energies
Article Semi-Analytical Modeling and Analysis of Halbach Array
Min-Seob Sim and Jong-Suk Ro * School of Electrical and Electronics Engineering, Chung-Ang University, Seoul 06974, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-820-5557
Received: 5 February 2020; Accepted: 5 March 2020; Published: 8 March 2020
Abstract: Analysis of Halbach array placed in open space by using finite element method involves substantial consumption of memory, time, and cost. To address this problem, development of a mathematical modeling and analytic analysis method for Halbach array can be a solution, but research on this topic is currently insufficient. Therefore, a novel mathematical modeling and analytic analysis method for Halbach array in open space is proposed in this study, which is termed as the Ampere model and the Biot–Savart law (AB method). The proposed AB method can analyze the Halbach array rapidly and accurately with minimal consumption of memory. The usefulness of the AB method in terms of accuracy and memory and time consumption is verified by comparing the AB method with finite element method in this paper.
Keywords: Halbach permanent magnet array; magnetic field analysis; mathematical model
1. Introduction Halbach array (HA) was proposed by K. Halbach to maximize the efficiency in a given volume of a permanent magnet (PM) via the combination of PMs, which are magnetized in diverse directions [1]. By combining PMs appropriately, an effective magnetic flux path can be obtained without using magnetic cores. The combination of the PMs is termed as HA. By the appropriate arrangement of PMs, which are magnetized in different directions, it is possible to divert the magnetic flux in unnecessary areas to the required area. This renders enhancement of the magnetic flux in the required area and reduction of the same in the unnecessary area. Hence, maximization of the efficiency of a magnetic field system is possible with the usage of restricted volume of PMs [2–8]. A high permeability core material is commonly used to obtain an effective magnetic flux path. However, core losses, such as hysteresis loss and eddy current loss occur in the core and lower the efficiency of the magnetic field system. On the other hand, when the HA is used for the magnetic system, an effective magnetic flux path can be obtained without using the core, and the efficiency of the magnetic field system can be increased. Nevertheless, the cost of an electric machine could be increased when HA is used due to the manufacturing difficulties of HA arising from factors such as the bonding difficulties of PMs, which are magnetized in different directions, etc. However, as there are many merits of HA, it is widely used for diverse kinds of electric machines, such as, motor [9–15], maglev [16,17], eddy current brake [18–22], actuator [23,24], magnetic bearing [25–27], energy harvester [28–31], etc. Most studies on the analysis of HA have been focused on the case where the HA is placed with high permeability cores in a small air gap [9–29]. Moreover, HA was modelled and analyzed through the methods of the magnetic circuit method [9], image method [10,12], Fourier series [15], and Layer method [18,24]. The above methods are not applicable if the device contains no ferromagnetic core
Energies 2020, 13, 1252; doi:10.3390/en13051252 www.mdpi.com/journal/energies Energies 2020, 13, 1252 2 of 11
Energies 2020, 13, x 2 of 12 and has wide airgap. However, in some cases, the HA can be placed in open space and most of the flux can pass through a long distance in air such as when it is used in energy harvester around the magnetic flux can pass through a long distance in air such as when it is used in energy harvester transmission line [30,31]. around the transmission line [30,31]. The finite element method (FEM) can deal with the problem, but it needs much time and The finite element method (FEM) can deal with the problem, but it needs much time and memory memory to analyze. These days, the use of more efficient computational methods is of extreme to analyze. These days, the use of more efficient computational methods is of extreme importance to importance to reduce energy consumption in computing centers [32], and FEM cannot satisfy this reduce energy consumption in computing centers [32], and FEM cannot satisfy this issue. issue. To address this problem, a novel and useful mathematical modeling and analytic analysis method To address this problem, a novel and useful mathematical modeling and analytic analysis is proposed in this research for HA placed in open space. Specifically, the mathematical model is method is proposed in this research for HA placed in open space. Specifically, the mathematical derived for PM and HA by considering the Ampere model, Gilbert model, and Ampere’s circuital law. model is derived for PM and HA by considering the Ampere model, Gilbert model, and Ampere's The governing equations and analytical method for the analysis of HA are derived in this paper by circuital law. The governing equations and analytical method for the analysis of HA are derived in applying the Ampere model and the Biot–Savart law. Hence, the proposed method in this study is this paper by applying the Ampere model and the Biot–Savart law. Hence, the proposed method in termed as Ampere model and the Biot–Savart law (AB) method. this study is termed as Ampere model and the Biot–Savart law (AB) method. The usefulness of the proposed mathematical model and analysis method in terms of accuracy The usefulness of the proposed mathematical model and analysis method in terms of accuracy and time consumption is validated by comparison with the finite element method (FEM). and time consumption is validated by comparison with the finite element method (FEM). 2. Structure and Working Principle of HA 2. Structure and Working Principle of HA There are diverse kinds of arrangements for HA depending on its applications. The arrangement of theThere HA used are diverse in this study kinds is of a basicarrangements one that is for widely HA depending used, as shown on its in applications. Figure1c, which The correspondsarrangement toof the the case HA that used uses in the this magnetic study is flux a basic flowing one in that the is z widelydirection used, area. as shown in Figure 1c, which corresponds to the case that uses the magnetic flux flowing− in the −z direction area.
