Design Methodology of a Dual-Halbach Array Linear Actuator with Thermal-Electromagnetic Coupling

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1 sensors Article Design Methodology a Dual-Halbach Array Linear Actuator with Thermal-Electromagnetic Coupling Paulo Roberto Eckert 1, *, Aly Ferreira Flores Filho 1, Eduardo Perondi 2, Jeferson Ferri 2 and Evandro Goltz 3 1 Post-Graduate Program in Electrical Engineering, Federal University Rio Grande do Sul, Av. Osvaldo Aranha 103, Porto Alegre, RS , Brazil; aly.flores@ufrgs.br 2 Department Mechanical Engineering, Federal University Rio Grande do Sul, Rua Sarmento Leite 425, Porto Alegre, RS , Brazil; eduardo.perondi@ufrgs.br (E.P.); jeferson.ferri@ufrgs.br (J.F.) 3 Technology Centre, Federal University Santa Maria, Av. Roraima 1000, Santa Maria, RS , Brazil; evandro@inf.ufsm.br * Correspondence: paulo.eckert@ufrgs.br; Tel.: ; Fax: Academic Editors: Slawomir Wiak and Manuel Pineda Sanchez Received: 31 December 2015; Accepted: 19 February 2016; Published: 11 March 2016 Abstract: This paper proposes a design methodology for linear actuators, considering rmal and electromagnetic coupling with geometrical and temperature constraints, that maximizes force density and minimizes force ripple. The method allows defining an actuator for given specifications in a step-by-step way so that requirements are met and temperature within device is maintained under or equal to its maximum allowed for continuous operation. According to proposed method, electromagnetic and rmal models are built with quasi-static parametric finite element models. The methodology was successfully applied to design a linear cylindrical actuator with a dual quasi-halbach array permanent magnets and a moving-coil. The actuator can produce an axial force 120 N and a stroke 80 mm. The paper also presents a comparative analysis between results obtained considering only an electromagnetic model and rmal-electromagnetic coupled model. This comparison shows that final designs for both cases differ significantly, especially regarding its active volume and its electrical and magnetic loading. Although in this paper methodology was employed to design a specific actuator, its structure can be used to design a wide range linear devices if parametric models are adjusted for each particular actuator. Keywords: design methodology; dual quasi-halbach actuator; linear actuators design; moving-coil actuator; parametric analysis; rmal-electromagnetic coupling; tubular actuators 1. Introduction The advantages linear actuators over rotation-to-translation conversion systems when linear motion is needed are well documented in literature [1]. Applications linear actuators are mainly in industry and transportation due to its force density, efficiency, levels acceleration, and precision. An application that has also drawn particular interest to linear actuators is mechanical vibration control [2,3]. For such an application, linear actuators that have interesting characteristics are cylindrical actuators equipped with quasi-halbach arrays permanent magnets (PMs) [4,5]. The advantages quasi-halbach arrays are well known, and include applicability for a wide range electromechanical devices [6]. Even so, re are still recent developments and designs for specific applications under research, such as, for example, for limited angle torque actuators [7] and for assemblies in which flux density can be altered by a mechanical operation [8]. In this context, this paper proposes a methodology applied to design a special cylindrical actuator with dual quasi-halbach arrays and a moving-coil coreless armature that was first proposed by Sensors 2016, 16, 360; doi: /s

2 produced by PMs and currents in three-phase windings are in quadrature, electromagnetic force produced in axial direction is maximized. With regard to PMs quasi-halbach arrays, ones with axial magnetization can be easily manufactured; however, ones with radial magnetization, when using NdFeB sintered Sensors material, 2016, 16, i.e., 360 with high B-H product, should be segmented and magnetized in a parallel direction to 2 28 avoid a more complicated and expensive magnetizing fixture. In [10], it was shown that such a method slightly reduces performance if ring is segmented with an appropriate number Yan elements. et al. [9]. Figure 1a shows a 3D view constitutive elements actuator. It consists a device Some with features a stator formed actuator by presented two layersin Figure PMs 1 mounted are attractive on when surface compared to inner or and outer permanent back-irons, magnet in order tubular to actuators, produce two such concentric as, for example: quasi-halbach moving-coil arrays, reduces in accordance moving mass with magnetization actuator, pattern increasing shown acceleration by arrows and inside speed; re PMsis inno Figure cogging 1b. force, Thus, considering a multipolarthat distribution re are magnetic no slots, which field significantly in axialcontributes direction to in reducing air-gap force volume ripple; occupied core losses by are zero moving-coil because is produced. it is a coreless The three-phase device, leading ABC to windings higher efficiency; are supported quasi-halbach mechanically array by on a both reel, sides forming a coil windings increases magnetic flux density and allows for a distribution with low harmonic distortion concentrically mounted in relation to quasi-halbach arrays and having free relative movement in magnetic gap, which increases force density and reduces ripple force, respectively. The high in axial direction. The three-phase currents in windings generate a multipolar magnetic field levels flux density can be explained by fact that quasi-halbach array reduces local leakage distribution in axial direction, analogous to one produced by PMs. When magnetic fields flux, i.e., from PMs to back-iron, while double array helps to decrease interpolar leakage flux. In produced [11] byauthors PMs and show currents that if in same three-phase volume PMs windings is employed are inby quadrature, this topology, in electromagnetic comparison forceto produced a topology inwith outer axial or direction inner PMs is maximized. only, dual PMs array provides a better force density. Figure 1. Linear tubular moving-coil actuator with a four pole dual quasi-halbach array: (a) 3D Figure 1. Linear tubular moving-coil actuator with a four pole dual quasi-halbach array: isometric section view indicating structural elements; and (b) 2D half-section indicating design (a) 3D isometric section view indicating structural elements; and (b) 2D half-section indicating variables. design variables. With regard to PMs quasi-halbach arrays, ones with axial magnetization can be easily manufactured; however, ones with radial magnetization, when using NdFeB sintered material, i.e., with high B-H product, should be segmented and magnetized in a parallel direction to avoid a more complicated and expensive magnetizing fixture. In [10], it was shown that such a method slightly reduces performance if ring is segmented with an appropriate number elements. Some features actuator presented in Figure 1 are attractive when compared to or permanent magnet tubular actuators, such as, for example: moving-coil reduces moving mass actuator, increasing acceleration and speed; re is no cogging force, considering that re are no slots, which significantly contributes to reducing force ripple; core losses are zero because it is a coreless device, leading to higher efficiency; quasi-halbach array on both sides windings increases magnetic flux density and allows for a distribution with low harmonic distortion in magnetic gap, which increases force density and reduces ripple force, respectively. The high levels flux density can be explained by fact that quasi-halbach array reduces local leakage flux, i.e., from PMs to back-iron, while double array helps to decrease interpolar leakage flux. In [11] authors show that if same volume PMs is employed by this topology, in comparison to a topology with outer or inner PMs only, dual PMs array provides a better force density.

3 Sensors 2016, 16, Design parameters topology presented in Figure 1 were discussed in [12], where authors concluded, based on an analytical model, that topology has better force capability and lower force ripple with three-phase windings than ones with one-phase or with two-phase windings. Some geometrical analyses were performed based on an analytical electromagnetic model. In [13], a rmal-electromagnetic coupled design methodology is presented for project a short-duty slotted non-halbach array with axially magnetized PMs linear actuator. The duty cycle actuator required a transient rmal analysis because high levels transitory current density were employed. Designs specific actuators that consider rmal influence are addressed in literature, for example, in [14,15]. In [14], a coupled rmal-electromagnetic analysis slotless tubular permanent magnet machines was presented. Bianchi et al. [15], on or hand, presented an overall comparison on linear actuators such as interior and surface-mounted PMs, and slotted and slotless motors. These two references apply a rmal constraint as maximum temperature actuator; rmal influence is inserted as a constant overall heat transfer coefficient. In [16] a multiphysics approach to model a tubular permanent magnetic slotted actuator with radially magnetized non-halbach and interior surface mounted PMs based on finite element models is addressed. Encica et al. [17] described an optimization methodology employed in design an actuator with a similar topology as in [16] using rmal and electromagnetic lumped circuit models. The development a design methodology for electromagnetic devices using modern computational tools is an extensive task that involves a coupling between electromagnetic and rmal domains. However, one can expect a high degree conformity between models and results obtained experimentally. In this paper, a design methodology that encompasses modeling and analysis where specifications such as force and stroke must be met considering rmal-electromagnetic coupling for continuous operation is addressed. Specifications are met with maximization force density and minimization force ripple subjected to geometrical and rmal constraints. Thermal and electromagnetic 2D axisymmetric models are built and simulated with finite element stware packages. A post-processing analysis is carried out based on results parametric electromagnetic simulation. The proposed methodology can be employed to design a wide range linear devices, such as step motors, synchronous machines, DC machines, and reluctance machines. In order to employ it in different devices, electromagnetic parametric model and rmal model must be particular to each case. The main requirement to apply this methodology is that linear actuator should be a multipole device with long armature, although it could be adapted to short armature if steps that compute number poles and axial length are adjusted. However, proposed methodology cannot be directly employed to design linear single-phase oscillatory actuators, linear plunger solenoids, and single-phase solenoids, for instance, because those are examples non-multipolar devices. 2. Design Methodology The first step in developing proposed design methodology is rationalization an engineering problem considering a specific application that can be obtained by a flowchart, as shown in Figure 2. Identifying technical requirements axial force and stroke an actuator associated with dimensional constraints are main input elements for design. The main goal is to obtain geometric relationships device, in order to maximize force density and minimize force ripple. The proposed design methodology is mainly based on obtaining an absolute value for force density one pole pitch device, finding its best design with aid parametric analysis, and, afterwards, determining actuator geometrical relationships and dimensions that are able to meet design specifications.

