A 3-D Simulation of a Single-Sided Linear Induction Motor with Transverse and Longitudinal Magnetic Flux

applied sciences

Article A 3-D Simulation of a Single-Sided Linear Induction Motor with Transverse and Longitudinal Magnetic Flux

Juan Antonio Domínguez Hernández 1,*, Natividad Duro Carralero 1 and Elena Gaudioso Vázquez 2

1 Dpto. Informática y Automática, ETSI Informática, Universidad Nacional de Educación a Distancia, Juan del Rosal, 16, 28040 Madrid, Spain; [email protected] 2 Dpto. Inteligencia Artiﬁcial, ETSI Informática, Universidad Nacional de Educación a Distancia, Juan del Rosal, 16, 28040 Madrid, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-913-987-169

Received: 16 September 2020; Accepted: 5 October 2020; Published: 8 October 2020

Abstract: This paper presents a novel and improved configuration of a single-sided linear induction motor. The geometry of the motor has been modified to be able to operate with a mixed magnetic flux configuration and with a new configuration of paths for the eddy currents induced inside the aluminum plate. To this end, two slots of dielectric have been introduced into the aluminum layer of the moving part with a dimension of 1 mm, an iron yoke into the primary part, and lastly, the width of the transversal slots has been optimized. Specifically, in the enhanced motor, there are two magnetic fluxes inside the motor that circulate across two different planes: a longitudinal magnetic flux which goes along the direction of the movement and a transversal magnetic flux which is closed through a perpendicular plane with respect to that direction. With this new configuration, the motor achieves a great increment of the thrust force without increasing the electrical supply. In addition, the proposed model creates a new spatial configuration of the eddy currents and an improvement of the main magnetic circuit. These novelties are relevant because they represent a great improvement in the efficiency of the linear induction motor for low velocities at a very low cost. All simulations have been made with the finite elements method—3D, both in standstill conditions and in motion in order to obtain the characteristic curves of the main forces developed by the linear induction motor.

Keywords: single sided linear induction motor; transverse magnetic flux; thrust force; levitation force; airgap magnetic flux density; eddy currents

1. Introduction A linear induction motor (LIM) is a type of asynchronous machine that in comparison with the traditionally rotary induction machine does not develop an electromagnetic torque [1]. The electromagnetic forces in an LIM operate along two directions: horizontal and vertical, instead of rotating. Figure1 illustrates the process for generating the electromagnetic forces in an LIM [2].

Appl. Sci. 2020, 10, 7004; doi:10.3390/app10197004 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 7004 2 of 26 Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 26

FigureFigure 1.1. Electromagnetic forcesforces inin aa linearlinear inductioninduction motormotor (LIM).(LIM).

AsAs shownshown in in Figure Figure1, a,1, LIMa, LIM has has two two di ff erentdifferen parts,t parts, which which produce produce twodi twofferent different magnetic magnetic fields. Thefields. first The part first is part a ferromagnetic is a ferromagnetic steel (called steel (called stator stator or primary or primary part). part). Inside Inside this this part, part, is located is located an alternatingan alternating current current (AC) (AC) three-phase three-phase winding winding that that generates generates a a traveling traveling magnetic magnetic field. field. The The secondsecond partpart isis composedcomposed of of two two joined joined layers layers of of aluminum aluminum and and iron iron (called (called rotor rotor or secondaryor secondary part), part), where where the generatedthe generated eddy eddy currents currents produce produce a new a new magnetic magnetic field. field. The The interaction interaction between between the the two two magneticmagnetic fieldsfields createdcreated inin bothboth partsparts causescauses thethe motionmotion ofof thethe secondarysecondarypart. part. TheThe relevancerelevance ofof LIMsLIMs inin thethe industryindustry isis hugehuge becausebecause therethere areare aa lotlot ofof applicationsapplications inin didifferentfferent areasareas [3[3],], such such as transportationas transportation systems systems (propulsion (propulsion of wheel-on-rail of wheel-on-rail vehicles vehicles and magnetic and levitationmagnetic high-speedlevitation high-speed trains), vertical trains), drives vertical (passenger drives elevator (passenger or an elevator withoutor an elevator ropes), industrialwithout ropes), drives (impactindustrial machine drives and (impact machine machine tools), or and automotive machine control tools), and or robotic. automotive Additionally, control these and motors robotic. are relevantAdditionally, in military these applicationsmotors are relevant [4], such in as military electromagnetic applications aircraft [4], such launch as electromagnetic or electromagnetic aircraft rail gunlaunch systems. or electromagnetic Besides, LIMs rail are verygun importantsystems. Besides, in all magneto-hydrodynamic LIMs are very important applications in all magneto- because underhydrodynamic low-velocity applications conditions, because an LIM canunder be usedlow-velocity as an electromagnetic conditions, pumpan LIM where can thebe liquidused metalas an representselectromagnetic the moving pump part.where the liquid metal represents the moving part. AA generalgeneral classificationclassification ofof LIMsLIMs [[1],1], divides these electrical induction machines intointo two didifferentfferent familiesfamilies dependingdepending onon twotwo criteria:criteria: thethe configurationconfiguration of thethe magneticmagnetic fluxflux andand thethe geometrygeometry ofof thethe machine.machine. The firstfirst family can bebe divideddivided intointo twotwo categoriescategories dependingdepending onon thethe numbernumber ofof primaryprimary parts.parts. Hence, an LIMLIM thatthat operatesoperates withwith onlyonly aa primaryprimary partpart isis calledcalled single-sidedsingle-sided linearlinear inductioninduction motormotor (SSLIM)(SSLIM) andand anan LIMLIM withwith twotwo primaryprimary partsparts isis calledcalled double-sideddouble-sided linearlinear inductioninduction motormotor (DSLIM).(DSLIM). TheThe secondsecond familyfamily cancan bebe alsoalso divideddivided intointo twotwo didifferentfferent categories.categories. First,First, anan LIMLIM withwith aa magneticmagnetic fluxflux thatthat circulatescirculates alongalong thethe directiondirection ofof thethe movementmovement calledcalled aa longitudinallongitudinal fluxflux linearlinear inductioninduction motormotor (LFLIM). (LFLIM). Second, Second, an an LIM LIM with with a magnetica magnetic flux flux closed closed to ato perpendicular a perpendicular plane plane to the to directionthe direction of the of movementthe movement called called a transverse a transverse flux flux linear linear induction induction motor motor (TFLIM) (TFLIM) [5,6]. [5,6]. TheThe mostmost popularpopular topologytopology inin thethe literatureliterature isis LFLIM.LFLIM. However,However, TFLIMTFLIM presentspresents somesome advantagesadvantages [[1],1], suchsuch asas aa magneticmagnetic fluxflux thatthat crossescrosses thethe airgapairgap twice,twice, aa reductionreduction ofof thethe consumedconsumed magnetizationmagnetization current,current, andand aa reductionreduction inin thethe weightweight ofof thethe device.device. InIn thethe literature,literature, therethere areare severalseveral techniquestechniques toto increase increase the the effi ciencyefficiency of LIM of ´LIM´ss which which can be can classified be classified into the threeinto the following three strategies.following Thestrategies. first strategy The first is calledstrategy winding is called skew winding and itsk isew oriented and it is to oriented reduce theto reduce ripple whosethe ripple effects whose are unwishedeffects are [ 7unwished]. The second [7]. strategyThe second is focused strategy on employingis focused theon third-orderemploying harmonicsthe third-order current harmonics injection tocurrent improve injection the thrust to improve and thethe levitationthrust and forcethe levitation [8] and theforce third [8] and strategy the third modifies strategy the modifies geometric the parametersgeometric parameters of the LIM of [9– the13]. LIM [9–13]. OurOur paperpaper is focusedis focused on theon thirdthe strategythird strategy that allows that us allows to modify us theto modify magnetic the flux magnetic configuration flux insideconfiguration the armature inside on the the armature LIM and on redesign the LIM the and paths redesign available the paths for the available eddy currents for the induced eddy currents inside theinduced aluminum inside plate. the aluminum The starting plate. point The is a starting prototype point of an is LIMa prototype that will of be an called LIM model that will 1. Model be called 1 is anmodel SSLIM 1. Model with a 1 configuration is an SSLIM with of TFLIM, a configuration whose geometry of TFLIM, has whose been modified geometry in has both been parts modified (primary in andboth secondary parts (primary part) and in order secondary to improve part) in the order efficiency to improve of the the LIM. efficiency Changes of in the the LIM. geometry Changes have in beenthe geometry introduced have to configurebeen introduced a relevant to threeconfigure dimensions a relevant main three magnetic dimensions circuit main (3D-MMC). magnetic Our circuit goal is(3D-MMC). to search twoOur usefulgoal is pathsto search in perpendicular two useful paths planes in perpendicular available for planes the magnetic available fluxes. for the This magnetic novel configurationfluxes. This novel will beconfiguration called mixed will flux be magnetic called mixed linear flux induction magnetic motor linear or hybridinduction flux motor magnetic or hybrid linear inductionflux magnetic motor linear (MFLIM) induction where longitudinalmotor (MFLIM) and transverse where longitudinal fluxes are operating and transverse simultaneously. fluxes are operating simultaneously. Appl. Sci. 2020, 10, 7004 3 of 26

The objective of the paper will be carried out using three different models that will be explained in the following sections. In model 2, the aluminum layer geometry will be modified. Next, in model 3, the back iron belonging to the primary part will be redesign as well. The last model, model 4, will present a geometry capable to generate a high thrust force without increasing the fixed electric supply used in model 1, due to an increment of the width of the transversal slots in the primary part. All models are simulated with the three-dimensional finite elements method (FEM-3D) used to design electromagnetic devices [14,15]. This paper is organized as follows. Section2 describes the original model (model 1), its geometry, its electric and magnetic properties, and the simulation conditions using FEM-3D. In Section3, model 2, model 3, and model 4 are explained. Section4 details the simulation results for each model in standstill conditions and also take into account the motion in the TFLIM. Finally, the main conclusions obtained are presented in Section5.