Figure 1. (a) PM arrays magnetized in the +z and z direction; (b) permanent magnet (PM) arrays Figure 1. (a) PM arrays magnetized in the +z and− −z direction; (b) permanent magnet (PM) arrays magnetized in +x and x direction; (c) Halbach array (HA) through the combination of (a) and (b). magnetized in +x and −x direction; (c) Halbach array (HA) through the combination of (a) and (b) The model of Figure1c is the combination of Figure1a,b to maximize and utilize the magnetic flux in theThez direction.model of Figure Figure 11ca is shows the combination a combination of Figure of PM 1a, elementsb to maximize magnetized and utilize in +z and the magneticz directions. flux − − Figurein the 1−bz direction shows the. Figure arrangement 1a shows of a PM combination elementsmagnetized of PM elements in + magnetizedx and x directions. in +z and −z directions. − FigureBy 1b combining shows the the arrangement PM elements of demonstratedPM elements magnetized in Figure1a,b, in the+x and magnetic −x directions. fluxes generated by the modelsBy incombining Figure1a,b the are PM superimposed, elements demonstrated as illustrated in Figure in Figure 1a,1b,c. the Hence, magnetic the magnetic fluxes gener fluxated in the by +thez direction models in is canceledFigure 1a, outb are and superimposed, the flux in the as zillustrateddirection in is augmented.Figure 1c. Hence, In other the words,magnetic by flux using in − thethe arrangement+z direction is of ca HAnceled shown out and in Figure the flux1c, in it isthe possible −z direction to add is augmented. the magnetic In flux other flowing words, in by the using+z direction,the arrangement which is of not HA utilized, shown toin theFigurez direction 1c, it is possible where the to magneticadd the magnetic flux is used. flux flowing in the +z − direction,A given which PM is has not a utilized, limited amount to the −z of direction total magnetic where the flux. magnetic Therefore, flux it is is used. helpful to use the HA structureA given to maximize PM has thea limited magnetic amount flux in of the total utilized magnetic area flux. within Therefore, a given PM it is volume. helpful Into other use the words, HA thestructure cost and to volume maximize of PMthe can magnetic be minimized flux in the by applyingutilized area the HA. within a given PM volume. In other words, the cost and volume of PM can be minimized by applying the HA. 3. The Proposed AB Method for PM 3. The Proposed AB Method for PM This section may be divided by subheadings. It should provide a concise and precise description of theThis experimental section may results, be divided their by interpretation, subheadings. It as should well asprovide the experimental a concise and conclusionsprecise description that can of bethe drawn. experimental results, their interpretation, as well as the experimental conclusions that can be drawn.
3.1. Mathematical Model of PM
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3.1.Energies Mathematical 2020, 13, x Model of PM 3 of 12
There are are two two mathematical mathematical models models of PM of forPM the for analysis the analysis of its magnetic of its magnetic characteristics: characteristics: Ampere modelAmpere and model Gilbert and model. Gilbert model. As shown in Figure 22a,a, inin thethe AmpereAmpere model,model, thethe PMPM isis assumedassumed asas anan equivalentequivalent AmperianAmperian current or bound current flowing flowing on the surface of PM,PM, and the magnetic flux flux is generated by the Amperian current.