4 Sensors 2016, 16, Sensors 2016, 16, Figure 2. Flowchart proposed design methodology linear cylindrical actuators with electromagnetic-rmal coupling. An An initial initial approach approach is is proposed proposed in in which which a design a design is obtained is obtained from from a pure a electromagnetic pure electromagnetic model, model, in this paper in this referred paper to referred as uncoupled to as uncoupled model. This model. model provides This model initial provides geometric initial dimensions geometric to dimensions allow for a simulation to allow for a simulation rmal model. A rmal rmal model. constraint, A rmal in form constraint, a maximum in form allowable a maximum temperature allowable at armature, temperature is applied, at which armature, is is basis applied, for adjusting which is effective basis for current adjusting density effective accordingly, current and density operation accordingly, characteristics and PMs operation for characteristics so called coupled PMs model. for so called coupled A similar model. approach could be conducted with a multiphysics rmal-electromagnetic coupled simulation A similar by means approach finite could element be conducted models with for both a multiphysics domains, which rmal-electromagnetic is already available coupled in some simulation commercial by packages. means However, finite element such an models analysis for would both domains, be an extensive which and is already time-consuming available task in some and commercial it does not provide packages. a routine However, to design such a an device. analysis would be an extensive and time-consuming task and it does not provide a routine to design a device. 3. Design Cylindrical Linear Actuator with Dual-Halbach Array 3. Design Cylindrical Linear Actuator with Dual-Halbach Array In this section, each design steps presented at flowchart Figure 2 is discussed in detail. In The this steps section, are each followed in design order tosteps design presented a specificat cylindrical flowchart linear actuator, Figure 2 which is discussed used in as detail. a case study. The steps are followed in order to design a specific cylindrical linear actuator, which is used as a case study Determination Actuator Requirements 3.1. Determination Actuator Requirements This is first step in design process, i.e., identifying which are output parameters that linear This is actuator first has step toin meet. design In process, design i.e., electrical identifying rotating which machines, are output first parameters specification that is generally linear actuator rated has torque, meet. which In allows design us electrical to determine rotating active machines, volume first oncespecification electrical and is generally magnetic loading rated aretorque, specified which [18]. allows For conventional us to determine rotating machines, active volume loadingonce characteristics electrical and are magnetic available loading references, are specified such[18]. as [18], For based conventional extensive rotating empirical machines, data loading availablecharacteristics or resultant from are available in references, such as [18], based on extensive empirical data available or resultant from heat transfer analysis. However, in linear machines with unconventional topologies and complex

5 Sensors 2016, 16, heat transfer analysis. However, in linear machines with unconventional topologies and complex heat transfer systems, scarce information is available, especially about compromise between electrical loading and heat exchange ability depending on temperature constraints materials. In case linear machines, similar to rotating ones, rated effective axial force that actuator has to produce in continuous operation must be specified. It is important to highlight that short-stroke linear actuators hardly operate with constant speed, different from rotating machines; refore, axial force y produce may also vary. In this sense, it is important to point out that effective force must be set as rated, once it is directly proportional to effective current density, to which main source losses is quadratically proportional, i.e., Joule losses, imposing a performance limited by temperature. Anor essential output parameter that must be specified is stroke actuator, which is required to determine axial length armature L zw and total length actuator L zt. In this paper, techniques for specification force and stroke will not be discussed; instead, an actuator will be designed for a general application that requires a rated effective axial force F r 120 N and stroke S 80 mm Definition Actuator Electromagnetic Topology Although general idea proposed methodology is applicable for design a wide range linear actuator, re are particularities applied to parametrization, as presented in Section 3.3, and to rmal and electromagnetic models for one pole pitch, τ p, which are exclusive to each linear actuator topology. For this reason, definition actuator topology must be a step in design process. The selection topology can depend on application, on actuator performance or cost, etc. In this paper, a cylindrical actuator with a three-phase moving coil and two arrays quasi-halbach PMs that has some interesting features is addressed Building and Simulation Quasi-Static Parametric 2D FEM for One Pole Pitch The parameterization model is essential, once it enables variation dimensional parameters that define topology electromagnetic actuator. By means this geometric model and results numerical parametric simulation, it is possible to analyze behavior various figures merit for device performance in terms interdependency and sensitivity variables. The first step in building this model is identification geometric variables that define topology, as shown by Figure 1. The active volume one pole pitch is delimited by R ipmi, R opmo ( inner and outer radii, respectively), and τ p. Some variables have dimensional constraints related to manufacturing issues parts or to limitation space in a specific application. Ring-shaped PMs have a limitation for both R ipmi and R opmo, due to need for radial magnetization and dimensional limitations magnetizer. Even though techniques segmentation rings and parallel magnetization are employed [10], in this particular case re is a need for a gas mass flow channel for cooling device, which also requires R ipmi > 0. The rmal heat transfer mechanism is discussed in Section 3.8. The axial length PMs with radial magnetization, τ r, may be normalized according to pole pitch τ p, i.e., τ r /τ p ; refore, axial length PMs with axial magnetization, τ z, is τ z ˆ 1 τr τ p (1) τ p In a cylindrical device finite length, it is possible to observe a relationship between axial length and diameter. This relation can be named as form factor, and, in this case, it is given by ratio pole pitch to external radius outer PMs, according to n f orm τ p R opmo (2)

6 Sensors 2016, 16, Due to aspects previously mentioned, it is interesting to design actuator using a hollow shaft. Therefore, form factor must be set to allow devices with annular cross sections. The cylindrical case results when R ipmi = 0, which excludes possibility radial magnetization and cooling air flow. To maintain consistency in definition form factor, volumetric comparison is required between cylindrical and ring cases, which results in an equivalent radius and provides a proper definition for form factor given by n f orm τ p b pr opmo2 R ipmi 2 q (3) The result expression is dimensionless and provides a comparison devices with different scales. The form factor is suitable for use as a parametric variable because it condenses all dimensional variables that define active volume for one pole pitch device. A parametric variable that is defined as ratio between radial length coil L rcoil and radial active space, i.e., L N CPMs rcoil (4) pr opmo R ipmi q Anor parametric variable is defined as ratio between radial length inner PMs and total radial length PMs: N PMi L rpmi pl rpmi ` L rpmo q (5) where L rpmi = (R opmi R ipmi ), and L rpmo is radial length outer PMs, i.e., (R opmo R ipmo ). A parametric variable that affects only radial length back-irons is given by n BI R ipmi 2 R i 2 R ipmi τ r R o 2 2 R opmo (6) R opmo τ r This variable relates cross section area back-irons with half cylindrical surface area PMs with radial magnetization in contact with back-irons. If n BI = 1, areas mentioned are equal to each or. The n BI factor can be initially estimated by magnetic properties materials. For devices with coreless armatures and quasi-halbach arrays, a good rule is B r 2B sat ď n BI ď B r B sat (7) where B r is remanent maximum flux density PMs and B sat is saturation flux density back-iron st ferromagnetic material. Because topology uses quasi-halbach arrays, influence back-irons in device performance is less significant in comparison to actuators with non-halbach arrays. Simulation results flux density in back-iron regions must be analyzed and radial length adjusted in order to avoid saturation. Considering a position-dependent synchronous electric drive, e.g., with pulse width modulation, sinusoidal currents excitation in sections windings are defined according to I A? 2J rms A cond F f sin pθ e q I B? 2J rms A cond F f sin pθ e ` 2π{3q I C? 2J rms A cond F f sin pθ e 2π{3q (8) where J rms is effective current density through copper conduction area, A cond = τ p L rcoil /3 is geometric conduction area one coil in model, F f is fill factor windings, and θ e is

7 Sensors 2016, 16, position-dependent electrical angle in radians. By means se equations, current density applied to a FEM produces same magnetomotive force that would be observed in experimental case. Considering a study in quasi-static domain, it is possible to study behavior electromagnetic force depending on relative position between stator and translator, p z, which can be defined in terms electrical angle according to p z pθ e q τ pθ e π (9) By means Equations (8) and (9), initial relative position p z (0) corresponds to electrical angle at which current in phase A is zero, but currents in phases B and C are identical, so coil must be positioned with alignment phases B and C with pole faces PMs with radial magnetization and axial length τ r. Therefore, generated fields are in quadrature, producing maximum force in relation to applied current. Using equations presented in this subsection, it is possible to build parametric models geometry in 2D or 3D spaces. The geometry is axisymmetric, so 2D models satisfactorily represent behavior spatial distribution fields, even if technique segmenting rings into a minimum eight sections with parallel magnetization is applied [10]. However, if number segments is lower than eight, and asymmetry in circumferential direction becomes relevant, 2D model must be replaced by a 3D model. Thus, model would be more complex and simulation would become time consuming, but main structure proposed methodology would still be applicable. In order to create and simulate 2D FEM for one pole pitch, commercial package Ansys Maxwell V was employed. The boundary conditions at axial boundaries model were set as symmetric Master/Slave, i.e., B slave = B master. Thus, spatial distribution field in se regions behaves as if y had infinite neighboring poles. This approach neglects end effects, however; at this stage, number poles device to be designed is not known, so adverse effects should be avoided. The number poles needed to meet force requirement in continuous operation is computed in Sections 3.6 and The magnetic end poles are set as radial magnetized PMs with half axial length, i.e., end poles present radial length τ r /2 in order to minimize back-iron flux. Table 1 presents design variables, and Figure 3 shows influence four parametric variables with variation between its lower and upper limits, according to evaluated values. Table 1. Design variables evaluated for case study. Variable Evaluated Values Step N CPMs N PMi τ r /τ p n f orm In Figure 3, geometric dimensions affected by parametric variables highlighted in bold below its respective figure are indicated. The parametric variable N CPMs affects radial length coils L rcoil and radial length inner and outer arrays PMs, L rpmi and L rpmo, respectively. The radial lengths PMs are also altered by N PMi. The parametric variable given by ratio τ r /τ p has effect over radial length inner back-iron L rpmi, radial length outer back-iron L rpmo, axial length radially magnetized PMs τ r, and axially magnetized PMs τ a. The radial lengths back-irons are also a function n f orm. In addition, τ p is also affected by n f orm.