2. Original Developed Model In this section, the original model is described, called model 1. Its technical features (electric and magnetic coefficients, parameters of the primary winding, and dimensions) and the principal reasons for its construction are detailed. In Section3, all modifications made to its geometry will be justified. Model 1 was built in the 90s by the Spanish Polytechnic University in Madrid [16]. The objective was the definition of a new formulation that could explain the exact behavior of a TFLIM. This model included the improvements between the theoretical and experimental results obtained in the laboratory. All experimental tests made in [16] were done with the moving part blocked, therefore its velocity was considered equal to 0 m/s. The two main conclusions of these experiments were the following: first, it was necessary to modify the geometry of the motor to generate a longitudinal magnetic flux into the TFLIM and to minimize the anti-thrust force generated by the lateral teeth. Second, the analysis of the model was simulated with FEM-3D, in order to consider the longitudinal and transversal end effects into the characteristic forces of the LIM. The main features of the motor are described in the following subsections. Section 2.1 details the geometric parameters of the TFLIM. Section 2.2 shows a graphical representation of the primary winding. Section 2.3 explains the justification of the electric and magnetic properties of the materials and the type of B-H curve assigned to the ferromagnetic regions of the LIM that used in FEM-3D.

2.1. Geometric Parameters The modeled TFLIM was composed of a primary part built with 31 magnetic sheets with an E-shaped design [5]. In the moving part, there were two available layers. The ﬁrst layer was an aluminum plate and the second layer was made of ferromagnetic material, located over the ﬁrst. Table1 shows the principal geometric parameters of the TFLIM. Figure2 shows a transversal view of one ferromagnetic sheet that includes its principal dimensions. More details can be found in reference [16].

Table 1. Main geometric parameters of the LIM.

Geometric Parameters of the LIM (mm) Thickness Al. Thickness Fe. Width Al. Width Fe. Length Al. Length Fe. 10 25 300 195 990 970 Mechanic Airgap: g 5 Width Longitudinal Slot 8.5 Slot Pitch 16.5 Length Fixed Part 503.5 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 26 Appl. Sci. 2020, 10, 7004x FOR PEER REVIEW 4 of 26

Figure 2. Dimensions of a ferromagnetic sheet (mm) [16]. Figure 2. Dimensions of a ferromagnetic sheet (mm) [16]. Figure 2. Dimensions of a ferromagnetic sheet (mm) [16]. 2.2. Graphical View of the Winding in the Primary Part 2.2. Graphical View of the Winding in the Primary Part 2.2. GraphicalFigure3 showsView of a the 3-D Winding view of in the the di Primaryfferent Part parts of the model 1. The length of the primary part, Figure 3 shows a 3-D view of the different parts of the model 1. The length of the primary part, whereFigure the winding 3 shows is a located, 3-D view is shorterof the different than the parts secondary of the ormodel moving 1. The part. length The dynamicof the primary end-e ffpart,ects where the winding is located, is shorter than the secondary or moving part. The dynamic end-effects ofwhere this configurationthe winding is wouldlocated, be is di shorterfferent than if the the dimensions secondary of or the moving aluminum part. The and dynamic the back end-effects iron in the of this configuration would be different if the dimensions of the aluminum and the back iron in the secondaryof this configuration part were shorterwould be than different in the primaryif the dimensions part. of the aluminum and the back iron in the secondary part were shorter than in the primary part. secondary part were shorter than in the primary part.

Figure 3. Three-dimensionalThree-dimensional view view of model 1. Figure 3. Three-dimensional view of model 1. The primary primary AC AC three-phase three-phase winding winding built built with with FEM-3D FEM-3D is represented is represented in Figure in Figure 4. It4. shows It shows the The primary AC three-phase winding built with FEM-3D is represented in Figure 4. It shows the distributionthe distribution of each of each phase phase inside inside the the primary primary co core.re. The The red red coils coils at at the the beginning beginning of of the the winding distribution of each phase inside the primary core. The red coils at the beginning of the winding belong toto phasephase A. A. The The green green coils coils correspond correspond to phase to phase C and C the and blue the coils blue located coils atlocated the end at representthe end belong to phase A. The green coils correspond to phase C and the blue coils located at the end representphase B. Under phase theseB. Under conditions, these conditions, a traveling a magnetic traveling field magnetic is developed field isby developed the primary by the winding. primary represent phase B. Under these conditions, a traveling magnetic field is developed by the primary winding.Thus, the primary winding is characterized by two parameters: q that indicates the number winding. of slots per pole and phase (q is equal to 4) and the pole pitch, τp, that corresponds to a value of 200 mm [16]. The traveling magnetic field is defined by a linear velocity called synchronous velocity, vSynchronous (expressed in m/s) according to Equation (1). This velocity depends on the pole pitch and on the frequency linked to the AC three-phase system. In this paper, the LIM is supplied by a fixed frequency equal to 50 Hz, which could be selected by a converter and in consequence, the synchronous velocity will be variable [17]. v = 2 τp f = 20 m/s (1) Synchronous · · Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 26

Appl. Sci. 2020, 10, 7004 5 of 26 Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 26

Figure 4. Scheme of the AC three-phase winding with four layers.

Thus, the primary winding is characterized by two parameters: q that indicates the number of slots per pole and phase (q is equal to 4) and the pole pitch, τp, that corresponds to a value of 200 mm [16]. The traveling magnetic field is defined by a linear velocity called synchronous

velocity, (expressed in m/s) according to Equation (1). This velocity depends on the pole pitch and on the frequency linked to the AC three-phase system. In this paper, the LIM is supplied by a fixed frequency equal to 50 Hz, which could be selected by a converter and in consequence, the synchronous velocity will be variable [17]. FigureFigure 4. 4. SchemeScheme of of the the AC AC three-phase three-phase winding winding with with four four layers. layers.

v =2∙τ ∙ f = 20m/s (1) Thus,Once the travelingprimary winding magnetic is field characterized is generated, by the two thrust parameters: and levitation q that forces indicates are developed the number by theof Once the traveling magnetic field is generated, the thrust and levitation forces are developed by slotsmoving per pole part ofand the phase LIM. ( Thisq is equal secondary to 4) and part the is characterizedpole pitch, τp,by that a linearcorresponds velocity, tov amoving value_part of, 200 diff erentmm the moving part of the LIM. This secondary part is characterized by a linear velocity, , [16].from The the vSynchronoustraveling. Bothmagnetic velocities field depend is defined on a parameter by a linear called slip,velocity s, defined called by synchronous Equation_ (2), different from the . Both velocities depend on a parameter called slip, s, defined by velocity,and explained in [17 ]. (expressed In our work, in m/s) initially, according the movingto Equation part (1). was This blocked. velocity So, depends the velocity on the of pole the pitchmovingEquation and part on(2), wasthe and frequency equal explained to 0 mlinked/ s,in and [17]. to the the In slip ACour equal three-work, tophase one.initially, Undersystem. the these movingIn this dynamic paper, part conditions, wasthe LIMblocked. is thrust supplied So, force the bywasvelocity a fixed into theof frequency the motor moving zone, equal part and to 50 thewas Hz, levitation equal which to forcecould0 m/s, belongs be and selected the to theslip by repulsive aequal converter to zone. one. and FigureUnder in consequence,5 theserepresents dynamic the synchronousoperationconditions, point thrust velocity selected force will forwas be thevariable into LIM the in [17]. motor this situation. zone, and Once the standstilllevitation condition force belongs can be to studied the repulsive widely, thezone. TFLIM Figure can 5 alsorepresents be simulated, the operation taking intopoint account selected all for slips the in LIM the rangein this from situation. 1 to 0. Once standstill condition can be studied widely, thev TFLIM can=2∙τ also be∙ f simulated,= 20m/s taking into account all slips in(1) the . = vSyncronous vmoving_part v = rangeOnce from the 1 travelingto 0 magnetic fields is generated,− the/ thrustSyncronous and 1levitation forces are developed by (2) | {z } the moving part of the LIM. This secondary − part is characterized_ by a linear velocity, _, = vmoving_part=0 m/s =1 different from the . Both velocities depend on a parameter called slip, s, defined (2)by _ / Equation (2), and explained in [17]. In our work, initially, the moving part was blocked. So, the velocity of the moving part was equal to 0 m/s, and the slip equal to one. Under these dynamic conditions, thrust force was into the motor zone, and the levitation force belongs to the repulsive zone. Figure 5 represents the operation point selected for the LIM in this situation. Once standstill condition can be studied widely, the TFLIM can also be simulated, taking into account all slips in the range from 1 to 0. − = _ =1 (2) _ /

Figure 5. CurveCurve forces forces versus slip in an LIM [ 1818].]. 2.3. Electric and Magnetic Properties of the Materials Used in FEM-3D To simulate the models with FEM-3D, we considered two electrical and magnetic properties: electrical resistivity and magnetic permeability. Table2 shows the values selected for the simulations. The electrical resistivity of the materials in the moving part is given by its inverse value, the electric conductivity.

Figure 5. Curve forces versus slip in an LIM [ 18]. Appl. Sci. 2020, 10, 7004 6 of 26

Table 2. Electrical and magnetic properties.

Electrical and Magnetic Properties of the Materials Esed in FEM-3D Aluminum Conductivity (Sm–1) 3.73 107 × Relative Permeability 1 Steel Conductivity (Sm–1) 0 Relative Permeability 2500 Copper Resistivity (Ωmm2/m) 2.37 10–2 × Relative Permeability 1

With respect to the magnetic permeability (B-H curve), soft ferromagnetic materials were used for the laminated cores of electrical machines. The shape of the B-H curve was simplified to a straight line with a constant slope given by the relative permeability, µr, according to Equation (3), as explained in [18]: → → → µ = µr = →/µ0 dH = cte B = µ µr H (3) rd dB · → 0· · where µ is the differential relative permeability; µ : 4 π 10 7 H is the magnetic permeability of free rd 0 · · − m space; →B is the magnetic field density vector (T); H→ is the magnetic field intensity (A/m). With these parameters, the behavior of the motor had two relevant consequences: the zero electric conductivity of the steel implied the absence of eddy currents in the ferromagnetic parts (second layer in the moving part and primary part) and the linear B-H curve implied that there are not regions inside of the LIM working under magnetic saturation conditions. For these reasons, the iron losses are neglected in the simulations with FEM-3D, according to Equation (4). See references [19,20].

PFe = Peddy + Ph = 0 W (4) |{z} |{z} = S m Linear B H Curve σFe 0 / − | {z } | {z } = =0 W. 0 W. where PFe is the power losses in core iron (W); P eddy is the power eddy currents losses (W); Ph is the power hysteresis losses (W). A practical recommendation to reduce eddy currents losses is to build the ferromagnetic cores by coated double-sided areas with a thin layer of insulation, usually oxide insulation. To quantify this, the stacking factor Ki was used as a ratio deﬁned by Equation (5), as explained in [21]. d Ki = l/dl + 2 ∆i = 1 (5) | {z ·} ∆i=0 mm. where d is the thickness of the bare sheet (mm); ∆i is the thickness of one-sided insulation (mm). l Nevertheless, to reduce the computing time and the number of nodes in the domain of the FEM-3D, theses insulated areas were not considered (i.e., ∆i = 0 mm). All ferromagnetic regions in our work used a value of stacking factor equal to one; Ki = 1.

