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Semi-Analytical Modeling and Analysis of Halbach Array

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Semi-Analytical Modeling and Analysis of Halbach Array 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 thethe case HA that used uses in thethis magnetic study is fluxa basic flowing one inthat the is zwidelydirection 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 volumemaximize of PMthe canmagnetic 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 Energies 2020, 13, 1252 3 of 11 3.1.Energies Mathematical 2020, 13, x Model of PM 3 of 12 There areare two two mathematical mathematical models models of PMof forPM thefor analysis the analysis of its magneticof its magnetic characteristics: characteristics: Ampere modelAmpere and model Gilbert and model.
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