8 Sensors 2016, 16, Sensors 2016, 16, Figure Figure D 2D parametric parametric models models one one pole pole pitch pitch presenting presenting limits limits parametric parametric variation variation according according to Table 1: Variations (a) NCMPs = 0.2; (b) NCPMs = 0.5; (c) NPMi = 0.3; (d) NPMi = 0.7; (e) τr/τp = 0.5; (f) τr/τp to Table 1: Variations (a) N CMPs = 0.2; (b) N CPMs = 0.5; (c) N PMi = 0.3; (d) N PMi = 0.7; (e) τ r /τ p = 0.5; = 0.85; (g) nform = 0.4; and (h) nform = 1.4. While one parametric variable was varied ors were kept (f) τ r /τ p = 0.85; (g) n f orm = 0.4; and (h) n f orm = 1.4. While one parametric variable was varied constant with NCPMs = 0.4, NPMi = 0.6, τr/τp = 0.75, and nform = 1.2. The variable nbi was also kept constant ors were kept constant with N CPMs = 0.4, N PMi = 0.6, τ r /τ p = 0.75, and n f orm = 1.2. The variable and equals 0.4 in all cases. n BI was also kept constant and equals 0.4 in all cases Electromagnetic Material Properties Electromagnetic Material Properties The electromagnetic material properties should correspond to properties materials employed The electromagnetic in construction material properties actuator, should if such correspond data are available. to In properties this case study, materials employed properties inwere construction defined as default actuator, by Ansys ifmaxwell such data library, are available. with values In this at case a temperature study, properties 25 C, were once defined main as default scope bythis Ansys paper Maxwell is about library, design with values methodology at a temperature itself, and not 25 C, comparison once main scope with experimental this paper isresults aboutat this design stage. methodology itself, and not comparison with experimental results at this stage.

9 Sensors 2016, 16, The main material properties are as follows: PMs made sintered NdFeB with nominal maximum energy product 278 kj/m 3, B r (T 0 ) = 1.23 T, H c (T 0 ) = 890 ka/m, where T 0 is temperature 25 C, and µ r = 1.099; back-iron pieces were set with nonlinear B-H curve steel 1010 with B sat = 2.0 T; coil made annealed copper with bulk conductivity σ cooper 5.8 ˆ 10 7 S/m and µ r = ; and reel was set to use a high-temperature polymeric material, named Teflon, with µ r = 1 and zero conductivity Electrical Loading The initial electrical loading should be an estimation what effective current density could be applicable in continuous operation mode. For conventional rotating machines, this is available in references, e.g., [18], but for linear actuators with more complex heat transfer systems this is hardly known, if no appropriate study were carried out. In this case, it was assumed that effective current density is similar to one used in electrical machines with natural convection, i.e., J rms = 3 A/mm Geometrical Constraints The geometrical constraints applied to this specific case study are summarized in Table 2. Table 2. Design constraints case study. Variable Evaluated Values Inner radius internal PMs (R ipmi ) 18 mm Outer radius external PMs (R opmo ) 38 mm Inner and outer mechanical gap (M Ci and M Co ) 1 mm Radial length Reel (L rreel ) 1 mm Radial length inner and outer PMs ě3 mm Back-iron factor (n BI ) 0.4 Ring-shaped PMs have limitations applied to inner radius internal array R ipmi and to outer radius external array R opmo. By assigning a constant value to se variables, active region to be analyzed is limited radially. The limitation applied to R opmo can be justified due to restrictions physical space where actuator must be installed or due to magnetizer restriction. In this work, it was set considering limitation imposed by magnetizer fixture available, i.e., model X-Series from Magnet Physik, which presents a maximum outer radius approximately 38 mm. The limitation imposed on R ipmi is justified by need for radial magnetization and to enable air flow for cooling actuator. The radial magnetization on ring-shaped PMs would be physically impractical on a cylinder if R ipmi is zero; refore, a limitation imposed by magnetizer fixture available, if ideal radial magnetization is desired, may apply. On or hand, to enable air flow passing through a hollow shaft in inner back-iron, discussed in Section 3.8, with a transversal area with same order magnitude transversal area mechanical air-gaps, requires R ipmi to be estimated accordingly. In this case study, it is considered that radially magnetized PMs are active with segmentation and parallel magnetization [10], so that no restriction by magnetizer fixture is directly applicable. Thus, considering R opmo dimension, and needed air flow, R ipmi was initially set as 18 mm; however, if necessary, R ipmi could be reconsidered during project. It should be noted that even if segmentation and parallel magnetization are also applied to PMs outer array, restriction R opmo imposed by magnetizer is still applicable in order to manufacture axially magnetized PMs in a ring shape without segmentation. The mechanical gaps depend on many factors, such as bearing backlash, manufacturing tolerances, rmal expansion materials, etc. In this paper, inner and outer gaps were set as constant with a value that is typically found in literature and in datasheets manufactures and suppliers linear actuators; however, such value might be minimized, if an in-depth study about factors that affects mechanical gaps were performed.

10 Sensors 2016, 16, The radial length reel should be as short as possible, once its presence increases magnetic gap. In this topology, reel is necessary to provide more stiffness to moving coils, and its radial length was defined based on simulation mechanical stress. The constraints applied to radial lengths PMs are discussed in detail in Section 3.7. The back-iron factor was set in order to satisfy condition given by Equation (7); thus, all simulations considered n BI = 0.4. If desired, after defining all or variables, n BI could be optimized in order to possibly reduce some ferromagnetic material inner and outer back-iron pieces. Observing flowchart presented in Figure 2, it is possible to see that if some conditions, addressed in Section 3.7, are not met, new geometrical constraints must be applied. These new geometrical constraints may apply especially to R opmo or to R ipmi, depending on how flexible each one those is Electromagnetic Uncoupled Parametric Analysis Analysis simulation results using 2D or 3D graphics a space defined by four parametric variables presented in Table 1, plus force density and force ripple, can be a difficult and extensive task. An approach that was adopted in this work to analyze results in five dimensions is illustrated in Figure 4. This figure condenses five variables (four parametric variables, which are independent variables, and one objective function variable, which is a dependent variable) on a single graph. It represents a top-oriented view a combination 40 3D graphs, in a way that color map indicates dependent variable, which, in case Figure 4, is force density. The vertical axis holds two independent variables: τ r /τ p, which is indicated, and N CPMs on minor vertical grid each rectangle, which is not indicated in Figure 4 so that this figure is not visually overloaded. The horizontal axis also holds two independent variables, i.e., N PMi, as indicated, and n f orm on minor horizontal grid each rectangle. Therefore, each rectangles in Figure 4 represents a variation parametric variables n f orm, on horizontal axis, and N CPMs on vertical axis. As an example, a 3D view one rectangles Figure 4, one with τ r /τ p = 0.75 and N PMi = 0.60, is shown in Figure 5. The form presentation five variables in a single figure allows one to compare results for entire domain, which represent combination all parametric variables according to Table 1, in Sensors a comprehensive 2016, 16, 360 way Force Density [N/m 3 ] x r / p N PMi Figure 4. Results Figure 4. Results force density force density for Jfor rms Jrms = 3= A/mm 3 2 andθe θ= e π/6, = π/6, uncoupled uncoupled design for design four for four parametric variables: τr/τp, NPMi, NCPMs, and nform. parametric variables: τ r /τ p, N PMi, N CPMs, and n f orm. 3 x /m 3 ]

11 N PMi Figure 4. Results force density for Jrms = 3 A/mm Sensors 2016, 16, and θe = π/6, uncoupled design for four parametric variables: τr/τp, NPMi, NCPMs, and nform. 3 x Force Density [N/m 3 ] N CPMs n form Figure Figure D graph with results force density uncoupled design for two parametric variables: NCPMs and nform. Results for Jrms = 3 A/mm 2, θe = π/6, τr/τp = 0.75, and NPMi N = CPMs and n f orm. Results for J rms = 3 A/mm 2, θ e = π/6, τ r /τ p = 0.75, and N PMi = From Figure 4, and with more detail in Figure 5, it is possible to infer that parametric variable The force density F NCPMs has a higher influence d Figures 4 and 5 is given by over force density than nform. This can be explained by a compromise between electrical and magnetic loading, which is directly F related to NCPMs. On or hand, nform is related to interpolar leakage flux, F d z1p which is more significant for lower values this parametric (10) π R opmo2 R 2 5 ipmi τ p variable. A maximum force density N/m 3 is obtained with NCPMs = 0.4 and nform 1.2, where with a Fdifference z1p is axial approximately force produced 30% by one relative pole pitch. to Inminimum actuator simulated with Halbach results arrays, observed. back-irons do not have a significant effect, in fact less than 10% if y are not used at all [11]. Yet, in this work it was decided to keep m to increase performance and for mechanical support. However, y were not taken into account for computation force density, because n BI was not considered in parametric analysis; refore, re would be no guarantee that force density is maximized if it were computed considering back-irons. It should be noted that shape, or color map, as presented, force density would not be affected by its initial electrical loading uncoupled model, because effective current density is constant; however, its absolute value would be linearly proportional. From Figure 4, it can be seen that variable τ r /τ p has a greater influence on force density in relation to N PMi. This can be explained by increased volume PMs with radial magnetization as τ r /τ p increases, since it leads to higher levels radial component induction in magnetic gap, while N PMi slightly alters total volume PMs and distribution this volume between internal and external PMs. Consequently, mean radius coil is shifted and so its volume is altered; however, this effect is more significantly observed with N CPMs. From Figure 4, and with more detail in Figure 5, it is possible to infer that parametric variable N CPMs has a higher influence over force density than n f orm. This can be explained by a compromise between electrical and magnetic loading, which is directly related to N CPMs. On or hand, n f orm is related to interpolar leakage flux, which is more significant for lower values this parametric variable. A maximum force density 2.92 ˆ 10 5 N/m 3 is obtained with N CPMs = 0.4 and n f orm ě 1.2, with a difference approximately 30% relative to minimum simulated results observed.