3. Modified LIM Models and Implementation with FEM-3D In this section, we describe the modifications made in model 1 to obtain model 2, model 3, and model 4. In addition, we describe the main issues that we have to take into account to simulate them with FEM-3D. The TFLIM prototype (model 1), described in Section2, will be modified to improve the thrust force. To this end, three different new models were developed: Appl. Sci. 2020, 10, 7004 7 of 26

Model 2, where the moving part was redesigned. More concretely, the aluminum layer was • Appl. Sci.modified 2020, 10, with x FOR the PEER introduction REVIEW of two narrow slots of air or diamagnetic material (µr = 1). Each7 of slot 26 has a width of 1 mm and they are located symmetrically at 108 mm from the edge of the aluminum 3. Modifiedlayer (see LIM Figure Models6). Theand newImplementation regions, insulated with FEM-3D electrically, allowed us to have three regions available inside the aluminum layer with new paths for the eddy currents, so that each zone of In this section, we describe the modifications made in model 1 to obtain model 2, model 3, and aluminum was located above lateral and central teeth. Consequently, the thrust force originated model 4. In addition, we describe the main issues that we have to take into account to simulate them with overFEM-3D. the lateral The TFLIM teeth developed prototype along(model the 1), direction described of in the Section motion. 2, will be modified to improve Model 3, where the primary part included a ferromagnetic yoke located under the central teeth. the• thrust force. To this end, three different new models were developed: •The Model high of 2, this where ferromagnetic the moving block part waswas 50redesign mm, itsed. width More was concretely, 83 mm, and the italuminum was extended layer along was 503.5 mm that corresponded with the length of the LIM. With this new block, the motor produced modified with the introduction of two narrow slots of air or diamagnetic material (µr = 1). a longitudinalEach slot has flux a width that operatedof 1 mm throughand they theare mainlocated magnetic symmetrically circuit withat 108 the mm transversal from the edge flux. So,of the the combined aluminum action layer of these(see Figure fluxes provided6). The new a higher regions, thrust insulated force. Figureelectrically,6 shows allowed the changes us to includedhave three in model regions 2 and available model inside 3. the aluminum layer with new paths for the eddy currents, Model 4, where the initial dimension of the width of the transversal slots in the primary part • so that each zone of aluminum was located above lateral and central teeth. Consequently, the (35thrust mm) wasforce incremented originated over along the the lateral direction teeth developed of the x-axis along (see the Figure direction7) in of order the motion. to reach a •maximum Model 3, thrust where force. the primary The width part of included the transversal a ferromagnetic slots was yoke increased located from under 10 the to 40central mm. Thisteeth. model The incorporated high of this all ferromagnetic the changes proposedblock was in 50 the mm, previous its width models. was It83 was mm, important and it was to commentextended that along the width 503.5 ofmm the that lateral corresponded teeth was not with going the length to be modified of the LIM. although With this the new dimension block, of thethetransversal motor produced slots wasa longitudinal increased alongflux that the op x-axis.erated During through this the study main tomagnetic optimize circuit the thrust with forcethe the transversal width of theflux. lateral So, the teeth combined was fixed action to 27 of mm. these fluxes provided a higher thrust force. Figure 6 shows the changes included in model 2 and model 3.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 26 Figure 6. ChangesChanges included in models 2 and 3.

• Model 4, where the initial dimension of the width of the transversal slots in the primary part (35 mm) was incremented along the direction of the x-axis (see Figure 7) in order to reach a maximum thrust force. The width of the transversal slots was increased from 10 to 40 mm. This model incorporated all the changes proposed in the previous models. It was important to comment that the width of the lateral teeth was not going to be modified although the dimension of the transversal slots was increased along the x-axis. During this study to optimize the thrust force the width of the lateral teeth was fixed to 27 mm.

Figure 7. Transversal slots incremented from 10 to 40 mm [16]. Figure 7. Transversal slots incremented from 10 to 40 mm [16].

For the implementation of the models with FEM-3D, the time domain selected corresponds to a transitory state. In all simulations, the frequency f is fixed at 50 Hz. The computation process

time was calculated using three types of times: (called cycled time) is the period of the AC voltage signal employed during the simulations [s]; (s) (called simulation time), is the number of cycles necessary to achieve the stationary state which allows us to evaluate the

advantages of the proposed changes without increasing the computation effort; (called step time simulation) is the fraction of the necessary to converge to the solution (s). The is defined as three times the , which is divided into 10 steps to fix (see Equation (6)). These times were selected experimentally, after several simulations that achieved the correct model.

=3∙ = 0.06 . 1 = =0.02 .→ (6) = 10 = 0.002 . Finally, Figure 8 resumes all necessary conditions to simulate correctly all models proposed:

Figure 8. Simulation conditions imposed during the simulation.

4. Model Simulations with FEM-3D This section describes the simulations of all models with FEM-3D. All simulations were carried out with Altair Flux 3D, a finite element software focused on electromagnetic, electric, and thermal design of electric machines. It allowed for optimization of the performance of the machine, efficiency, dimensions, and weight. Section 4.1 establishes the steps to define the electrical supply of the LIM. Section 4.2 presents an analysis of the magnetic field density along the airgap. Section 4.3 shows the simulation results for the different models in standstill conditions and finally, Section 4.4 includes the simulation results with motion. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 26

Appl. Sci. 2020, 10, 7004 Figure 7. Transversal slots incremented from 10 to 40 mm [16]. 8 of 26

For the implementation of the models with FEM-3D, the time domain selected corresponds For the implementation of the models with FEM-3D, the time domain selected corresponds to a to a transitory state. In all simulations, the frequency f is fixed at 50 Hz. The computation process transitory state. In all simulations, the frequency f is fixed at 50 Hz. The computation process time was time was calculated using three types of times: (called cycled time) is the period of the AC calculated using three types of times: Tcycle (called cycled time) is the period of the AC voltage signal voltage signal employed during the simulations [s]; (s) (called simulation time), is employed during the simulations [s]; T (s) (called simulation time), is the number of cycles the number of cycles necessary to achievesimulation the stationary state which allows us to evaluate the necessary to achieve the stationary state which allows us to evaluate the advantages of the proposed advantages of the proposed changes without increasing the computation effort; (called changes without increasing the computation effort; Tstep (called step time simulation) is the fraction of step time simulation) is the fraction of the necessary to converge to the solution (s). The the Tcycle necessary to converge to the solution (s). The Tsimulation is defined as three times the Tcycle, is defined as three times the , which is divided into 10 steps to fix (see which is divided into 10 steps to fix Tstep (see Equation (6)). These times were selected experimentally, Equation (6)). These times were selected experimentally, after several simulations that achieved after several simulations that achieved the correct model. the correct model. ( = = Tsimulation =3∙3 Tcycle =0.06 0.06s .. Tcycle = 1/f = 0.02 s. · (6) = =0.02 .→→ Tstep = Tcycle/10 = 0.002 s. (6) = 10 = 0.002 . Finally, Figure8 resumes all necessary conditions to simulate correctly all models proposed: Finally, Figure 8 resumes all necessary conditions to simulate correctly all models proposed:

Figure 8. Simulation conditions imposed during the simulation. Figure 8. Simulation conditions imposed during the simulation. 4. Model Simulations with FEM-3D 4. Model Simulations with FEM-3D This section describes the simulations of all models with FEM-3D. All simulations were carried out withThis section Altair Flux describes 3D, a the finite simulations element software of all mo focuseddels with on FEM-3D. electromagnetic, All simulations electric, were and thermalcarried outdesign with of Altair electric Flux machines. 3D, a finite It allowed element for software optimization focused of theon performanceelectromagneti ofc, the electric, machine, and e thermalfficiency, designdimensions, of electric and machines. weight. Section It allowed 4.1 establishesfor optimization the steps of the to performance define the electrical of the machine, supply ofefficiency, the LIM. dimensions,Section 4.2 presents and weight. an analysis Section of 4.1 the establishes magnetic the field steps density to define along the the electrical airgap. Section supply 4.3 of shows the LIM. the Sectionsimulation 4.2 presents results for an the analysis different of the models magnetic in standstill field density conditions along and the finally,airgap. Section Section 4.4 4.3 includes shows the the simulationsimulation results results for with the motion. different models in standstill conditions and finally, Section 4.4 includes the simulation results with motion. 4.1. Definition of the Electrical External Circuit Coupling To start the simulation with FEM-3D, the electrical circuit coupling necessary to operate with the LIM was set up. To this end, the following steps were necessary:

1. Step 1: To fix the magnetic field density in the airgap, Bg, using an analytical method to estimate its value [22]. 2. Step 2: To select the configuration of the electrical circuit coupling to supply the device (Y-connected or D-connected system) and to generate the traveling magnetic field defined in Section 2.2. 3. Step 3: To use an iterative simulation process to calculate the parameters necessary to define the electrical circuit coupling. The first parameter is the value of the symmetrical line voltage, VLINE. The other parameters are the number of turns per phase and its resistance value: Nphase and Rphase. Appl. Sci. 2020, 10, 7004 9 of 26

Both are necessary to have the coils deﬁned correctly. The initial value of Rphase is ﬁxed at around 2.6 Ω to start the iterative process.

In the first step, the magnetic field density in the airgap was considered equal to the average value obtained from Equations (7) and (8), (see reference [22]). Both equations allowed for the obtaining of the value of the first harmonic of Bg, called Bg1, considering only the geometric parameters of the LIM and not their electrical quantities. So, there were two possibilities to calculate Bg1, using Equations (7) or (8). 1 B = 0.464 τ /6 (7) g1_I · p B b 0.32 Bg1_II = ts/kis (1 os/τs) Bts = 5.47 f − (8) | · − {z · } 0.32 B =5.47 f − /kis (1 bos/τs ) g1 · · − where B is the flux density in the primary tooth (T); k is the primary stacking factor equal to 1; b is ts is os the primary slot opening equal to 8.5 mm; τs is the slot pitch equal to 16.5 mm; τP is the pole pitch equal to 200 mm, and f is the frequency equal to 50 Hz. At the end, the value of Bg1 is given by Equation (9) and is equal to 0.761 T.

1 5.47 f 0.32 b · − os 6 Bg1_I + Bg1_II k 1 τ + 0.464 τp B = = is · − s · (9) g1 2 2 In the second step, the coils from the primary winding were supplied by a Y-connected three-phase voltage. This situation is described by the Equation system (10), (11), and (12) (see reference [21]), which generates a traveling magnetic ﬁeld along the desired direction.