(a) (b)
Figure 2. AssumptionAssumption of of PM PM as as ( (aa)) an an Ampere Ampere model model and and (b (b) )a a Gilbert Gilbert model, model, where where :: AmperianAmperian ⊗ current, ofof which flowingflowing directiondirection isis intointo thethe page; page; : Amperian: Amperian current, current, of of which which flowing flowing direction direction is
outis out of of the the page; page;: North-pole: North-pole of theof the magnetic magnetic charge; charge:; South-pole: South-pole of the of magneticthe magnetic charge. charge. ⊕ As shownshown inin FigureFigure2 b,2b, the the Gilbert Gilbert model model assumes assumes the the PM PM as an as equivalentan equivalent magnetic magnetic charge. charge. The polarityThe polarity of the of magneticthe magnetic charge charge in the in Gilbertthe Gilbert model model corresponds corresponds to the to Norththe North and and South South pole pole of PM. of ThePM. magneticThe magnetic charge charge is defined is defined as a as unipolar a unipolar charge, charge, not not a dipole, a dipole, and and is distributed is distributed in thein the area area of theof the corresponding corresponding polarity polarity in in the the PM. PM. Therefore, Therefore, in in the the Gilbert Gilbert model, model, thethe magneticmagnetic fieldfield generated by the PM can be calculated as done in the case of calculating the electric field field of electric charge by Coulomb’s law. The Gilbert model is reasonable fo forr the demonstration of of the the magnetic field field outside outside the the PM. PM. However, the magnetic field field inside the PM of the Gilbert model is quite different different from the actual magnetic field,field, as shown in FigureFigure2 2b.b. Therefore, the Ampere model can be useful for the analysis andand design of PM machine which uses magnetic fieldfield near PM. In this study, the Ampere model is used for the derivation of the analytic method for the analysis of PM.
3.2. Application of Amperian Current into PM The Ampere model assumes the PM as an equivalentequivalent current loop. Therefore, in this paper, an analytic method method for for the the analysis analysis of the of magnetic the magnetic field generated field generated by PM is by proposed PM is via proposed the assumption via the ofassumption the equivalent of the current equivalent loop current of PM andloop by of using PM and the by Biot–Savart using the law.Biot–Savart law. The sum of the Amperian currents flow flowinging through the surface of the PM can be determined using Ampere’s circuital law. In other words, the total current I , which is the summation of each using Ampere's circuital law. In other words, the total current Im, which is the summation of each Amperian current i of PM in the Ampere model, can be obtained by integrating the magnetic field field intensity H over the length l as shown in Equation (1). X Z I i H→ dl → (1) Im =m i = H dl (1) · The total current Im can be calculated using an equivalent magnetic circuit model and by using the materialThe total property current BIm-Hcan curve be calculatedof the PM. using an equivalent magnetic circuit model and by using the materialThrough property the B-H B-H curve curve of of magnetic the PM. material, the linear relationship between the residual magneticThrough flux the density B-H curve Br and of magnetic the coercive material, force the H linearc can relationship be found. between Moreover, the by residual modeling magnetic the fluxpermanent density magnetBr and the in coercive the form force of aH c Nortoncan be found. equivalent Moreover, circuit, by the modeling magnetomotive the permanent force magnet of the permanent magnet can be calculated [33]. Therefore, the total current Im can be obtained by multiplying the coercive force Hc by the length L in the magnetization direction of the PM.
Energies 2020, 13, 1252 4 of 11 in the form of a Norton equivalent circuit, the magnetomotive force of the permanent magnet can be calculated [33]. EnergiesTherefore, 2020, 13, x the total current Im can be obtained by multiplying the coercive force Hc by the length4 of 12 L in the magnetization direction of the PM.
IHLmc (2) Im = Hc L (2) · 3.3. Amperian Current Distribution Modelling in HA 3.3. Amperian Current Distribution Modelling in HA In this study, the most common and widely used arrangement of HA shown in Figure1c is In this study, the most common and widely used arrangement of HA shown in Figure 1c is considered. For the verification of the proposed analysis method, the model of Figure3a, which utilizes considered. For the verification of the proposed analysis method, the model of Figure 3a, which utilizes the minimum number of PMs (three PMs) for maximizing the z direction magnetic flux, is used. the minimum number of PMs (three PMs) for maximizing the −−z direction magnetic flux, is used.
(a) (b)
Figure 3. (a) Arrangement of the HA for the verification verification of the proposed analytical method where the magnetized direction is represented represented by by ⇨;( ; b(b)) Modeling Modeling of of the the HAHA withwith thethe distributeddistributed currentscurrents by applying the Amperian model.model.