12 Sensors 2016, 16, Sensors 2016, 16, The minimization force ripple is a desirable feature with regard to system linearity, The minimization force ripple is a desirable feature with regard to system linearity, reducing problems related to positioning control and minimizing inclusion oscillations reducing problems related to positioning control and minimizing inclusion oscillations by by actuator when speed is not zero. Depending on electric drive technique, re are actuator when speed is not zero. Depending on electric drive technique, re are timedependent induced harmonics that may generate force ripple during dynamic operation; however, time-dependent induced harmonics that may generate force ripple during dynamic operation; however, se harmonics are not taken into account in this study, once force ripple is computed based on a se harmonics are not taken into account in this study, once force ripple is computed based on a magnetostatic model. magnetostatic model. The The force force ripple ripple can can be be estimated estimated statically statically calculating calculating difference difference between between forces forces obtained obtained with with current current in in 2-phase 2-phase and and 3-phase, 3-phase, corresponding to, to, e.g., e.g., θ e and θ e = π/6, respectively, in relation θe = 0 and θe = π/6, respectively, in relation to to average average value value between se two cases, according to following equation: 33-2 F 2 F F ripple (11) (11) F 3 ` F 2 Figure 6 6 shows an an evaluation force F(θe) e ) normalized in relation to toits itsmean value computed using using its its maximum and and minimum, i.e., 2F(θ 2F(θe)/(F3 e )/(F 3 + F2). F 2 ). Force results were obtained over over a θe a θrange e range degrees for for four four τ r τr/τp p variations with NCPMs N CPMs = 0.4, = 0.4, NPMi N PMi = 0.6, = nform 0.6, n= f 1.2 orm and = 1.2 Jrms and = 3 JA/mm rms = 3 2. A/mm From 2. From Figure Figure 6, it 6, is itpossible is possible to infer to infer that that maximum and and minimum values values force force occur occur at two at two particular electrical electrical angles angles as as discussed discussed in in previous previous paragraph; thus, thus, a fair a fair estimation estimation variation variation force canforce be carried can be carried out by out evaluating by evaluating F2 F 2 and Fand F3 3 and computing and computing force force ripple ripple according according to to Equation Equation (11). Figure (11). 6Figure also shows 6 also shows that for that τ r /τ for τr/τp p = 0.75 = 0.75 force force ripple ripple observed observed significantly significantly decreased; decreased; in in fact fact it is 0.44%, it is 0.44%, whereas whereas for τ r /τ for τr/τp p = 0.55 = 0.55 it is it 4.87%, is 4.87%, i.e., i.e., more more than than 10% 10% higher higher for for former former case. case. 3 2 Normalized Force r / p =0.55 r / p =0.65 / =0.75 r p / =0.85 r p Electrical angle (degrees) Figure Figure Normalized force computed over a θe a θrange e range 60 degrees 60 degrees for four for τr/τp fourvariations τ r /τ p variations with NCPMs with N CPMs = 0.4, = NPMi 0.4, = N0.6, PMi nform = 0.6, = 1.2, n f orm and = Jrms 1.2, = 3 and A/mm J rms 2. = 3 A/mm 2. Parametric simulation results F3 and F2 applied to Equation (11) are shown in Figure 7, where Parametric simulation results F 3 and F 2 applied to Equation (11) are shown in Figure 7, results absolute value static force ripple are presented. The results suggest that τr/τp is closely where results absolute value static force ripple are presented. The results suggest that τ r /τ p related to force ripple, and with τr/τp = 0.75 produces approximately zero force ripple regardless is closely related to force ripple, and with τ r /τ p = 0.75 produces approximately zero force ripple values or variables. regardless values or variables.

13 Sensors 2016, 16, Sensors 2016, 16, Ripple Force [%] r / p N PMi Figure Figure Results Results absolute absolute percentage percentage static static force force ripple uncoupled uncoupled design design for for four four parametric parametric variables: variables: τ τr/τp, NPMi, NCPMs, and nform. r /τ p, N PMi, N CPMs, and n f orm Definition Dimensional Values for One Pole Pitch Uncoupled Design 3.5. Definition Dimensional Values for One Pole Pitch Uncoupled Design Based on result obtained in Section 3.4, and considering geometrical constraints Table 2, Based dimensional on values result obtained for geometrical in Section parameters 3.4, and considering actuator in geometrical radial direction constraints can be Table determined. 2, dimensional For this topology, values for geometrical parametric parameters variables for uncoupled actuator inmodel radial chosen direction were can be determined. τr/τp = 0.75, NPMi = For 0.6, this topology, NCPMs = 0.4 and parametric nform = 1.2. Results variables for for uncoupled model are model summarized chosen were in τ r /τ Section p = 0.75, 4. Dimensional N PMi = 0.6, Nresults CPMs = in 0.4 axial anddirection n f orm = 1.2. are computed Results forin uncoupled next subsection. model are summarized in Section 4. Dimensional results in axial direction are computed in next subsection Computation Axial Active Length, Total Length, and Number Poles Uncoupled Design 3.6. Computation Axial Active Length, Total Length, and Number Poles Uncoupled Design The active volume actuator that is able to cope with effective force specifications can be obtained The active by dividing volume rated actuator force Fr (Section that is able 3.1) by to cope force withdensity effective Fd obtained force specifications for one pole pitch can be obtained (Section by3.4). dividing The axial active rated force length F r (Section actuator 3.1) by Lz is n force calculated density Fby d obtained relating it for with oneits pole active pitch (Section volume, 3.4). so The that axial active length actuator L z is n calculated by relating it with its active volume, so that Lz F r L z 2 2 (12) F Fd ( R ) d πpr opmo2 R 2 (12) ipmi q The The axial axial length armature LLzW is simply sum Lz L z and specified stroke S S (Section 3.1), 3.1), whereas total total axial axial length L LzT must be at least sum Lz L z with twices: S: # Lz W z S L zw L z ` S Lz T 2S L zt L z ` 2S (13) (13) The number poles P actuator can also be determined based on previous results. It was defined that P must be an integer even number, and that each inner and outer magnets extremities count as one pole, even though y present radial length τr/2. Therefore, P is given by

14 Sensors 2016, 16, The number poles P actuator can also be determined based on previous results. It was defined that P must be an integer even number, and that each inner and outer magnets extremities count as one pole, even though y present radial length τ r /2. Therefore, P is given by P 2ceil L z ` 2n f orm br 1 (14) 2 2 opmo2 R ipmi where operator ceil rounds element to nearest even integer towards positive infinity [19]. In order to obey condition imposed by Equation (14), L z or n f orm must be recalculated. In this case, it can be observed from Figure 5 that n f orm slightly affects F d over a range 0.8 up to 1.4. That means it could be adjusted without having a significant effect on overall results. n f orm can be recalculated according to n f orm L z b 2 pp 1q R opmo2 R ipmi It is important to observe that as n f orm was recalculated, so must τ p be, according to Equation (3). If choice was to adjust L z, this could lead to an unnecessary increase in active volume device. If n f orm, calculated by Equation (15), returns a value outside range , e.g., it results from fact that P was adjusted to nearest even number towards positive infinity, but it was very close to an even number towards zero. In this situation, P should be set as closest even number to zero, if dimensional design conditions verified in next subsection are met, and final n f orm calculated with Equation (15). For case study, values for L z, L zw, and L zt were mm, mm, and mm, respectively. P was found to be 4 with a final n f orm Validity Parameters Uncoupled Design This step in designing process verifies wher results obtained so far are consistent in sense that final shape actuator presents acceptable dimensions and parameters, which are defined by following inequalities: $ & % L z L zw ě 0.5 P ě 4 L rpmi ě 3 mm L rpmo ě 3 mm The first inequality, which is ratio between axial active length and windings axial length, is set so that at least half total length windings is active during operation. This ratio is related to efficiency actuator, once re are Joule losses along total length windings, but only portion that is placed within active length produces force. On or hand, second inequality in Equation (16) requires that minimum number poles actuator is four. This criterion is established in order to limit way that end effect affects overall performance actuator. End effect is intrinsic to linear machines and is more significant if device has a low number poles. The third and fourth inequalities Equation (16) limit radial length permanent magnets to a minimum, which are imposed to prevent PMs from breaking during assembly, which would easily happen if radial length PMs is too small. The shaded area in Figure 8 indicates a domain parametric variables N PMi and N CPMs in which radial length both inner and outer PMs are bigger than 3 mm, whereas areas that are not shaded indicate wher inner or outer magnets present a radial length smaller than specified. (15) (16)