V = VLINE √2/√3 cos(ω t) = VLINE √2/√3 cos(2 π f t) (10) Phase_a · · · · · · · ·

V = VLINE √2/√3 cos(ω t 2 π/3) = VLINE √2/√3 cos(2 π f t 2 π/3) (11) Phase_b · · · − · · · · · · − · V = VLINE √2/√3 cos(ω t + 2 π/3) = VLINE √2/√3 cos(2 π f t + 2 π/3) (12) Phase_c · · · · · · · · · · where VLINE is the voltage established in a Y-connected symmetrical three-phase system (V); VPhase_a, VPhase_b, VPhase_c are the supply voltages per phase (V); ω is the angular frequency (rad/s) and t is the instantaneous time of simulation (s). In the third step, VLINE and Nphase were calculated by the simulations. When the value of Bg reached the value of the magnetic field density established in the first step, VLINE and Nphase were fixed. The measure of Bg was determined with a plane that crosses the mechanical airgap by its middle section (a detailed study of Bg is made in Section 4.2). The results obtained were the following: " # turns V [V] = 660 N = 22 (13) LINE phase coil phase ·

Once Nphase was known, it was possible to calculate the equivalent resistance of the winding linked to a one-phase according to the Equation (14):

R [Ω] = (N ρCu L /Swire Cu) n = 1.29 Ω (14) Phase phase· · Coil _ · Phase where LCoil is the average length of a single coil equal to 526.5 mm; Swire_Cu is the section of the copper conductor equal to 1.7 mm2; ρ is the resistivity assigned to the copper wire equal to 2.37 10 2 Cu × − (Ω mm2/m) (see Table2), and n is the number of coils per phase equal to 8. · phase Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 26

∙ ∙ Ω = ∙ = 1.29 Ω (14) _ where is the average length of a single coil equal to 526.5 mm; Swire_Cu is the section of the copper 2 conductor equal to 1.7 mm ; is the resistivity assigned to the copper wire equal to 2.37 × 10 (Ω∙mm /m) (see Table 2), and nphase is the number of coils per phase equal to 8.

Appl. Sci. 2020, 10, 7004 10 of 26 4.2. Analysis of the Magnetic Field Density in the Airgap To analyze the magnetic field density in the airgap, we represent in Figure 9 its spatial evolution 4.2. Analysis of the Magnetic Field Density in the Airgap (third component) along the airgap, Bgz with FEM-3D. The distribution of Bgz was calculated over the centralTo teeth analyze and the the magnetic lateral teeth, field with density three in thepaths airgap, located we exactly represent in the in Figure middle9 itsplane spatial of the evolution airgap. The(third blue component) line corresponds along the to airgap,the componentBgz with z FEM-3D. of the magnetic The distribution field density of B whengz was it calculated crosses the over airgap the overcentral the teeth central and teeth. the lateral The other teeth, two with lines three (yellow paths and located green), exactly that inare the superimposed, middle plane belong of the airgap.to the thirdThe blue component line corresponds of the magnetic to the component field density z of when the magnetic it crosses field the lateral density teeth. when For it crossesall lengths the airgapof the airgap,over the measured central teeth. in m Thein Figure other 9, two the lines value (yellow of Bgz over and the green), lateral that teeth are was superimposed, approximately belong half to with the respectthird component to the value of of the B magneticgz over the fieldcentral density teeth whenand it itwas crosses in the the opposite lateral teeth.direction For (Figure all lengths 9). of the airgap,A similar measured result in m was in Figureobtained9, the in value[16] that of Binclgz overuded the the lateral measurements teeth was approximatelyin the laboratory half of with the magneticrespect to field the value density of Batgz theover top the of central the lateral teeth an andd central it was inteeth the using opposite explorer direction coils (Figure located9). around each tooth. We have used these results from [16] to check the validity of our method.

Spatial Distribution of the Magnetic Field Density in the airgap [Bgz ] 0.8

0.6

0.4

0.2

0.0

-0.2

-0.4 Magnetic Field Density [T] -0.6 0.00.10.20.30.40.5 Longitudinal Length of the airgap [m] Bgz_Central_Teeth Bgz_Right_Teeth Bgz_Left_Teeth

Figure 9. DistributionDistribution of of the the magnetic magnetic field ﬁeld density along the airgap.

ConsiderA similar the result following was obtained points to in design [16] that the included magnetic the field measurements density along in the the airgap: laboratory of the magnetic• fieldFigures density 9 and at the10 show top of that the the lateral useful and magnetic central teeth flux usingonly operated explorer across coils located the transverse around each tooth. Wesections have usedand the these longitudinal results from magnetic [16] to check flux the was validity canceled. of our Equation method. (15) shows this Considerconfiguration the following of pointsthe transverse to design magnet the magneticic flux operating field density into along the TFLIM. the airgap: ∅ Figures9 and 10∅ show that≈∅ the useful→∅ magnetic flux≈∅ only operated≈ across the transverse sections(15) • 2 and the longitudinal magnetic flux was canceled. Equation (15) shows this configuration of the wheretransverse ∅ is magnetic the magnetic flux operating flux that into is the involved TFLIM. in the electromechanical conversion (Wb);

∅ is the magnetic flux closed along transverse sections (Wb); ∅ is the magnetic flux that crosses the central∅ useteeth f ul (Wb);∅Tranversal ∅ ∅right is the magnetic∅le f t flux∅Central thatteeth /crosses2 the right-sided(15) ≈ → teeth ≈ teeth ≈ teeth (Wb), and ∅ is the magnetic flux that crosses the left-sided teeth (Wb). where• ∅Theuse fhigh ul is thevalues magnetic reached flux for thatmagnetic is involved flux dens inity the in electromechanical Figure 10 were obtained conversion due to (Wb); the ∅Tranversallinearis the behavior magnetic of fluxthe closedB-H curve along that transverse did not sectionsinclude (Wb);the saturation∅Centralteeth effectis the inside magnetic the flux thatferromagnetic crosses the central regions. teeth These (Wb); values∅right teethwereis theunreachable magnetic inside flux that real crosses ferromagnetic the right-sided steel, teeth (Wb),but the and goal∅le f tofteeth thisis the simulation magnetic was flux to that demonstrate crosses the that left-sided the only teeth existing (Wb). magnetic flux The highwas values the transverse. reached for To magnetic validate fluxthis densitysimulation in Figure in an experimental10 were obtained test it due is necessary to the linear to • behaviordecrease of the B-Hthe excitation curve that in didorder not to include work near the to saturation the saturation effect zone inside from the the ferromagnetic B-H curve. regions. These values were unreachable inside real ferromagnetic steel, but the goal of this simulation was to demonstrate that the only existing magnetic flux was the transverse. To validate this simulation in an experimental test it is necessary to decrease the excitation in order to work near to the saturation zone from the B-H curve. Appl. Sci. 2020, 10, 7004 11 of 26 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 26

FigureFigure 10. 10. TransversalTransversal view of magnetic flux ﬂux in model 1.

4.3. Simulation Simulation Results Results in in Standstill Standstill Conditions Conditions The results results obtained obtained from from the the simulations simulations are are deta detailediled in inthe the next next subsections. subsections. These These results results are presentedare presented by showing by showing the thethree three principal principal magnitudes magnitudes of ofa aLIM: LIM: forces, forces, eddy eddy currents currents into into the aluminum plate, and cons consumedumed electric currents.

4.3.1. Simulation of the Original Model 4.3.1. Simulation of the Original Model In this section, we describe the experiences with model 1. Figure 11 shows the principal forces in In this section, we describe the experiences with model 1. Figure 11 shows the principal forces this model: thrust, levitation, and transversal. In this case, the levitation force (red line) was higher in this model: thrust, levitation, and transversal. In this case, the levitation force (red line) was higher thanthan the thrust forceforce (blue(blue line).line). IfIf thethe averageaverage value value of of the the forces forces was was taken taken as as a representativea representative value value of oftheir their evolution, evolution, the the levitation levitation force force reached reached around around 400 N,400 in N, comparison in comparison with thewith thrust the thrust force which force whichwas nearly was nearly to 80 N. to It 80 denotes N. It denotes that the that levitation the levitation force is veryforce important is very important in a LIM, in instead, a LIM, itinstead, does not it doesexist not in a exist rotatory in a rotatory induction induct motor.ion The motor. transverse The transverse force (green forceline) (green was line) not was relevant not relevant in this model,in this model,its value its is value only 12is only N, because 12 N, because the moving the moving and ﬁxed and parts fixed were parts aligned. were aligned.

Forces Developed by Model 1 650 550 450 350 250 Forces [N] Forces Forces [N] Forces 150 50 -50 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Time [s] Fz_Vertical Force Fy_Thrust Fx_Transversal Force

Figure 11. ForcesForces in in model model 1.

Equation (16) explains the generation of the thrust force. It is important to remark that during the Equation (16) explains the generation of the thrust force. It is important to remark that during simulation, the layer of iron located over the aluminum plate was assigned an electrical conductivity the simulation, the layer of iron located over the aluminum plate was assigned an electrical equal to zero, σFe = 0 S/m. Consequently, eddy currents were canceled inside this region, therefore conductivity equal to zero, σFe = 0 S/m. Consequently, eddy currents were canceled inside this region, therefore there was no thrust force. Thus, the thrust force was only generated by the aluminum layer. Appl. Sci. 2020, 10, 7004 12 of 26

