, the force of the new magnet array is higher and the RMS(∆Fx) and RMS(∆Fz) values are also relatively low, which are 23.11% and 32.82% of the conventional magnet array. To sum up, through the analysis of the above discussion, the new magnet array with a sinusoidal magnetic field is obtained. The curved surface based on the optimization results has good applicability to different dimensions of the permanent magnet. The new magnet array can be designed flexibly according to the feature. In contrast to the conventional magnet array, the new magnet array has little influence on the mass and air gap and can significantly reduce the force ripple. The new magnet array still maintains the small force ripple in the small air gap.

5. Conclusions A new 1D Halbach magnet array with a curved surface of the permanent magnet is proposed in this paper. The curved surface design method of the permanent magnet is realized based on the superposition principle. The shape of the optimized curved surface is similar to a sine or power function and fitted by a polynomial. The expression of magnetic flux density is simple and the same as the first harmonic of the conventional magnet array with rectangular permanent magnets. A good sinusoidal magnetic field is obtained in a very small air gap. The optimization results have good applicability to different dimensions of the permanent magnet. The effect on the mass of the permanent magnet array is very small. The magnetic flux density is slightly decreased when the air gap remains the same as the conventional magnet array. The new magnet array can significantly reduce the force ripple in comparison to the conventional magnet array. The new magnet array is especially suitable for precision positioning apparatus with strict requirements for a small air gap. The method of surface design of permanent magnet can be used for similar 2D permanent magnet arrays. In future work, a combined global-local optimization method will be further investigated to improve the optimization and identify the exact solution. Furthermore, the model structural design methodology will be studied to determine the fitted terms automatically.

Supplementary Materials: The following are available online at https://www.mdpi.com/article/10 .3390/s21072522/s1. Author Contributions: Conceptualization, G.L., and W.X.; methodology, G.L., and X.X.; software, S.H.; validation, G.L., and S.H.; formal analysis, G.L.; investigation, S.H.; resources, X.X.; data cura- tion, G.L.; writing—original draft preparation, G.L.; writing—review and editing, W.X.; visualization, G.L.; supervision, W.X.; project administration, W.X.; funding acquisition, W.X. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Shandong Provincial Natural Science Foundation (grant no. ZR2020ME204, ZR2019PEE031), and this work was supported by National Engineering Laboratory of Offshore Geophysical and Exploration Equipment. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available in the supplementary material, Data.zip. Conﬂicts of Interest: The authors declare no conﬂict of interest. Sensors 2021, 21, 2522 17 of 17

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