15 Sensors 2016, 16, become larger. The radial length inner and outer PMs is also higher than 3 mm, once in Section 3.4 NPMi and NCPMs were defined as 0.6 and 0.4, respectively, which, according to Figure 8, lies within allowed domain. If former discussed conditions were not attained, it would be suggested to Sensors 2016, alter 16, NPMi, 360which presents very little sensitivity in relation to design objective, instead changing NCPMs. 0.7 L rpmo < 3 mm 0.6 N PMi L < rpmi 3 mm N CPMs Figure 8. Limitation radial length PMs given by inequalities 3 and 4 in Equation (16) as a Figure 8. Limitation radial length PMs given by inequalities 3 and 4 in Equation (16) as function parametric variables NPMi and NCPMs, where LrPMi = (RoPMi RiPMi) is radial length a function parametric variables N inner PMs and LrPMo = (RoPMo RiPMo) is PMi and N radial CPMs, where L length outer rpmi = (R PMs. opmi R ipmi ) is radial length inner PMs and L rpmo = (R opmo R ipmo ) is radial length outer PMs. The inequalities given by Equation (16) could be different from ones that were set; y are For basically case criteria under imposed study, by it was designer. found that P = 4 and L z /L zw = 0.59; thus, both criteria were simultaneously met in first place and no change in geometrical constraints was required Definition Thermal Topology If, for example, resultant L z /L zw were lower than 0.5, n geometrical constraints ought to be The heat sources associated with losses an actuator can be many kinds; however, in this reviewed, as first inequality Equation (16) would not be complied with. In this case wher particular actuator, main source losses is Joule losses at conductors. Very low levels R opmo should be decreased or R induced current appear in ipmi increased, so that active axial length actuator would PMs, which can be neglected. This specific topology presents an become intrinsic larger. low Thelevel radial iron length losses because innerre andis outer no relative PMs ismovement also higher between than 3back-irons mm, onceand in Section PMs 3.4 N PMi and it Npresents CPMs were a coreless defined armature. as 0.6 and Still, 0.4, re respectively, may be induced which, current according back-irons to Figure by 8, lies variation within allowed domain. flux produced If former by discussed armature, but conditions in lower levels were not than attained, would be it observed would be within suggested actuators to alter N PMi, with which cored presents armature veryand little low sensitivity compared into relation Joule losses, to design so it can objective, be neglected. insteadfriction changing losses at N CPMs. The bearings inequalities are also not given considered. by Equation (16) could be different from ones that were set; y are basically criteria In order imposed to achieve by higher designer. force and power density, device must present an improved capacity heat exchange. In proposed methodology, rmal analysis plays an important role, 3.8. Definition since it allows Thermal for determining Topology absolute force density and, reafter, active volume for a given specification. The heat In this sources work, associated it was considered with losses that a forced an actuator air flow was canimposed be many to a hollow kinds; shaft, however, which in this particular insufflates actuator, actuator main with source air at ambient losses temperature. is Joule losses In order at to improve conductors. heat exchange Very low in this levels induced topology, currenta appear forced inlet flow PMs, with which a speed can be 0.5 neglected. m/s enters This hollow specific shaft topology from presents bottom, passes an intrinsic low level through iron inner losses gap, because n passes re through is no relative outer movement gap and outflows between at back-irons top actuator and PMs with and it a higher temperature. This characterizes a forced convection at hollow shaft and at inner and presents a coreless armature. Still, re may be induced current in back-irons by variation outer gaps, which significantly improves heat transfer. flux produced It must by be clear armature, that decision but in lower about wher levels than forced would or natural be observed heat transfer within is applied actuators must with cored armature be made in andaccordance low compared with to Joule mechanical losses, characteristics so it can be neglected. actuator. Friction Depending losses at bearings on are also not operation considered. conditions, actuator may be completely sealed, or, in a different situation, it might be Inopen order and toits achieve own movement higher force produces and power an air flow density, that characterizes device must forced present convection. an improved In any case, capacity heat it exchange. must be defined In at this proposed step because methodology, this significantly rmal affects analysis designing plays an process. important role, since it allows for determining absolute force density and, reafter, active volume for a given specification. In this work, it was considered that a forced air flow was imposed to a hollow shaft, which insufflates actuator with air at ambient temperature. In order to improve heat exchange in this topology, a forced inlet flow with a speed 0.5 m/s enters hollow shaft from bottom, passes through inner gap, n passes through outer gap and outflows at top actuator with a higher temperature. This characterizes a forced convection at hollow shaft and at inner and outer gaps, which significantly improves heat transfer. It must be clear that decision about wher forced or natural heat transfer is applied must be made in accordance with mechanical characteristics actuator. Depending on operation conditions, actuator may be completely sealed, or, in a different situation, it might be open and

16 Sensors 2016, 16, its own movement produces an air flow that characterizes forced convection. In any case, it must be defined Sensors 2016, at this 16, 360 step because this significantly affects designing process Building and Simulation Thermal 2D FEM Full Axial Length Actuator The rmal analysis was performed using FEA by means commercial package ANSYS Fluent. The forms heat transfer considered in simulation were by radiation, by conduction, and by by forced and natural convection. The models for convection and radiation used were Boussinesq and Rosseland, respectively, which are appropriate for for this this topology [20]. [20]. For For given given air flow, air flow, Reynolds Reynolds number number is relatively is relatively low, low, in in order order [21]; [21]; refore, refore, a laminar a laminar flow flow can can be be considered. Additionally, some important assumptions were made: an atmosphere temperature 300 K is is set; set; results are are for for steady state condition; gravity is is 9.81 m/s 2 ;; model is isaxisymmetric; actuator operates with with its its moving axis axis at at a a vertical position; and and temperature windings windings was was set set as as maximum maximum allowable allowable temperature temperature (discussed (discussed in Section in Section 3.9.2), 3.9.2), i.e., i.e., K. K. The The prile static temperature over actuator, considering rmal properties materials and and temperature windings windings (presented (presented in in Sections Sections and and , respectively), respectively), resultant resultant from from simulation simulation is shown is shown in Figure in Figure 9. Although 9. Although axial length axial length actuator actuator is presented is presented in a horizontal in a position horizontal here, position for convenience, here, for convenience, gravity wasgravity set parallel was set to parallel axialto direction. axial direction. Figure Figure Results Results 2D 2D axisymmetric axisymmetric rmal simulation full full length length actuator actuatordesign designwith with dimensions dimensions found found for for uncoupled uncoupled model model but but with with a constant temperature temperature 353 Kat at windings Thermal Material Properties The rmal properties materials which actuator is is composed and and properties properties air air that that were were considered to to perform simulation presented in in Section are are givenin in Table3. 3. These properties were obtained from Ansys Fluent library. Table 3. Thermal properties actuator materials and air used in rmal simulation. Table 3. Thermal properties actuator materials and air used in rmal simulation. Thermal Thermal Density Specific Conductivity Viscosity Material Viscosity Material Emissivity Emissivity Expansion (kg/m³) Heat(J/kg-K) (W/m-K) (kg/m-s) (kg/m³) Heat(J/kg-K) (W/m-K) (kg/m-s) Coefficient (1/K) NdFeB (1/K) - Steel NdFeB Teflon Steel Teflon Air Copper Air Copper

17 Sensors 2016, 16, Windings Maximum Allowable Temperature The maximum allowable temperature conductors at full load could be determined according to ir insulation class. Neverless, if conductor s maximum temperature is respected, maximum operation temperature PMs should also be respected. Sintered NdFeB PMs with a nominal maximum energy product 278 kj/m 3 and a maximum operation temperature 80 C represent one worst cases such materials commercially available [22]. This material specification results in an actuator with lowest force density, thus device s inferior performance limit can be assessed. For this reason, maximum allowable temperature at windings was set as 80 C. As re is no physical contact between windings and PMs, and considering forced convection present in inner and outer air-gap, setting windings temperature as 80 C guarantees that PMs temperature will not reach its maximum. The temperature must be set at windings because that is heat source and this allows for a proper simulation heat exchange phenomena Permanent Magnets Mean Temperature The mean operating temperature PMs at full load can be obtained from rmal simulation presented in Section From simulation, it can be observed that re is a low variance distributed temperature over PMs, which results from fact that y present relatively high rmal conductivity. Results showed that mean temperature inner PMs is 66 C, whereas mean temperature outer PMs is 68 C. In case actuators that do not employ PMs, this step design process and one in Section 3.11 should be neglected, e.g., for reluctance linear motors Calculation Thermal Corrected H c and B r Sintered NdFeB permanent magnets present a high B-H energy product, a recoil permeability that can be considered constant, a magnetic permeability similar to free space, a high coercive field, and a high remanent induction. These characteristics are desired in order to produce an actuator with elevated force density; however, properties PMs are significantly affected by temperature and this should be considered in design process. Manufacturers specify that both H c and B r can be corrected as a function operation temperature in a linear way while operating under maximum allowable temperature [22]. For PMs employed in this case study, B r is reduced by 0.12%/ C, α Br = , and H c is decreased by 0.7%/ C, α Hc = 0.007, according to # B r ptq B r pt 0 qr1 α Br pt T 0 qs H c ptq H c pt 0 qr1 α Hc pt T 0 qs Considering operating temperature inner and outer PMs obtained at preceding subsection, properties H c and B r, and, consequently, relative permeability µ r, are given in Table 4. Table 4. Corrected values inner and outer PM properties as a function operating temperature. (17) Property Inner PMs Outer PMs B r [T] H c [ka/m] µ r [-] It should be observed that as operating temperatures inner and outer magnets are slightly different from each or, properties PMs are also slightly differently affected. In a topology such as dual-halbach array actuator, this could lead to an optimal design with a higher volume inner or outer PMs, i.e., performance could be affected by parametric variable N PMi, especially