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 26 there was no thrust force. Thus, the thrust force was only generated by the aluminum layer. The total The total thrust force was defined by the summation of the electromagnetic forces created over each thrust force was defined by the summation of the electromagnetic forces created over each tooth. tooth. =_ +_ = _ ( /) : ___ (16) : ___ Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 26 (16) where _ is the thrust force in the aluminum layer (N); _ is the thrust force in the iron layer The total thrust force was defined by the summation of the electromagnetic forces created over each (N);where →F is the thrust is the force positive in the thrust aluminum force layergenerated (N); →F by theis themagnetic thrust forceflux that in the crosses iron layer the tooth.___y_ Al y_Fe central(N); →F teeth (N), andis the___ positive thrust is the force negative generated thrust by theforce magnetic created flux by the that magnetic crosses the flux central that y_Al_central_teeth = + = crosses the lateral teeth (N). Term 2 in_ Equation_ (15) shows the principal_ problem to obtain a great → teeth (N), and F y_Al_lateral_teeth is the negative( thrust /) force created: ___ by the magnetic flux that crosses(16) the thrust force into the model, that is, the anti-thrust force generated over the lateral teeth. As shown in lateral teeth (N). Term 2 in Equation (15) shows the principal: problem___ to obtain a great thrust force Figure 10, when the lines of the transversal magnetic flux cross the lateral teeth, they go in the into the model, that is, the anti-thrust force generated over the lateral teeth. As shown in Figure 10, oppositewhere direction _ is tothe that thrust when force they in the cross aluminum the cent layerral teeth. (N); According_ is the thrust to the force Lorentz’s in the forceiron layer [20], the when the lines of the transversal magnetic flux cross the lateral teeth, they go in the opposite direction negative(N); thrust___ force was is calculated the positive by thrust the vector force generatedproduct between by the magnetic the induced flux thateddy crosses currents the and to that when they cross the central teeth. According to the Lorentz’s force [20], the negative thrust the magneticcentral teeth field (N), density and ___ into the regions is the of negative the alumin thrustum force layer created located by the over magnetic the lateralflux that teeth. forcecrosses was calculated the lateral by teeth the (N). vector Term product 2 in Equation between (15) the shows induced the principal eddy currents problem and to theobtain magnetic a great field However, over the central teeth occurred the opposite situation because there, a positive thrust force densitythrust into force the regionsinto the model, of the aluminumthat is, the anti-thrust layer located force over generated the lateral over teeth.the lateral However, teeth. As over shown the in central was developed, see term 1 of Equation (15). Using FEM-3D was not possible to obtain the separated teethFigure occurred 10, thewhen opposite the lines situation of the transversal because there, magnetic a positive flux cross thrust the forcelateral was teeth, developed, they go in see the term 1 contribution in the thrust force from the central and lateral teeth. In our paper, the thrust force of Equationopposite (15). direction Using to that FEM-3D when wasthey notcross possible the central to teeth. obtain According the separated to the Lorentz’s contribution force in[20], the the thrust developed corresponded to the contributions of all teeth. forcenegative from the thrust central force and was lateral calculated teeth. by In the our vector paper, product the thrust between force the developed induced eddy corresponded currents and to the Figurethe magnetic 12 shows field the density spatial into distribution the regions of of the the induced alumin umeddy layer currents located insi overde thethe aluminumlateral teeth. plate, contributions of all teeth. , whoseHowever, values over are the given central in teeth A/m occurred2. Under the the opposite motion situation condition because of secondary there, a positive part blocked thrust force we can Figure 12 shows the spatial distribution of the induced eddy currents inside the aluminum plate, simplifywas the developed, eddy currents see term vector 1 of Equation according (15). to Usin Equationg FEM-3D (17) was [19], not where possible to is obtain the aluminum the separated electric →J 2 conductivityAl, whosecontribution values (S/m) in and arethe giventhrust the in electricforce A/m from field. Under the (V/m). central the Eddy motion and current lateral condition steeth. into of theIn secondaryour aluminum paper, partthe plate thrust blocked present force we only can developed corresponded to the contributions of all teeth. asimplify loop along the the eddy transversal currents vector direction according in x-axis. to Equation (17) [19], where σAl is the aluminum electric Figure 12 shows the spatial distribution of the induced eddy currents inside the aluminum plate, conductivity (S/m) and →E the electric field (V/m). Eddy currents into the aluminum plate present only 2 , whose values are given in A/m . Under the≈ motion∙ condition of secondary part blocked we can(17) a loop along the transversal direction in x-axis. simplify the eddy currents vector according to Equation (17) [19], where is the aluminum electric conductivity (S/m) and the electric field (V/m). Eddy currents into the aluminum plate present only →J σ →E (17) a loop along the transversal direction in x-axis.Al ≈ Al·

≈ ∙ (17)

Figure 12. Spatial distribution of eddy currents in model 1.

Figure 13 shows theFigure electricFigure 12. 12. currents,Spatial Spatial distribution distribution which pres of eddy eddyent ancurrents currents exclusive in inmodel model phenomenon 1. 1. of an LIM. Due to the finite length of the motor, the main magnetic circuit did not have a symmetry for each phase. This phenomenonFigure 13 isshows called the the electric longitudinal currents, end-effewhich presct.ent Therefore, an exclusive the phenomenon maximum value of an ofLIM. the Due electric to the finite length of the motor, the main magnetic circuit did not have a symmetry for each phase. This phenomenon is called the longitudinal end-effect. Therefore, the maximum value of the electric Appl. Sci. 2020, 10, 7004 13 of 26

Appl. Sci.Figure 2020, 1013, xshows FOR PEER the REVIEW electric currents, which present an exclusive phenomenon of an LIM.13 Dueof 26 to the finite length of the motor, the main magnetic circuit did not have a symmetry for each phase. currentThisAppl. phenomenon Sci. will 2020 be, 10 ,different x FOR is called PEER due REVIEW the to longitudinal the relative end-e posifftionect. of Therefore, each phase the maximuminside the valueprimary of the winding.13 electric of 26 Precisely,current will the be RMS different values due of tothe the electric relative currents position in ofphase each A, phase phase inside B, and the phase primary C were winding. 84.811, Precisely, 93.583, current will be different due to the relative position of each phase inside the primary winding. andthe RMS87.938 values A, respectively. of the electric currents in phase A, phase B, and phase C were 84.811, 93.583, and 87.938 Precisely, the RMS values of the electric currents in phase A, phase B, and phase C were 84.811, 93.583, A, respectively. and 87.938 A, respectively. Electric Currents in TFLIM Model 1 140 Electric Currents in TFLIM Model 1 90140

4090

-1040 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -60-10 Current [A] 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -110-60 Current [A]

-160-110 Time [s] -160 Current Phase A CurrentTime [s] Phase B Current Phase C Current Phase A Current Phase B Current Phase C

Figure 13. Electric current per phase in model 1. FigureFigure 13. 13. ElectricElectric current per per phase phase in in model model 1. 1. A system where the electric currents per phase are all different (IPhase_a ≠ IPhase_b ≠ IPhase_c) is called A system where the electric currents per phase are all different (I I I ) is called an unbalancedA system three-phase where the electricsystem currents[3] and presentsper phase two are relevant all different consequences.Phase_a (IPhase_a, ≠ IPhase_bPhase_b First, ≠, I Phase_casPhase_c presented) is called in Figureanan unbalanced unbalanced 14, the spatial three-phase three-phase phasor systemsystem of the [3] [3magnetomot] and and presents presentsive two twoforce relevant relevant (MMF) consequences. consequences.was not uniform First, First, as [21] presented as because presented in it describedin Figure 14,14an, theellipsoid the spatial spatial instead phasor phasor ofof of athe thecircumference. magnetomot magnetomotive iveFor force this force reason,(MMF) (MMF) wasthe was instantaneousnot not uniform uniform [21] velocity [21 because] because of itthe it firstdescribeddescribed harmonic an an ellipsoid of ellipsoid the traveling instead instead of magnetic of a circumference.a circumference. wave never For For reached this this reason, reason, the thesynchronou the instantaneous instantaneouss velocity. velocity velocity The of valueof the the first of harmonicfirst harmonic of the of traveling the traveling magnetic magnetic wave wave never ne reachedver reached the the synchronous synchronou velocity.s velocity. The The value value of of the the synchronous velocity calculated in Equation (21), = 20 /, corresponded to its synchronousthe synchronous velocity velocity calculated calculated in Equation in Equation (21), v(21), = 20 m=/ 20s, corresponded/, corresponded to its averageto its average value not to the instantaneous value. This effectSynchronous produced a huge variation in the principal valueaverage not tovalue the instantaneousnot to the instantaneous value. This value. effect This produced effect produced a huge variationa huge variation in the principal in the principal forces of forces of the LIM. Second, it is important to take into account the differences among electric currents theforces LIM. of Second, the LIM. it isSecond, important it is important to take into to accounttake into the account differences the differences amongelectric among currentselectric currents when we when we controlled the LIM and when the insulation of the primary winding conductors was controlledwhen we the controlled LIM and the when LIM the and insulation when the of theinsulation primary of winding the primary conductors winding was conductors selected. was selected.selected.

Figure 14. MMF generated by an unbalanced system of electric currents [21]. Figure 14. MMF generated by an unbalanced system of electric currents [21]. Figure 14. MMF generated by an unbalanced system of electric currents [21]. In order to increment the thrust force, the goal of this paper was to increase term 1 and to reduce In order to increment the thrust force, the goal of this paper was to increase term 1 and to reduce term 2 in Equation (16), by adding changes into the geometric parameters of model 1. term In2 in order Equation to increment (16), by theadding thrust changes force, theinto goal the ofgeometric this paper parameters was to increase of model term 1. 1 and to reduce term 2 in Equation (16), by adding changes into the geometric parameters of model 1. Appl. Sci. 2020, 10, 7004 14 of 26

4.3.2.Appl. Eﬀect Sci. 2020 of Adding, 10, x FORTwo PEER SlotsREVIEW of Dielectric into the Secondary Part 14 of 26

4.3.2. Effect of Adding Two Slots of Dielectric into the Secondary Part ModelAppl. Sci. 2020 2 incorporated, 10, x FOR PEER twoREVIEW slots into the aluminum layer (see Figure6) with a width14 of of 126 mm. These twoModel new 2 regions incorporated of air two allowed slots forinto athe decrease aluminum in thelayer negative (see Figure thrust 6) with force a width generated of 1 mm. over the lateral4.3.2.These teeth. Effect two With new of theseAdding regions new Two of three air Slots allowed regions of Dielectric for of a aluminum decrease into the in Secondary electrically the negative Part insulated thrust force by thegenerated slots, the over generated the eddylateral currentsModel teeth. into 2 incorporatedWith the these aluminum new two threeslots layer intoregions had the a aluminum diof ﬀalerentuminum behaviorlayer electrically (see withFigure insulated respect 6) with to aby width model the slots,of 1. 1 Figuremm. the 15 showsThesegenerated the principaltwo eddynew regionscurrents forces inof into thisair allowedthe model. aluminum for Considering a decrease layer ha din their a the different averagenegative behavior values,thrust forcewith the incrementrespectgenerated to modelover in the the 1. thrust forcelateralFigure with respect 15teeth. shows With to the model these principal 1new was forcesthree about inregions 20this N model. (anof al incrementuminum Considering electrically around their average 23.7%). insulated values, However, by thethe incrementslots, the levitation the forcegeneratedin decreased the thrust eddy to force 200 currents with N (a respect decrementinto the to aluminum model around 1 was layer 49.46%). about had 20 a differentN (an increment behavior around with respect 23.7%). to However, model 1. Figurethe levitation 15 shows force the decreased principal toforces 200 N in (a this decrement model. Considering around 49.46%). their average values, the increment in the thrust force with respect to model 1 was about 20 N (an increment around 23.7%). However, the levitation force decreased Forcesto 200 N (aDeveloped decrement around by Model 49.46%). 2 420 320 Forces Developed by Model 2 420 220 320 120 220 20 Forces [N] Forces 120 -80 20