As shown in FigureFigure3 3b,b, thethe HAHA demonstrateddemonstrated inin FigureFigure3 a3a is is converted converted into into the the Ampere Ampere model model via current distributiondistribution flowingflowing atat aa constantconstant intervalinterval onon thethe PMPM surface.surface. O means the origin with coordinates (0,0,0)(0,0,0) inin CartesianCartesian coordinatecoordinate system.system. The mathematical modeling of a PM with the Ampere model cancan be derived using Figure4 4.. ForFor the mathematicalmathematical expression of i(k), Equation (3) is proposedproposed in thisthis paperpaper byby assumingassuming the current distribution as a linear function of kk,, asas shownshown inin FigureFigure5 5.. TheThe totaltotal numbernumber ofof AmperianAmperian currentscurrents isis 2n + 1, n00 andand nn11 areare constant constant coefficients, coefficients, and and IImm isis the the total total sum sum of of the the currents. currents. I I i =i (n() n+ n n kk ) mm (3(3)) k k 0 011 · | | ·212nn+ 1
Im cancan be be obtained obtained using using Equation Equation (2). Because the sum of ii(k)(k) isis equalequal toto IImm, we can obtain Equation Equation (4). kn I k=n m X ()n01 n kIm Im (4) kn(n + n k ) 21n = Im (4) 0 1 · | | · 2n + 1 k= n By rearranging Equation (4) for n−0 and n1, we can derive Equation (5) for the constants n0 and n1. By rearranging Equation (4) for n and n , we can derive Equation (5) for the constants n and n . 0 1(2nn 1)(1 ) 0 1 n 0 (5) 1 (2n +nn(1)(1 1) n0) n = − (5) 1 n(n + 1)
1
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Figure 4. Modeling of a PM shown in Figure 3b with the Amperian model by using distributed FigureFigurecurrents, 4. 4. ModelingModeling where i(k) of is a athe PM PM k shown- th shown current in inFigure. Figure3bwith 3b with the Amperian the Amperian model model by using by distributed using distributed currents, currents,where i( kwhere) is the i(k)k-th is current.the k-th current.
Figure 5. The plot of the Amperian current distribution of Figure4 with the linear function. Figure 5. The plot of the Amperian current distribution of Figure 4 with the linear function. 3.4. MagneticFigure Field 5. The Analysis plot of the Using Amperian the AB current Method distribution of Figure 4 with the linear function. 3.4. TheMagnetic magnetic Field fluxAnalysis density Using at the a position AB Method that is at a distance r from the current flowing through 3.4. Magnetic Field Analysis Using the AB Method the diThefferential magnetic length fluxdl candensity be obtained at a position as Equation that is at (6) a bydistance using ther from Biot–Savart the current law, flowing where µ through0 is the permeabilitytheThe differential magnetic of length vacuum, flux densitydl canI is be the at obtained a intensity position as of thatEquation the is current, at a (6) distance by and usingrˆ ris from the a direction Biot the– currentSavart vector law, flowing for where an through arbitrary μ0 is the position [34]. thepermeability differential oflength vacuum, dl can I beis the obtained intensity as Equationof the current, (6) by and using r the is aBiot direction–Savart vector law, where for an μ arbitrary0 is the → permeabilityposition. [3 4of] vacuum, I is the intensity of the current,µ0I dl andrˆ r is a direction vector for an arbitrary d→B(r) = × (6) position. [34] 4π · r2 I dl r Figure6 illustrates the magnetic flux densitydB() r 0B(r , x ) at an arbitrary position that is at(6 a) line2 line 04Idlr r vertical distance r and height x fromdB the() r center point of the linear current I having finite length(6) L. line line 4 r2 Based on Figure6, Equation (7) is derived for the calculation of the magnetic flux density B(r , x ) Figure 6 illustrates the magnetic flux density B( rline , xline ) at an arbitrary position thatline is lineat a generatedFigure by6 illustrates the line current the magnetic I through flux the density application B( r of, Equationx ) at an (6) arbitrary [34]. position that is at a vertical distance and height from the centerline pointline of the linear current I having finite length vertical distance and height from the center point of the linear current I having finite length L. Based on Fig→ure 6, Equation µ(7)0I is derivedxline for+ Lthe/2 calculation of xtheline magneticL/2 flux density B( , B(rline, xline) = ( q q − ) (7) L. Based on Figure 6, Equation 4(7)πr isline derived 2for the calculation2 − of the2 magnetic flux2 density B( , ) generated by the line current I throughrline +the (x application+ L/2) of rEquationline + (x line(6) [3L4/].2 ) ) generated by the line current I through the application of Equation (6) [3−4].