18 Sensors 2016, 16, if anor rmal exchange method was applied, and it would lead to more significant differences Sensors 2016, 16, between operating temperature PMs. The The PM PM properties properties presented presented in in Table Table 4 must must be be corrected corrected at at 2D 2D FEM FEM for for one one pole pole pitch, pitch, so so that that toger toger with with maximum maximum effective effective current current density, density, an an absolute absolute value value force force density density can can be be calculated calculated in in which which rmal rmal and and geometrical geometrical constraints constraints are are taken taken into into account. account Determination Overall Heat Transfer Coefficient Determination Overall Heat Transfer Coefficient It is necessary to determine overall heat transfer coefficient because it directly affects It is necessary to determine overall heat transfer coefficient because it directly affects maximum effective current density that can be applied to windings in order to keep maximum maximum effective current density that can be applied to windings in order to keep maximum temperature within limits. temperature within limits. With With systems systems that that present present composite composite forms forms heat heat transfer, transfer, such such as as actuator actuator under under study, study, it it is ten is ten convenient convenient to work to work with an with overall an overall heat transfer heat transfer coefficient coefficient U, which U, is which defined is by defined an expression by an analogous expression to analogous Newton s to law Newton s cooling law [21]: cooling [21]: q x q UA s T T (18) (18) x where T ΔT is is overall temperature difference, AAs s is heat exchange surface area and qx q x is heat rate rate (W). Results obtained from Section 3.10 allow for calculating U and are presented in Figure 10. s U (W/(K.m 2 )) N PMi N CPMs Figure Figure Overall Overall heat heat transfer coefficient as as a a function parametric variables NNPMi and NCPMs. N CPMs. The parametric variable τr/τp does not affect because it does not alter volume electric The parametric variable τ r /τ p does not affect U because it does not alter volume electric conducting material. On or hand, nform affects volume conducting material for one pole conducting material. On or hand, n pitch because axial length one pole varies f orm affects volume conducting material for one pole with nform. Even so, this variable does not affect radial pitch because axial length one pole varies with n length conductors and PMs and, moreover, once f orm. Even so, this variable does not affect total active volume actuator is radial length conductors and PMs and, moreover, once total active volume actuator is determined (Section 3.19), nform is used to define number poles device. Thus, it should be determined (Section 3.18), n noted that U only varies with f orm is used to define number poles device. Thus, it should be parameters NPMi and NCPMs, as shown in Figure 10. noted that U only varies with parameters N PMi and N CPMs, as shown in Figure Maximum Effective Current Density Maximum Effective Current Density The Joule losses PJoule produced by windings can be determined by The Joule losses P Joule produced by windings can be determined by PJoule VwFf J rmsmax (19) 2 P Joule V w F f ρj rmsmax (19) where Vw is total volume windings, Ff is windings fill factor, ρ is resistivity copper, and JrmsMax is maximum effective current density at windings maximum allowable temperature. As only Joule losses are considered, since or kinds losses are negligible, heat rate is equal to PJoule; thus, maximum effective current density is given by 2

19 Sensors 2016, 16, where V w is total volume windings, F f is windings fill factor, ρ is resistivity copper, and J rmsmax is maximum effective current density at windings maximum allowable temperature. Sensors 2016, 16, As 360 only Joule losses are considered, since or kinds losses are negligible, 19 heat 27 rate is equal to P Joule ; thus, maximum effective current density is given by d UA UAs T J T J rmsmax rmsmax V wf f f ρ (20) (20) Only parameters NPMi and NCPMs affect JrmsMax, for an equivalent reason to that explained for Only parameters N PMi and N CPMs affect J rmsmax, for an equivalent reason to that explained U. A 3D plot JrmsMax is shown in Figure 11. for U. A 3D plot J rmsmax is shown in Figure J rmsmax (A/mm 2 ) N CPMs N PMi Figure Figure Maximum effective current density applicable at windings in in order to to reach maximum allowable temperature C C at at windings as a function parametric variables N CPMs and N PMi. NCPMs and NPMi. Maximum effective current densityis is observedfor for parameters NPMi N = 0.3 and NCPMs PMi = 0.3 and N= CPMs 0.2, which = 0.2, which correspond correspond to to relatively relatively lowest lowest volume volume windings. windings. On or On hand, or with hand, NPMi with = 0.7 Nand NCPMs PMi = 0.7 and = 0.5, N CPMs design = 0.5, presents design presents relatively highest relatively volume highest windings. volume In windings. short, designs In short, with designs a relatively with a low relatively volume low volume windings allow windings higher allow levels higher current levelsdensity, current while density, opposite while is true opposite for designs is true for with designs relatively with relatively high volume high volume windings. windings. This is This explained is explained by byfact that fact that a higher a higher volume volume windings windingswould wouldproduce producemore morejoule Joulelosses, losses, whereas whereasits itsoverall overallheat heattransfer transfer coefficientdoes doesnot increase increase at at aa proportional proportional rate, rate, so so its its effective effective current current density density must must be be proportionally proportionally adjusted Simulation Quasi-Static Parametric 2D FEM for One Pole Pitch The The quasi-static parametric model discussed in Section 3.3 can be used at this step. The only modification should should be be parameters BBr Hc r and H c inner and outer magnets, based on results presented presented in in Table Table This This approach provides a fast fast and quite accurate approximation once very little variation variation temperature temperature PMs PMs is is observed due due to to fact that re is no changein in radial length length air-gaps air-gaps and and in in maximum allowable temperature windings while performing parametric parametric simulation. simulation. There There is is a slight slight difference in in air flow through inner inner and and outer outer gap gap associated with parameters NCPMs and NPMi associated with parameters N once those modify volume air-gap. It was CPMs and N PMi once those modify volume air-gap. It was observed observed from from simulation simulation that that this this difference difference in in air-gap air-gap volume volume alters alters mean mean temperature temperature magnets magnets by less by less than than 3%. 3%. From flowchart Figure 2, it is possible to observe that maximum effective current density for each design is not an input this step. This results from fact that force density has a linear relationship with current density. So, in order to simplify analysis, model was simulated with 1 A/mm 2, and, reafter, absolute maximum force density for each design was obtained in

20 Sensors 2016, 16, From flowchart Figure 2, it is possible to observe that maximum effective current density for each design is not an input this step. This results from fact that force density has a Sensors linear 2016, relationship 16, 360 with current density. So, in order to simplify analysis, model was simulated with 1 A/mm 2, and, reafter, absolute maximum force density for each design was obtained a inpost-process, a simply simply by bymultiplying each eachcorresponding value following results obtained in Section This is done as a post-process, in order to speed up method. If results do not meet specific design criteria, assessed in Section 3.18, geometrical constraints for one pole pitch must must be be reviewed. For For this this case case study, study, parameters parameters RoPMo Ror opmo RiPMi orshould R ipmi be should altered be altered and this and should this should be applied be applied to tormal rmal and and electromagnetic models, models, as indicated as indicated Figure in Figure Electromagnetic Coupled Analysis A similar approach to to one one presented in Section in Section 3.4 is 3.4 carried is carried out at out thisat step, this i.e., step, first, i.e., first, results results for forcefor density force density are evaluated are evaluated as a function as a function parametric parametric variables variables τ r /τ p, N PMi τr/τp,, NNPMi, CPMs, NCPMs, and n f and orm. After nform. After considering considering ripple ripple force, force, ratio ratio τ r /τ τr/τp p is is chosen and, considering geometrical constraints, NPMi is isdefined. A 3D graph force density for for chosen τ r τr/τp /τ p and NNPMi is presented detailing results as a function parametric variables NCPMs and nform. n f orm. It is important to reinforce that, at this stage, effective force density is corrected in a post-process routine using Matlab script for reasons discussed in previous subsection. Each design evaluated presents a maximum effective current density, resulting in maximum allowable temperature at windings. This means that force density values presented in Figure 12 are absolute maximum considering energy drop at PMs and rmal constraint. Force Density [N/m 3 ] x r / p N PMi 1.4 Figure 12. Results force density for effective current density according to to Figure 11 coupled design for four parametric variables: τr/τp, τ NPMi, NCPMs, and nform. r p, N PMi, N CPMs, and n f orm. The results presented in Figure 12 are significantly different from those presented in Figure 4, especially because parametric variable NCPMs, which is directly related to volume windings. It is important to emphasize that in Figure 4 force density for uncoupled model is obtained with constant effective current density; on or hand, in Figure 12, in coupled model, effective current density is adjusted according to Equation (20). This becomes even more evident when comparing Figures 5 and 13, where independent parametric variables are clearly identified, and force density dependency shows that re is a shift in maximum point in relation to NCPMs, from 0.4 for uncoupled model to 0.25 for coupled model. The parametric