Forces [N] Forces -180 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -80 Time [s] Fz_Vertical Force Fy_Thrust Fx_Transversal Force -180 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Time [s] Figure 15. Forces in model 2. Fz_VerticalFigure Force 15. ForcesFy_Thrust in modelFx_Transversal 2. Force

Figure 16 presents a new distribution of eddy currents. The new slots generated new paths into Figure 16 presents a new distributionFigure 15. of Forces eddy i currents.n model 2. The new slots generated new paths into thethe aluminumaluminum layer. layer. Now, Now, there there were were three three loops loops of ofeddy eddy currents currents along along the thex-axis. x-axis. With Withthis this configuration,Figure 16 itpresents was possible a new thatdistribution over each of tooth eddy (central currents. and The lateral), new slots the vectorgenerated product new betweenpaths into conﬁguration, it was possible that over each tooth (central and lateral), the vector product between →J theand aluminum generated layer. a force Now, along there the were direction three of loops the mo ofvement, eddy currents but at the along same the time, x-axis. the Withlevitation this → and Bconfiguration,forcegenerated decreased a it force considerably.was possible along the that direction over each of tooth the movement, (central and butlateral), at the the same vector time, product the levitationbetween force decreasedand considerably. generated a force along the direction of the movement, but at the same time, the levitation

Figure 16. Spatial distribution of eddy currents in model 2.

FigureFigure 16. 16.Spatial Spatial distribution distribution of of eddy eddy currents currents in inmodel model 2. 2. Appl. Sci. 2020, 10, 7004 15 of 26

Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 26 Figure 17 represents the consumed electric currents in model 2. Every phase decreased approximatelyFigure 17 2represents A, compared the toconsumed model 1. electric This eﬀcurrentsect supposes in model a little 2. raiseEvery in phase the eﬃ decreasedciency of approximatelythe LIM. 2 A, compared to model 1. This effect supposes a little raise in the efficiency of the LIM.

Electric Currents in TFLIM Model 2 150

100

0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -50 Current [A]

-100

-150 Time [s] Current Phase A Current Phase B Current Phase C

Figure 17. Electric current per phase in model 2.

4.3.3. Effect Effect of Adding a Ferromagnetic Yoke inin thethe PrimaryPrimary PartPart Model 3 developed a new strategy different different from model 2. Until Until now, now, there was only a main traverse magneticmagnetic circuit. circuit. In In model model 3, the3, the primary primary part part was modifiedwas modified with awith ferromagnetic a ferromagnetic yoke located yoke locatedunder the under central the central teeth (see teeth Figure (see 6Figure) in order 6) in to order generate to generate a main a longitudinal main longitudinal magnetic magnetic circuit. circuit. With Withthis new this ferromagneticnew ferromagnetic region, region the, lines the lines of the ofmagnetic the magnetic field field density density crossed crossed the airgapthe airgap from from the primarythe primary part part to the to the secondary secondary part, part, through through these thes twoe two main main magnetic magnetic circuits. circuits. Model Model 3 was 3 was called called the themixed mixed flux magneticflux magnetic linear linear induction induction motor motor or hybrid or fluxhybrid magnetic flux magnetic linear induction linear induction motor (MFLIM). motor (MFLIM).Equation (18)Equation defines (18) the defines result magneticthe resultflux magnetic in this flux model: in this model: ∅ =∅ +∅ (18) ∅use f ul = ∅Longitudinal + ∅Tranversal (18) where ∅ is the magnetic flux involved in the electromechanical conversion (Wb); ∅ is where ∅use f ul is the magnetic flux involved in the electromechanical conversion (Wb); ∅Tranversal is the the transverse magnetic flux (Wb), and ∅ is the longitudinal magnetic flux (Wb). With this transverse magnetic flux (Wb), and is the longitudinal magnetic flux (Wb). With this new new magnetic flux, the motor increased∅Longitudinal the thrust force and decreased the levitation force due to the magnetic flux, the motor increased the thrust force and decreased the levitation force due to the great great attraction between both parts. Figure 18 plots the principal forces in model 3. With this new attraction between both parts. Figure 18 plots the principal forces in model 3. With this new primary primary part, the motor developed a total thrust of around 125.054 N. This value represents a wide part, the motor developed a total thrust of around 125.054 N. This value represents a wide increment increment with respect to models 1 and 2. In relative terms, the thrust force raised around 56.2% in with respect to models 1 and 2. In relative terms, the thrust force raised around 56.2% in respect to respect to model 1 and around 26.6% compared to model 2. The attraction between both parts model 1 and around 26.6% compared to model 2. The attraction between both parts increased, so the increased, so the levitation force reached a value of 65.383 N, which implied a decrement of 84%. levitation force reached a value of 65.383 N, which implied a decrement of 84%. Forces Developed by Model 3

250 150 50 -50 -150 Forces [N] Forces -250 -350 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Time [s] Fy_Thrust Force Fz_Vertical Force Fx_Transversal Force

Figure 18. Forces in model 3. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 26

Figure 17 represents the consumed electric currents in model 2. Every phase decreased approximately 2 A, compared to model 1. This effect supposes a little raise in the efficiency of the LIM.

Electric Currents in TFLIM Model 2 150

100

0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -50 Current [A]

-100

-150 Time [s] Current Phase A Current Phase B Current Phase C

Figure 17. Electric current per phase in model 2.

4.3.3. Effect of Adding a Ferromagnetic Yoke in the Primary Part Model 3 developed a new strategy different from model 2. Until now, there was only a main traverse magnetic circuit. In model 3, the primary part was modified with a ferromagnetic yoke located under the central teeth (see Figure 6) in order to generate a main longitudinal magnetic circuit. With this new ferromagnetic region, the lines of the magnetic field density crossed the airgap from the primary part to the secondary part, through these two main magnetic circuits. Model 3 was called the mixed flux magnetic linear induction motor or hybrid flux magnetic linear induction motor (MFLIM). Equation (18) defines the result magnetic flux in this model:

∅ =∅ +∅ (18)

where ∅ is the magnetic flux involved in the electromechanical conversion (Wb); ∅ is the transverse magnetic flux (Wb), and ∅ is the longitudinal magnetic flux (Wb). With this new magnetic flux, the motor increased the thrust force and decreased the levitation force due to the great attraction between both parts. Figure 18 plots the principal forces in model 3. With this new primary part, the motor developed a total thrust of around 125.054 N. This value represents a wide Appl.increment Sci. 2020, 10with, 7004 respect to models 1 and 2. In relative terms, the thrust force raised around 56.2%16 in of 26 respect to model 1 and around 26.6% compared to model 2. The attraction between both parts 84%. Forces Developed by Model 3

250 150 50 -50 -150 Forces [N] Forces -250 -350 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Time [s] Fy_Thrust Force Fz_Vertical Force Fx_Transversal Force

FigureFigure 18.18. Forces in in model model 3. 3.

Appl.Appl. Sci. Sci. 2020 2020, 10, ,10 x, FORx FOR PEER PEER REVIEW REVIEW 16 of16 26of 26 Figure 19 shows another advantage of model 3, that is, the reduction of the consumed electric currents.FigureFigure Approximately, 19 19 shows show sanother another the value advantage advantag of thesee ofof currentsmodelmodel 3, was thatat around is,is, thethe reduction 15%reduction smaller of of the thanthe consumed consumed in model electr 1.electric currents. Approximately, the value of these currents was around 15% smaller than in model 1. ElectricElectric CurrentsCurrents in MFLIM ModelModel 3 3

115115

6565

1515

-35 -35 Current [A] Current [A] -85 -85 -135 -135 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Time [s] Time [s] Current Phase A Current Phase B Current Phase C Current Phase A Current Phase B Current Phase C

FigureFigure 19.19. Electric current per per phase phase in in model model 3. 3. Figure 19. Electric current per phase in model 3. InIn model model 3, 3, the the combined combined actionaction ofof bothboth magneticmagnetic fluxes ﬂuxes generated generated the the maximum maximum thrust thrust force force obtainedobtainedIn model in in all all 3, proposed proposedthe combined models. models. action FiguresFigures of both 20 and magn 21 illustratedillustratedetic fluxes the thegenerated longitudinal longitudinal the maximumand and transverse transverse thrust paths pathsforce obtainedforfor the the magnetic magneticin all proposed ﬁeld field density. density. models. Figures 20 and 21 illustrated the longitudinal and transverse paths for the magnetic field density.

Figure 20. Longitudinal magnetic field density. Figure 20. Longitudinal magnetic ﬁeld density. Figure 20. Longitudinal magnetic field density.

Figure 21. Transverse magnetic field density.

Figure 21. Transverse magnetic field density. Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 26

Figure 19 shows another advantage of model 3, that is, the reduction of the consumed electric currents. Approximately, the value of these currents was around 15% smaller than in model 1. Electric Currents in MFLIM Model 3

115

-35 Current [A] -85

-135 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Time [s] Current Phase A Current Phase B Current Phase C

Figure 19. Electric current per phase in model 3.

In model 3, the combined action of both magnetic fluxes generated the maximum thrust force obtained in all proposed models. Figures 20 and 21 illustrated the longitudinal and transverse paths for the magnetic field density.

Appl. Sci. 2020, 10, 7004 17 of 26 Figure 20. Longitudinal magnetic field density.

FigureFigure 21. TransverseTransverse magnetic field ﬁeld density.