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Figure 6. Illustration of the magnetic flux density B(rline,xline ) at the position (rline,xline ) by the line Figure 6. Illustration of the magnetic flux density B( rline , xline ) at the position ( , ) by the line current I with a finite length L. current I with a finite length L.
Figure7 illustrates the vector Bloop(x,y,z) of the magnetic flux density from the current loop at a point (x,y,z) relative to the center point I of the xloop LO/. 2 The current xI flows L / 2 through the loop, and loop B(,)() r x 0 line line has the width A and the lengthline B. line The4 functionr 2 of B (x,y 2,z) in Figure 27 can be 2 derived as Equation (8)(7) line rline( xloop L / 2) r line ( x line L / 2) by applying Equation (7). Figure 7 illustrates the vector Bloop(x,y,z) of the magnetic flux density from the current loop at a point (x,y,z) relative→ to the center point of the loop O. The current I flows through the loop, and loop Bloop(x, y, z) = Bx aˆx + By aˆy + Bz aˆz has the width A and the length B. ·The function· of B·loop(x,y,z) in Figure 7 can be derived as Equation (8) by applying Equationwhere (7). y+ B y B µ0I z 2 2 Bx = 2 q q − 4π A + 2 A 2 B 2 A 2 B 2 · (x 2 ) z · (x ) +z2+(y+ ) − (x ) +z2+(y ) − − 2 2 − 2 − 2 y+ B y B µ0I z 2 2 2 q q − 4π + A + 2 A 2 B 2 A 2 B 2 − · (x 2 ) z · (x+ ) +z2+(y+ ) − (x+ ) +z2+(y ) 2 2 2 − 2 x+ A x A µ0I z 2 2 By = 2 q q − 4π B + 2 A 2 B 2 A 2 B 2 · (y 2 ) z · (x+ ) +z2+(y ) − (x ) +z2+(y ) − 2 − 2 − 2 −2 x+ A x A µ0I z 2 2 (8) 2 q q − 4π + B + 2 A 2 B 2 A 2 B 2 − · (y 2 ) z · (x+ ) +z2+(y+ ) − (x ) +z2+(y+ ) 2 2 − 2 2 x+ A y+ B y B µ0I 2 2 2 Bz = 2 q q − 4π + A + 2 A 2 B 2 A 2 B 2 · (x 2 ) z · (x+ ) +z2+(y+ ) − (x+ ) +z2+(y ) 2 2 2 − 2 x A y+ B y B µ0I 2 2 2 Figure 7. Illustration of− the2 magnetic q flux density vectorq Bloop(x,y,z− ) at the position (x,y,z) by the 4π A + 2 A 2 B 2 A 2 B 2 − · (x 2 ) z · (x ) +z2+(y+ ) − (x ) +z2+(y ) rectangular current loop− of the current − I2. 2 − 2 − 2 y+ B x+ A x A µ0I 2 2 2 + 2 q q − 4π + B + 2 A 2 B 2 A 2 B 2 · (y 2 ) z · (x+ ) +z2+(y+ ) − (x ) +z2+(y+ ) 2 2 − 2 2 y B x+ A x A µ0I 2 2 2 − 2 q q − 4π B + 2 A 2 B 2 A 2 B 2 − · (y 2 ) z · (x+ ) +z2+(y ) − (x ) +z2+(y ) − 2 − 2 − 2 − 2
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Figure 6. Illustration of the magnetic flux density B( rline , xline ) at the position ( , ) by the line current I with a finite length L.
I x L/ 2 x L / 2 B(,)() r x 0 line line line line 4r 2 2 2 2 (7) line rline( x L / 2) r line ( x line L / 2)
Figure 7 illustrates the vector Bloop(x,y,z) of the magnetic flux density from the current loop at a point (x,y,z) relative to the center point of the loop O. The current I flows through the loop, and loop Energies 2020, 13, 1252 7 of 11 has the width A and the length B. The function of Bloop(x,y,z) in Figure 7 can be derived as Equation (8) by applying Equation (7).