21 Sensors 2016, 16, The results presented in Figure 12 are significantly different from those presented in Figure 4, especially because parametric variable N CPMs, which is directly related to volume windings. It is important to emphasize that in Figure 4 force density for uncoupled model is obtained with constant effective current density; on or hand, in Figure 12, in coupled model, effective current density is adjusted according to Equation (20). This becomes even more evident when comparing Figures 5 and 13 where independent parametric variables are clearly Sensors identified, 2016, 16, and 360 force density dependency shows that re is a shift in maximum point 21 in 27 relation to N CPMs, from 0.4 for uncoupled model to 0.25 for coupled model. The parametric variable NNCPMs CPMs directly affects volume Vw V w and andexternal surface area As A s windings, and se are used to calculate JrmsMax. J rmsmax. 3 x 10 5 Force Density [N/m 3 ] N CPMs n form Figure 13. 3D graph with results force density coupled design for two parametric variables: NCPMs N CPMs and andnform. n f orm Results. Results for for τr/τp τ r = /τ 0.75 p = and 0.75NPMi and = N0.60. PMi = Moreover, absolute maximum force density has also changed from 2.92Ripple ˆ 10 5 N/m Force 3 [%] for uncoupled model to 2.56 ˆ 10 5 N/m 3 for coupled model. This will directly affect total active 0.85 volume actuator and, consequently, its dimensional parameters. Even though figures maximum force density are close to each or, in this case, y result from a good estimation 6 maximum 0.80 electrical loading. The results could significantly differ, e.g., if: initial effective current density were set as 5 A/mm 2 for uncoupled model, which is ten employed in conventional electrical machines; if maximum allowable temperature at windings were set as 120 C, which 5 in general 0.75 is acceptable because its insulation class; or if NdFeB PMs operation temperature were higher. The absolute force ripple presented in Figure 14 shows behavior similar to that in Figure 4 7, i.e., for0.70 τ r /τ p = 0.75, percentage ripple force is approximately zero. These results were expected because this parametric variable is not affected by rmal issues because it does not alter volume windings or volume PMs r / p

22 Sensors 2016, 16, Figure 14. Results absolute percentage static force ripple coupled design for four parametric variables: τ r /τ p, N PMi, N CPMs, and n f orm Definition Dimensional Values for One Pole Pitch Coupled Design Based on results obtained in Section 3.15, and considering geometrical constraints given in Table 2, dimensional values for geometrical parameters actuator in radial direction can be determined. The parametric variables chosen were τ r /τ p = 0.75, N PMi = 0.6, N CPMs = 0.25, and n f orm = 1.0. Applying chosen parametric variables to equations in Section 3.3 in conjunction with parameters obtained in next section, it is possible to finalize design for coupled model in a similar way as for uncoupled model. Results for coupled model are summarized and compared to coupled model in Section Computation Axial Active Length, Total Length, and Number Poles Coupled Design This step is same as in Section 3.6, except for fact that this time results parametric variables are based on coupled model. For case study, values for L z, L zw, and L zt were mm, mm, and mm, respectively. P was found to be 6 with a final n f orm Verification if Parameters Coupled Design Are Valid The same criteria as adopted in Section 3.7 were applied at this step. For coupled model, it was found that p = 6 and L z /L zw = 0.624; thus, those two criteria were simultaneously met. The radial length inner and outer PMs is higher than 3 mm, once in Section 3.15 N PMi and N CPMs were defined as 0.6 and 0.25, respectively. That, according to Figure 8, lies within allowed domain.

23 Sensors 2016, 16, If any criteria given by Equation (16) are not met, geometrical constraints should be reviewed; this ought to be applied to rmal and electromagnetic coupled model in order to perform a second coupled analysis. 4. Results and Discussion This section discusses influence each parametric variable studied for case under study; it also presents final designs for coupled and uncoupled models and compares results. For topology under study, it was found that parametric variable τ r /τ p presents high sensitivity in relation to ripple force. Setting τ r /τ p = 0.75 was sufficient to take ripple force practically to zero, regardless value or parametric variables. It should be observed that value τ r /τ p for which ripple force becomes close to zero may change for different geometrical constraints. Differences between uncoupled and coupled models were found to be insignificant regarding how τ r /τ p affects ripple force. The parametric variable N PMi slightly affected design objectives, i.e., high force density and low ripple force. Results for N PMi from 0.5 up to 0.7 were almost same. In this case, an N PMi value that leads to a relatively lower volume magnets could be applied, which, in summary, represents setting a higher value for N PMi. However, N PMi is limited by constraint related to minimum radial length PMs. Also, as discussed in Section 3.11, N PMi could become more relevant for coupled model if differences between temperature inner and outer PMs were higher; thus, PMs with lower temperature should present relatively greater volume, which can be achieved by adjusting N PMi. The most relevant parametric variable in relation to force density is N CPMs, especially when coupled and uncoupled models are compared to each or. For uncoupled model, with a constant current density no matter volume active windings, force density was maximized for N CPMs equals to 0.4. On or hand, with coupled model, in which effective current density is corrected in order to keep temperature in actuator limited to its specified maximum, force density was best when N CPMs is equal to In a simple way, this implies that, if temperature limits are considered, electrical loading is decreased in detriment an increase in magnetic loading, once Joule losses are limited. The parametric variable n f orm plays an important role in defining number poles P actuator. In both uncoupled and coupled models, it is also relevant to find best force density; neverless, it showed low sensitivity in range 0.8 to 1.4. For low values n f orm, such as 0.4, re is significant leakage flux between adjacent poles because se poles become close to each or, thus reducing force density. The low sensitivity observed helps in designing process because n f orm can be defined in a wide range so that number poles can be set with a pre-defined axial active length. According to proposed methodology, number poles and final dimensions actuator are those found for coupled model, although results uncoupled model are indispensable to give a first approximation in order to enable rmal analysis. For sake comparison, Table 5 presents number poles and dimensions each design, while Figure 15 presents two axisymmetric finite element models, showing ir final shape. Table 5. Number poles and dimensional results for uncoupled and coupled model. Dimensions are presented in mm. Design P R i R opmi R ireel R ic R oc R ipmo R o τ r τ z L z L zw L zt Uncoupled Coupled

24 Table 5. Number poles and dimensional results for uncoupled and coupled model. Dimensions are presented in mm. Design P Ri RoPMi RiReel RiC RoC RiPMo Ro τr τz Lz LzW LzT Sensors Uncoupled 2016, 16, Coupled (a) (b) Figure 15. Axisymmetric view electromagnetic actuator for: (a) uncoupled model with P = Figure 15. Axisymmetric view electromagnetic actuator for: (a) uncoupled model with P = 4, 4, NCPMs = 0.4, NPMi 0.6, τr/τp = 0.75, and nform = , and nbi = 0.4; (b) coupled model with 6, N CPMs = 0.4, N PMi = 0.6, τ r /τ p = 0.75, and n f orm = , and n BI = 0.4; (b) coupled model with P = 6, NCPMs = 0.25, NPMi = 0.6, τr/τp = 0.75, = nform = , and nbi = 0.4. N CPMs = 0.25, N PMi = 0.6, τ r /τ p = 0.75, = n f orm = , and n BI = 0.4. Temperatures in actuator coupled model are those presented in Figure 9, once effective Temperatures current density in was actuator adjusted to obtain coupled those model results. are The those maximum presented effective in Figure current 9, once density effective for coupled current design density is was 3.71 adjusted A/mm 2, to which obtain is higher those results. than initially The maximum estimated; effective however, current it should density be for noted that coupled re design is a lower is 3.71 volume A/mm 2 windings, which is than higher previously than initially mentioned. estimated; Finite however, element it simulation should be noted that axisymmetric re is a lower model volume Figure windings 15b resulted than in previously an axial force mentioned Finite N, which element is close simulation to specifications, axisymmetric i.e., 120 model N. These Figure results 15b show resulted that in an end axial effect force has little significance, N, which is which close to is specifications, i.e., 120 N. These results show that end effect has little significance, which is consistent with assumption that PMs with radial magnetization at end poles should present an axial length τ r /2, as expected. In contrast, temperatures for uncoupled design were simulated with a current density 3 A/mm 2. In this case mean temperature windings was approximately 90 C, whereas mean temperature was approximately 74 C for inner PMs and approximately 76 C for outer PMs. The temperatures at PMs are close to maximum specified by manufacturer, i.e., 80 C [22], which can easily lead to demagnetization, especially if high transitory currents are applied during operation. A FEM simulation axisymmetric uncoupled model shown in Figure 15a, with 3 A/mm 2 without considering temperature effect over PMs, i.e., with B r = 1.23 T and H c = 890 ka/m, resulted in N axial force. Now, if same current density is applied, but PMs remanent induction and coercive force are adjusted according to ir temperature, as given by Table 6, axial force produced by actuator is 92.5 N. This result shows relevance temperature to PMs and, thus, to performance actuator, once axial force decreased by 22.9% in relation to specification. Table 6. PM properties uncoupled model, adjusted according to ir operating temperature with electrical loading 3 A/mm 2. Property Inner PMs Outer PMs B r [T] H c [ka/m] µ r [-] In fact, uncoupled model could operate at same temperature as coupled one; however, in this case, maximum effective current density should be 2.75 A/mm 2. Considering this effective current density and adjusting PMs parameters according to ir operating temperature, presented in Table 4, axial force produced with this design drops to 89.9 N, which represents a 25% reduction when compared to specifications.