In conclusion, model 3 included two different phenomena. First, the effect of the incorporation of Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 26 the two slots into the aluminum layer, which generated a great thrust force. Second, the introduction of the ferromagneticIn conclusion, yoke model under 3 included the central two different teeth, whichphenomena. was aFirst, correct the effect way toof createthe incorporation an LIM with a mixedof orthe hybrid two slots magnetic into the flux aluminum configuration. layer, which generated a great thrust force. Second, the introduction of the ferromagnetic yoke under the central teeth, which was a correct way to create an 4.3.4.LIM Effect with of a Varying mixed or the hybrid Width magnetic of the flux Transversal configuration. Slots Model 4 redesigns the width of the transversal slots of the primary part increasing its dimension 4.3.4. Effect of Varying the Width of the Transversal Slots along the x-axis (see Figure7). To this end, a sweep parameter was used which found the optimal value of thisModel dimension. 4 redesigns This the width change of intothe transversal the opening slots of of the the transversalprimary part slots increasing was necessary its dimension to show along the x-axis (see Figure 7). To this end, a sweep parameter was used which found the optimal how the thrust force increased and the electric field over the lateral teeth decreased when the opening value of this dimension. This change into the opening of the transversal slots was necessary to show of thehow slot the increased. thrust force The increased initial valueand the of electric the transversal field over the slots lateral was teeth 35mm decreased (see Figure when the2) and opening the final valueof was the slot 75 mm.increased. Figure The 22 initial shows value the of thrustthe transversal force for slots the was diff 35erent mm openings (see Figure in 2) a and transitory the final state. As shown,value was the 7 thrust5 mm. forceFigure did 22 shows not change the thrust significantly. force for the different openings in a transitory state. As

Evolution FyThrust Force for several increment of the transversal slot 310

200

90 Forces [N] Forces -20 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -130 Time [s] Thrust_IncSlotTrv_10 Thrust_IncSlotTrv_20 Thrust_IncSlotTrv_30 Thrust_IncSlotTrv_40

FigureFigure 22. 22.Thrust Thrust force force in in a a transitory transitory state for for di differentﬀerent values values of ofthe the transversal transversal slots. slots.

To studyTo study the evolutionthe evolution of the of forcesthe forces for eachfor each dimension dimension of theof the transversal transversal slots, slots, the the average average values values of Fy are summarized in Table 4. Besides, this table contains the average values obtained with of Fy are summarized in Table3. Besides, this table contains the average values obtained with model 3, model 3, whose results are a reference value. Model 4 incorporated all changes from models 2 and 3. whose results are a reference value. Model 4 incorporated all changes from models 2 and 3.

Table 4. Average value of thrust force and levitation force.

Final Width of the Transversal Slots (mm) Fy_Thrust Force (N) Fz_Levitation Force (N) 45 mm (Increment of 10 mm) 132.888 −13.704 55 mm (Increment of 20 mm) 138.867 −56.445 65 mm (Increment of 30 mm) 136.854 −96.439 75 mm (Increment of 40 mm) 135.479 −136.308 Model 3 125.054 65.383 (Reference Values)

Figure 23 collects all data from Table 4. The optimal thrust force was reached in model 4 for an increment of 20 mm in the transversal slots. This force was around 138 N and consequently, the opening of the transversal slots has been redefined with a value of 55 mm. Compared to model 3, the thrust force represents an improvement of around 10%. Not all increments of the transversal slots were valid to improve the thrust force because with an increment greater than 20 mm the leakage flux began to be relevant, and the linkage magnetic flux fell. Appl. Sci. 2020, 10, 7004 18 of 26

Table 3. Average value of thrust force and levitation force.

Final Width of the Transversal Slots (mm) Fy_Thrust Force (N) Fz_Levitation Force (N) 45 mm (Increment of 10 mm) 132.888 13.704 − 55 mm (Increment of 20 mm) 138.867 56.445 − 65 mm (Increment of 30 mm) 136.854 96.439 − 75 mm (Increment of 40 mm) 135.479 136.308 − Model 3 125.054 65.383 (Reference Values)

Figure 23 collects all data from Table3. The optimal thrust force was reached in model 4 for an increment of 20 mm in the transversal slots. This force was around 138 N and consequently, the opening of the transversal slots has been redefined with a value of 55 mm. Compared to model 3, the thrust force represents an improvement of around 10%. Not all increments of the transversal slots were valid to improve the thrust force because with an increment greater than 20 mm the leakage flux began to be relevant, and the linkage magnetic flux fell. Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 26

Appl. Sci. 2020, 10, x FOR PEER REVIEWEvolution Fy Thrust Force 18 of 26 140 138 Evolution Fy Thrust Force 136 134140 132138 136

Force [N] 130 128134 132 Fy_Thrust_Force [N]

10Force [N]mm.130 132.888 128 20 mm. Fy_Thrust_Force138.867 [N] 30 mm.10 mm. 136.854132.888 40 mm.20 mm. 135.479138.867

30 mm. 136.854 FigureFigure40 mm. 23. ThrustThrust forces forces versus versus an anincrem incrementent135.479 of the of transversal the transversal slots. slots.

TheThe levitation levitation force forceFigure presented presented 23. Thrust the the forces opposite opposite versus behavior behaan incremvior when entwhen of thethe the transversaltransve transversalrsal slots. slots slots increased. increased. IfIf model 3 wasmodel considered 3 was considered as a reference as a refere value,nce with value, the with new the dimension new dimension of the of transversal the transversal slots slots ﬁxed fixed in 55 mm, in 55 mm,The model levitation 4 operated force presented under conditions the opposite of attraction behavior forces when as the shown transversa in Figurel slots 24. increased.Figure 25 If model 4 operated under conditions of attraction forces as shown in Figure 24. Figure 25 plots the plotsmodel the 3average was considered values for as this a refere attractionnce value, force. with Moreover, the new the dimension evolution of of the Fz transversaldid not present slots fixedan average values for this attraction force. Moreover, the evolution of F did not present an optimum optimumin 55 mm, value model when 4 operatedthe transversal under slotsconditions were increa of attractionsed 20 mmforces in ascontrast shownz with in Figure the thrust 24. Figure force. 25 valueForplots whenthis value,the the average transversalmodel values 4 developed slots for this were an attraction average increased attractionforce. 20 mmMoreover, force in contrast of the−56.445 evolution with N. the of thrust Fz did force. not present For this an value, model 4 developed an average attraction force of 56.445 N. Evolution Fz Vertical Force for− several increment of the transversal slot Evolution Fz Vertical Force for several increment of the 150 transversal slot 50 -50150 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -150 50 -250-50

Forces [N] Forces 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Time [s] -350-150 -450-250 Forces [N] Forces Time [s] -550-350 -450 Vertical Force_Inc_SlotTrv_10 Vertical Force_Inc_SlotTrv_20 -550 Vertical Force_Inc_SlotTrv_30 Vertical Force_Inc_SlotTrv_40 Vertical Force_Inc_SlotTrv_10 Vertical Force_Inc_SlotTrv_20

Vertical Force_Inc_SlotTrv_30 Vertical Force_Inc_SlotTrv_40 Figure 24. Levitation force in the transitory state with different values of the transversal slots.

FigureFigure 24. Levitation24. Levitation force force in in the the transitory transitory statestate with different diﬀerent values values of ofthe the transversal transversal slots. slots. Evolution Fz Vertical Force 140 90 Evolution Fz Vertical Force 40 140 -10 90 -60 40 -110

Force [N] -10 -160 -60 Fz_Levitation_Force [N] 10 mm.-110 -13.704 Force [N] -160 20 mm. Fz_Levitation_Force-56.445 [N] 30 mm.10 mm. -96.439-13.704 40 mm.20 mm. -136.308-56.445

30 mm. -96.439 40 mm.Figure 25. Levitation forces versus incr-136.308ement of transversal slots.

Figure 25. Levitation forces versus increment of transversal slots. Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 26

Evolution Fy Thrust Force 140 138 136 134 132

Force [N] 130 128 Fy_Thrust_Force [N] 10 mm. 132.888 20 mm. 138.867 30 mm. 136.854 40 mm. 135.479

Figure 23. Thrust forces versus an increment of the transversal slots.

The levitation force presented the opposite behavior when the transversal slots increased. If model 3 was considered as a reference value, with the new dimension of the transversal slots fixed in 55 mm, model 4 operated under conditions of attraction forces as shown in Figure 24. Figure 25 plots the average values for this attraction force. Moreover, the evolution of Fz did not present an optimum value when the transversal slots were increased 20 mm in contrast with the thrust force. For this value, model 4 developed an average attraction force of −56.445 N. Evolution Fz Vertical Force for several increment of the transversal slot

150 50 -50 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -150 -250 Forces [N] Forces Time [s] -350 -450 -550 Vertical Force_Inc_SlotTrv_10 Vertical Force_Inc_SlotTrv_20 Vertical Force_Inc_SlotTrv_30 Vertical Force_Inc_SlotTrv_40

Appl. Sci. 2020, 10, 7004 19 of 26 Figure 24. Levitation force in the transitory state with different values of the transversal slots.

Evolution Fz Vertical Force 140 90 40 -10 -60 -110 Force [N] -160 Fz_Levitation_Force [N] 10 mm. -13.704 20 mm. -56.445 30 mm. -96.439 40 mm. -136.308

Figure 25. Levitation forces versus incr incrementement of transversal slots. Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 26 4.3.5.Appl. Comparative Sci. 2020, 10, x StudyFOR PEER of REVIEW the Main Results Obtained in Standstill Conditions 19 of 26 4.3.5. Comparative Study of the Main Results Obtained in Standstill Conditions In4.3.5. this Comparative section, a comparative Study of the Main study Resu betweenlts Obtained all models in Standstill is presented. Conditions Figures 26 and 27 show the In this section, a comparative study between all models is presented. Figures 26 and 27 show the evolutionIn of this the section, thrust anda comparative levitation study forces between for each all model. models Foris presented. model 4, Figures an opening 26 and of 27 the show transversal the evolution of the thrust and levitation forces for each model. For model 4, an opening of the transversal slotsevolution equal to 55of the mm thrust was and considered. levitation Eachforces modelfor each developed model. For model a thrust 4, an force opening greater of the than transversal the previous slots equal to 55 mm was considered. Each model developed a thrust force greater than the previous slots equal to 55 mm was considered. Each model developed a thrust force greater than the previous model.model. For theFor the dynamic dynamic conditions conditions imposed imposed ((ss = 11),), the the levitation levitation force force presented presented a great a great change change in in model. For the dynamic conditions imposed (s = 1), the levitation force presented a great change in eacheach model. model. In conclusion, In conclusion, one one can can say say that that anan improvement in in the the thrust thrust force force implied implied a reduction a reduction each model. In conclusion, one can say that an improvement in the thrust force implied a reduction in thein levitation the levitation force. force. In In fact, fact, in in model model 4, 4, the the levitationlevitation force force changed changed from from the the levitation levitation condition condition in the levitation force. In fact, in model 4, the levitation force changed from the levitation condition to attraction.to attraction. to attraction. Dynamic evolution of the Thrust Force Dynamic evolution of the Thrust Force 310 310 260 260 210 210 160 160 110 110 60

Thrust Force [N] 60

Thrust Force [N] 10 10 -40 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -40 0.00 0.01 0.02Time 0.03 [s] 0.04 0.05 0.06 Time [s] Thrust Model 1 [N] Thrust Model 2 [N] Thrust Model 1 [N] Thrust Model 2 [N]

Figure 26. Dynamic evolution of the thrust force in each model. FigureFigure 26. 26.Dynamic Dynamic evolution evolution of the the thrust thrust force force in ineach each model. model.