Figure 7. Illustration of the magnetic flux density vector Bloop(x,y,z) at the position (x,y,z) by the Figure 7. Illustration of the magnetic flux density vector Bloop(x,y,z) at the position (x,y,z) by the rectangular current loop of the current I. rectangular current loop of the current I. By applying Equation (8) for the model of Figure3b, the magnetic flux density vector B(x,y,z) at the position that is at (x,y,z) from the center of the bottom of PM can be derived as Equation (9).
→ → → → BHalbach(x, y, z) = BLe f t(x, y, z) + BCenter(x, y, z) + BRight(x, y, z) 1 0 0 n P → w1+w2 t tk = 0 1 0 Bloop(x + 2 , y, z 2 2n ) k= n { · − − − − 0 0 1 0 0 1 (9) → t w2k + 0 1 0 Bloop(z + , y, x + ) · 2 2n 1 0 0 − 1 0 0 − → w1+w2 t tk + 0 1 0 Bloop( x + , y, z + ) · − 2 2 − 2n } 0 0 1 − where BLeft(x,y,z), BCenter(x,y,z), and BRight(x,y,z) each represent the vector of the magnetic flux density from the left magnet, the center magnet, and the right magnet in Figure3.
Equation (9) can be simplified to Equation (10) when the magnetic flux density is calculated in the condition that x and y are zero. The magnetic flux density through the line where x and y are zero is calculated simply, and it is reasonable as a metric for the performance of HA at the initial design stage. Based on the applied methods, the proposed analysis method is termed as the Ampere model and the Biot–Savart law analysis method (AB method) in this paper. n P µ0ik z l B(z) = r 4π −kw 2 2 2 · kw 2 2 k= n z +( 2n ) 2 2 l − z +( 2n ) +( 2 ) z+t l + r kw 2 2 (z+t)2+ 2 · 2 kw2 l 2 2n (z+t) + 2n +( 2 ) l z+t + r kw 2 2 2 2 l · 2 kw 2 +( ) (z+t) + 2 + l 2n 2 2n ( 2 ) (10) l z r kw 2 2 2 − 2 +( l ) · 2 kw2 l 2 2n 2 z + 2n +( 2 ) 2( kt +z+ t ) − 2n 2 l + 2 w 2 q kt t 2 kt t 2 w2 2 l 2 ( 2n +z+ 2 ) +( 2 ) · ( +z+ ) +( ) +( ) 2n 2 2 2 kt t 2( n +z+ ) l + 2 2 q kt t 2 w2 2 2 w 2 2 ( +z+ ) +(w2+ ) · kt t 2 l 2n 2 2 ( 2n +z+ 2 ) +(w2+ 2 ) +( 2 ) Energies 2020, 13, 1252 8 of 11
4. Verification of the AB Method This study proposes the mathematical model and analytical method for the calculation of the magnetic flux density generated by the HA at an arbitrary position in an open area. For the verification of the proposed mathematical model and analysis method, a comparison of the magnetic flux density calculated by using both the proposed AB method and the finite element method (FEM) was carried out, as shown in Figure8. The widely used arrangement of the HA, shown in Figure3, is used for the verificationEnergies 2020, 1 model3, x with data tabulated in Table1. 9 of 12
(a) (b)
(c) (d)
(e) (f)
Figure 8. Comparison betweenbetween the the proposed proposed AB AB method method and and the the FEM FEM via via the the calculated calculated magnetic magnetic flux densityflux density data data of the of model the model demonstrated demonstrated in Figure in Fig3:(urea) 3: when (a) whenz varies z varies from from 0 to 10 0 to mm 10 with mm (withx,y) fixed(x,y) atfixed (0,0); at ((0,0)b) when; (b) zwhenvaries z fromvaries 0 from to 10 mm0 to with10 m (mx, ywith) fixed (x, aty) (5,fixed2.5); at (5, (c) − when2.5); (zc)varies when from z varies 0 to 10from mm 0 − withto 10 (mx,ym) fixedwith at(x, (0,y) fixed6); (d at) when (0,−6);z varies(d) when from z 0varies to 10 from mm with0 to (1x0,y m) fixedm with at (10,0);(x,y) fixed (e) when at (10,0);y varies (e) − fromwhen y6 varies to 6 mm from with −6 (tox,z 6) fixedmm with at (0,5); (x,z ()f fixed) when atx (0,5);varies (f) from when 8x tovaries 8 mm from with −8 (y to,z) 8 fixed mm atwith (0,5). (y,z) − − fixed at (0,5).