25 thus, to performance actuator, once axial force decreased by 22.9% in relation to specification. Table 6. PM properties uncoupled model, adjusted according to ir operating temperature Sensorswith 2016, electrical 16, 360 loading 3 A/mm Property Inner PMs Outer PMs The magnetic induction Br final [T] actuator design (shown in Figure 15b), evaluated via finite element analysis, is shown in Hc Figure [ka/m] 16. It can 585 be observed 572 that saturation is not reached in back-irons, although inner back-iron µr [-] presents a higher magnitude induction when compared to outer one. This suggests that inner and outer back-irons could be adjusted independently from each or. In fact, It should uncoupled be noted that model could resultsoperate shown inat Figure same 16 were temperature obtained by as applying coupled a nominal one; effective however, current in this density; case, refore, maximum small effective levels current asymmetry density should in be flux 2.75 lines A/mm and 2. inconsidering distribution this effective magnetic current induction density are and caused adjusting by armature PMs parameters reaction. Inaccording terms magnitude, to ir operating peak temperature, value induction presented in in Table middle 4, axial magnetic force produced gap is approximately with this design 0.55 drops T, whereas to 89.9 it N, iswhich nearlyrepresents 1.96 T in a inner 25% reduction back-iron. when compared to specifications. Sensors 2016, 16, or. It should be noted that results shown in Figure 16 were obtained by applying a nominal Figure Figure Magnitude Magnitude magnetic magnetic induction induction and and flux flux lines lines a section a section final final actuator actuator design design with effective current density; refore, small levels asymmetry in flux lines and distribution nominal with magnetic nominal effective induction effective current current are density. caused density. by armature reaction. In terms magnitude, peak value induction in middle magnetic gap is approximately 0.55 T, whereas it is nearly 1.96 T in ItThe is inner mentioned magnetic back-iron. induction Section final that actuator volume design back-irons (shown in may Figure be optimized 15b), evaluated after via finite active volume element analysis, It coupled mentioned is shown model in Section in isfigure determined that It can The volume be design observed back-irons methodology that saturation may be was optimized carried is not reached after out considering in active backirons, 0.4; however, n BI = volume although in Figure coupled inner 17, model back-iron force is determined. presents and force The a higher density design magnitude methodology final induction design was carried when out actuator, considering compared according to nbi = 0.4; however, in Figure 17, force and force density final design actuator, touter Figure one. 15b, This were suggests obtained that inner as a function and outer back-irons n BI. Thecould force be density, adjusted in this independently case, is computed from each by according to Figure 15b, were obtained as a function nbi. The force density, in this case, is computed considering a volume given byπ R o2 R 2 2 i 2 by considering a volume given by R L z. o Ri Lz. Force [N] x Force Density [N/m 3 ] n BI Figure 17. Force and force density final actuator design as a function parametric variable Figure 17. Force and force density final actuator design as a function parametric variable nbi obtained with nominal effective current density. n BI obtained with nominal effective current density. It can be observed that for nbi = 0 total force is approximately 90 N, while for nbi = 0.8 axial force is 122 N. More importantly, with nbi = 0.4 re is almost no force drop in relation to its maximum. For nbi smaller than 0.4, a magnetic potential drop is present in back-irons, which is reflected as a reduction in radial component magnetic induction in magnetic gap, resulting in lower levels force. On or hand, force density was found to be maximized for nbi = 0.15, with approximately

26 Sensors 2016, 16, It can be observed that for n BI = 0 total force is approximately 90 N, while for n BI = 0.8 axial force is 122 N. More importantly, with n BI = 0.4 re is almost no force drop in relation to its maximum. For n BI smaller than 0.4, a magnetic potential drop is present in back-irons, which is reflected as a reduction in radial component magnetic induction in magnetic gap, resulting in lower levels force. On or hand, force density was found to be maximized for n BI = 0.15, with approximately 1.28 ˆ 10 5 N/m 3. Neverless, at maximum force density, force is equal to N, which does not meet specification 120 N. Force density decreases significantly for larger values n BI once volume actuator increases and force is almost constant. In order to comply with force requirement without compromising force density, n BI could be set as 0.3, nearly 120 N. Thus, it implies that R i = mm and R o = mm. The presented design methodology was shown to be effective in designing a dual-halbach array linear actuator with rmal-electromagnetic coupling, once results obtained for final design meet specifications given in Section 3.1 while geometrical constraints Section and rmal constraints Section are considered. The main strengths design methodology presented can be highlighted as follows: it provides a step-by-step method for designing an actuator so that its overall dimensions can be obtained based on a one pole pitch electromagnetic analysis, which is simple and fast to compute; it applies rmal-electromagnetic coupling, which allows one to obtain simulation results that are expected to be in good agreement with experimental ones; and it considers rmal constraints applied to windings, ensuring that maximum temperature is not exceeded on eir PMs and windings, by means specification a maximum effective current density for continuous operation with safety. The main drawbacks methodology are as follows: use parametric analysis is a quite extensive work, even though simulation can be fast for 2D models; design procedure can be laborious if parameter verification defined by inequalities given in Sections 3.7 and 3.18 is not met in first place; and final design is not necessarily optimal, because parametric simulation is a discrete domain, although result can be very close to optimal if discretization parametric variables with higher sensitivity is increased. 5. Conclusions The methodology presented is effective and allows one to obtain in a step-by-step manner a final design considering rmal and geometrical constraints. Such a methodology is an important tool to assist in design process linear cylindrical actuators, and its general concepts could be applied to a wide range actuator topologies. The results study showed that if rmal effect is not considered, device may not meet specifications; noneless, in order to meet specifications, operating temperatures may exceed maximum values specified for materials. It is important to note that, when considering electromagnetic-rmal coupling, optimum design can be significantly different from a model that only considers electromagnetic behavior, especially with regard to ratio between volume windings to volume permanent magnets. Furr work should produce analytical models for rmal and electromagnetic analysis and apply a mamatical optimization method in order to find a global maximum in an automatic way instead performing parametric analysis. The inner and outer back-irons may also be taken into account separately in an optimization process in a future work.

27 Sensors 2016, 16, Acknowledgments: The authors acknowledge financial support Post-Graduate Program in Electrical Engineering Federal University Rio Grande do Sul to cover publishing in open access. Author Contributions: Paulo Eckert contributed to overall study, developed methodology, built electromagnetic model, performed parametric simulations, analyzed data, and wrote paper. Aly Ferreira Flores Filho and Eduardo Perondi supervised work, edited manuscript, and provided valuable suggestions to improve this paper. Jeferson Ferri carried out rmal simulation and wrote correspondent subsection. Evandro Goltz helped to develop methodology, assisted to build parametric model, analyzed data, and wrote four subsections paper. Conflicts Interest: The authors declare no conflict interest. Abbreviations The following abbreviations are used in this manuscript: PMs NdFeB DC FEM FEA Permanent magnets Neodymium Iron Boron Direct current Finite element model Finite element analysis References 1. Boldea, I. Linear Electric Machines, Drives, and MAGLEVs Handbook; CRC Press: Boca Raton, FL, USA, Gysen, B.L.J.; Paulides, J.J.H.; Janssen, J.L.G.; Lomonova, E.A. Active Electromagnetic Suspension System for Improved Vehicle Dynamics. IEEE Trans. Veh. Technol. 2010, 59, [CrossRef] 3. Martins, I.; Esteves, J.; Marques, G.D.; DaSilva, F.P. Permanent-Magnets Linear Actuators Applicability in Automobile Active Suspensions. IEEE Trans. Veh. Technol. 2006, 55, [CrossRef] 4. Wang, J.; Wang, W.; Atallah, K. A Linear Permanent-Magnet Motor for Active Vehicle Suspension. IEEE Trans. Veh. Technol. 2011, 60, [CrossRef] 5. Gysen, B.L.J.; Janssen, J.L.G.; Paulides, J.J.H.; Lomonova, E.A. Design Aspects an Active Electromagnetic Suspension System for Automotive Applications. IEEE Trans. Ind. Appl. 2009, 45, [CrossRef] 6. Zhu, Z.Q.; Howe, D. Halbach permanent magnet machines and applications: A review. IEE Proc. Electr. Power Appl. 2001, 148, [CrossRef] 7. Chen, S.-L.; Kamaldin, N.; Teo, T.; Liang, W.; Teo, C.; Yang, G.; Tan, K. Towards Comprehensive Modeling and Large Angle Tracking Control a Limited Angle Torque Actuator with Cylindrical Halbach. IEEE/ASME Trans. Mechatron. 2015, [CrossRef] 8. Bjørk, R.; Bahl, C.R.H.; Smith, A.; Pryds, N. Comparison adjustable permanent magnetic field sources. J. Magn. Magn. Mater. 2010, 322, [CrossRef] 9. Yan, L.; Hu, J.; Jiao, Z.; Chen, I.; Lim, C.K. Magnetic field modeling linear machines with double-layered Halbach arrays. In Proceedings IEEE International Conference on Fluid Power and Mechatronics, Beijing, China, August 2011; pp Eckert, P.R.; Goltz, E.C.; Flores Filho, A.F. Influence Segmentation Ring-Shaped NdFeB Magnets with Parallel Magnetization on Cylindrical Actuators. Sensors 2014, 14, [CrossRef] [PubMed] 11. Eckert, P.R.; Wiltuschnig, I.P.; Flores Filho, A.F. Design Aspects Quasi-Halbach Arrays Applied to Linear Tubular Actuators. In Proceedings 10th International Symposium on Linear Drives for Industry Applications, Aachen, Germany, August 2015; pp Yan, L.; Zhang, L.; Jiao, Z.; Hu, H.; Chen, C.-Y.; Chen, I.-M. A tubular linear machine with dual Halbach array. Eng. Comput. 2014, 31, Simpson, N.; Wrobel, R.; Mellor, P.H. A multi-physics design methodology applied to a high-force-density short-duty linear actuator. In Proceedings 2014 IEEE Energy Conversion Congress and Exposition (ECCE), Pittsburgh, PA, USA, September 2014; pp Wang, J.; Jewell, G.W.; Howe, D. Design optimisation and comparison tubular permanent magnet machine topologies. IEE Electr. Power Appl. 2001, 148, [CrossRef]

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