Table 1. Design variable and material properties of the HA for the verification of the AB method.
Design Variable Value w1 5 mm w2 5 mm l 5 mm t 5 mm μr 1.0998 Hc −890 kA/m
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Table 1. Design variable and material properties of the HA for the verification of the AB method.
Design Variable Value
w1 5 mm
w2 5 mm l 5 mm t 5 mm
µr 1.0998
Hc 890 kA/m −
As shown in Figure8, the di fference of the calculated magnetic flux density between the proposed AB method and FEM is within 8%. However, the calculation times are approximately 144 s and 50 ms, when using the FEM and AB method, respectively. The AB method used 21 current loops to create an equivalent circuit model for each permanent magnet. The FEM analysis was performed with 67,857 triangular elements which warrant grid-independence of the calculation results. The optimal design of HA requires extensive trial and error with high time- and cost-consumption. When the optimal design of the HA is carried out by using FEM, the time and cost increase exponentially as the model of the HA is complex. The proposed AB method does not require a lot of memory and time because it calculates the magnetic flux density only at the required position. Hence, when the proposed AB method is used for the optimal design of the HA; the time and cost can be reduced dramatically compared to the FEM, regardless of the complexity of the HA. Moreover, the induced function of magnetic flux density of the AB method is differentiable and is itself an evaluation function of the objective function for the optimal design of the HA. Therefore, the general function of magnetic flux density derived with the AB method can be used simply and directly for the objective function of the optimization through the differentiation and using the gradient method. This facilitates a significant reduction in time and cost in the optimal design of the HA. Briefly, the proposed AB method can reduce the memory, time, and cost required in the analysis and optimal design of the HA in an open area. Moreover, the AB method can be directly used for the objective function via simple differentiation and using the gradient method in the optimal design of the HA.
5. Conclusions The existing mathematical models and analysis methods for the analysis of the HA in open space are not sufficient due to the condition that the device has no ferromagnetic core and has wide airgap. Hence, this research is significant in that it proposes the AB method, a novel mathematical model and rapid and accurate analytical method for the HA. The proposed AB method contains the function for magnetic flux density defined by the design variables of the HA. Therefore, it is remarkable in that the relation between the design variables and characteristics of the HA can be estimated mathematically and directly via the AB method, and it makes the HA design procedure simpler. The magnetic flux density generated by the Halbach array was derived as a function of three-dimensional space, which gives a continuous solution at any position in the space around HA. In addition, the proposed equation can be applied to HA with more complex structures, as well as to HA consisting of three permanent magnets presented in the manuscript, along with modification of the function depending on the structure. In the analysis of the HA in open spaces by using FEM, many problems with respect to memory, time, and cost arise due to the increase in the number of mesh elements. Therefore, the rapid and accurate analysis of the HA being made possible with little memory usage by using the proposed AB Energies 2020, 13, 1252 10 of 11 method is noteworthy. These merits of the AB method compared to the FEM are augmented as the complexities of HA and open space are increased. The optimal design requires many trial-and-error procedures. Hence, when the AB method is adopted in the optimization of HA, the memory, time, and cost decrease dramatically compared to when FEM is used. The derived magnetic flux density function of the AB method is differentiable, and it can offer additional indicators such as gradient and extremum to designer. Hence, its optimization can consider the magnetic characteristics of magnet array effectively and improve its optimization performance and efficiency. Briefly, the proposed AB method is useful for reducing memory, time, and cost in the characteristic analysis and optimization of the HA. Hence, the AB method is useful and can be widely used in the analysis and optimal design of diverse types of electric machines, which use numerous arrangements and shapes of HAs. Moreover, the AB method can contribute to the development of diverse PM electric machines via the modification of its governing equations.
Author Contributions: Conceptualization, M.-S.S. and J.-S.R.; funding acquisition, J.-S.R.; investigation, M.-S.S.; methodology, M.-S.S. and J.-S.R.; project administration, J.-S.R.; resources, J.-S.R.; software, M.-S.S.; supervision, J.-S.R.; validation, M.-S.S.; visualization, M.-S.S.; writing—original draft, M.-S.S. and J.-S.R.; writing—review and editing, M.-S.S. and J.-S.R. All authors have read and agreed to the published version of the manuscript. Funding: This work was funded by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (2016R1D1A1B01008058) and by the Chung-Ang University Research Grants in 2018. Conflicts of Interest: The authors declare no conflict of interest.
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