Dynamic evolution of the Levitation Force Dynamic evolution of the Levitation Force

500 500 300 300 100 100 -100 0.00 0.01 0.02 0.03 0.04 0.05 0.06 -100 0.00 0.01 0.02 0.03 0.04 0.05 0.06

Force Leviation [N] Leviation Force -300 Time [s] Force Leviation [N] Leviation Force -300 -500 Time [s] -500 Force Levitation Model 1 [N] Force Levitation Model 2 [N] Force Levitation Model 1 [N] Force Levitation Model 2 [N] Force Levitation Model 3 [N] Force Levitation Model 4 [N] Force Levitation Model 3 [N] Force Levitation Model 4 [N]

FigureFigure 27. 27.Dynamic Dynamic evolution evolution of the levitation levitation force force in ineach each model. model. Figure 27. Dynamic evolution of the levitation force in each model. The dynamic evolution of the forces is important to design a control strategy over an MFLIM. The dynamic evolution of the forces is important to design a control strategy over an MFLIM. All models presented a similar transitory state with a maximum value of the thrust force in the All models presented a similar transitory state with a maximum value of the thrust force in the interval time (0.010–0.012 s). Figure 28 shows the average value of the principal forces in each model. interval time (0.010–0.012 s). Figure 28 shows the average value of the principal forces in each model. The transversal force along the x-axis was not relevant, although this component cannot be neglected The transversal force along the x-axis was not relevant, although this component cannot be neglected because an LIM can develop a motion through a curve path. because an LIM can develop a motion through a curve path. Appl. Sci. 2020, 10, 7004 20 of 26

The dynamic evolution of the forces is important to design a control strategy over an MFLIM. All models presented a similar transitory state with a maximum value of the thrust force in the interval time (0.010–0.012 s). Figure 28 shows the average value of the principal forces in each model. The transversal force along the x-axis was not relevant, although this component cannot be neglected becauseAppl. Sci. 2020 an LIM, 10, x canFOR developPEER REVIEW a motion through a curve path. 20 of 26

Appl. Sci. 2020, 10, x FOR PEER REVIEW 20 of 26 Comparison of Electromagnetic Forces 500 Comparison400 of Electromagnetic Forces 300500 200400 100300 200

Forces [N] Forces 0 -100100 Model 1 Model 2 Model 3 Model 4

Forces [N] Forces 0 Fy_Thrust -100 79.096 98.770 125.054 138.867 Model 1 Model 2 Model 3 Model 4 Fz_Vertical Force 404.038 204.160 65.383 -56.445 Fy_Thrust 79.096 98.770 125.054 138.867 Fx_Transversal Force 12.627 3.120 0.747 -2.54 Fz_Vertical Force 404.038 204.160 65.383 -56.445 Fx_Transversal Force 12.627 3.120 0.747 -2.54 FigureFigure 28.28. Comparison of the principalprincipal forcesforces inin eacheach model.model.

FigureFigure 2929 representsrepresentsFigure thethe consumedconsumed28. Comparison electricelectric of the currentscurren principalts ininforces allall models.inmodels. each model. TheThe obtainedobtained resultsresults ofof eacheach modelmodel showedshowed thatthat thethe electricelectric currentscurrents alwaysalways presentpresent thethe samesame behavior,behavior, see see Equation Equation (19): (19): Figure 29 represents the consumed electric currents in all models. The obtained results of each model showed that the electric currents always< present <the same behavior, see Equation (19): (19) IPhase__a < IPhase_c_< IPhase_b_ (19)

_ <_ <_ (19) Comparison of Electric Currents 100Comparison of Electric Currents 10080 6080 60 40 40 20 Current [A] 20 Current [A] 0 0 Model 1Model 2Model 3Model 4 Model 1Model 2Model 3Model 4 Current Phase A 84.811 82.491 67.496 70.513 Current Phase A 84.811 82.491 67.496 70.513 Current Phase B 93.583 91.208 79.605 80.178 Current Phase B 93.583 91.208 79.605 80.178 Current Phase C 87.938 84.919 73.583 76.044 Current Phase C 87.938 84.919 73.583 76.044

FigureFigure 29.29. Electric currents perper phasephase inin eacheach model.model. Figure 29. Electric currents per phase in each model. The current linked to phase B was higher than the currents in other phases with a maximum value TheThe current current linked linked to to phasephase BB waswas higher thanthan thethe currents currents in in other other phases phases with with a maximuma maximum in model 1 around 93.5 A. Instead, phase A always presented the most eﬃcient current phase with valuevalue in in model model 1 1around around 93.5 93.5 A.A. Instead,Instead, phase AA alwaysalways presented presented the the most most efficient efficient current current phase phase onlywithwith 67.496only only 67.496 67.496 A in A model A in in model model 3. The 3. 3. The The electric electricelectric current curren associatedtt associatedassociated with with withphase phase phase C Chad had had an an an intermediate intermediate intermediate value valuevalue betweenbetweenbetween thethe the othersothers others inin in allall all experiences.experiences. experiences. InIn order to to design design the thethe dimensions dimensionsdimensions of of the thethe copper coppercopper wire wirewire and andand the thethe restrestrest ofof of thethe the devicesdevices devices thatthat that protectprotect protect the thethe MFLIM,MFLIM,MFLIM, thethe currentcurrent of ofof phase phasephase B BB was waswas a a areference referencereference signal. signal.signal. Model ModelModel 3 33 impliedimplied a areduction reduction of of the the currentscurrents consumedconsumed perper phasephase with with respect respect to to the the initial initial model model around around 20.5%20.5% (phase (phase A), A), 15% 15% (phase (phase B),B), andand 16.3% (phase C).C). TheThe developed developed models models illustrated illustrated the the great great relevancerelevance of of the the location location of of the the coils coils ofof each phase insideinside the the armature armature winding winding and and showed showed also also that that thethe changes changes introduced introduced into into the the geometrygeometry are veryvery importantimportant for for the the consumed consumed electric electric current. current. Appl. Sci. 2020, 10, 7004 21 of 26 implied a reduction of the currents consumed per phase with respect to the initial model around 20.5% (phase A), 15% (phase B), and 16.3% (phase C). The developed models illustrated the great relevance of the location of the coils of each phase inside the armature winding and showed also that the changes introduced into the geometry are very important for the consumed electric current.

4.4. Simulation Results with Motion Appl. Sci.To 2020 determine, 10, x FOR the PEER influence REVIEWof the motion in the proposed models, in this section, it is presented21 of 26 the FEM-3D simulation of the models including motion in the moving parts. Thus, Table4 shows the 4.4. Simulation Results with Motion velocities and its corresponding slips (see Equation (2)) selected to study the effect of the motion in the analyzedTo determine models. Morethe influence specifically, of the we motion analyze in thethe thrustproposed force, models, the levitation in this section, force, and it is the presented electric thecurrents. FEM-3D It is simulation important of to the notice models that theincluding friction moti losseson arein the not moving taken into parts. account Thus, in Table the present 5 shows work. the velocities and its corresponding slips (see Equation 2) selected to study the effect of the motion in the analyzed models. MoreTable specifically, 4. Slips values we andanalyze linear the velocities thrust simulated force, the with levitation FEM-3D. force, and the electric currents. It is important to notice that the friction losses are not taken into account in the present Dynamic Conditions for the Motion Analysis work. Slip 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Velocity (m/s) 20 18 16 14 12 10 8 6 4 2 0 Table 5. Slips values and linear velocities simulated with FEM-3D.

Dynamic Conditions for the Motion Analysis Figure 30 shows the evolution of the thrust force of each model for all values shown in Table4. Slip 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 To understand this ﬁgure, it is necessary to consider the following regions inside: Velocity (m/s) 20 18 16 14 12 10 8 6 4 2 0 Region I (low velocities zone): 0.6 < s < 1 0 m < v < 8 m . With these velocities, • Figure 30 shows the evolution of the thrust force↔ ofs eachmoving model_part for alls values shown in Table 5. the behavior of the thrust force had the same evolution as in standstill conditions, where this To understand this figure, it is necessary to consider the following regions inside: •force obtained its highest value in model 4 (see Equation (20)). Region I (low velocities zone): 0.6 < < 1 ↔ 0 ⁄ <_ <8 ⁄ . With these velocities, the behavior of the thrust force had the same evolution as in standstill conditions, →F > →F > →F > →F (20) where this force obtainedy_model its highest_4 y _valuemodel_3 in modely_model 4_2 (see Equationy_model_1 (20)). (20) __ >__ >__ >__

Curve Thrust Forces vs slip 300

250

200

150

100 Thrust Force [N] Force Thrust 50

0 1.0 0.9 0.8 0.7 0.6 slip 0.5 0.4 0.3 0.2 0.1 0.0 Thrust Force Model 1 [N] Thrust Force Model 2 [N] Thrust Force Model 3 [N] Thrust Force Model 4 [N]

Figure 30. ThrustThrust force force versus versus slips.

Between this this range of slips, the developed thrust forces for each model increased their values until the slipslip reachedreached the the value value of of 0.6. 0.6. At At this this point, point, the the thrust thrust forces forces from from model model 2, model 2, model 3, and 3, model and model4 reached 4 reached their maximum their maximum value around value around 226.2, 262.447, 226.2, 262.447, and 284.778 and 284.778 N, respectively. N, respectively. • Region II (Medium velocities zone): 0.5 < < 0.6 ↔ 8 ⁄ < <10⁄ . Region II (Medium velocities zone): 0.5 < s < 0.6 8 m < v _< 10 m . This region was • ↔ s moving_part s characterizedThis region by was a non-uniformity characterized by evolution a non-uniformi of the thrustty evolution force.Model of the 2,thrust model force. 3, and Model model 2, model 3, and model 4 presented their inflection points between these velocities and after that continued with an evolution similar to region I (see Equation (21)). (21) __ >__ >__ However, the thrust force developed in model 1 continued to increase from 8 to 10 m/s, where the thrust force was nearly 294 N (with a slip around 0.5). This value was the highest thrust force obtained in the present work. • Region III (High velocities zone): 0<<0.5↔10 ⁄ <_ <20 ⁄. In this region, the evolution of the thrust force decreased in all models. In general, model 1 presented a higher thrust force than others. The behavior of these forces in model 2, model 3, and model 4 were very similar. The thrust forces decreased as the velocity of the secondary Appl. Sci. 2020, 10, 7004 22 of 26

4 presented their inﬂection points between these velocities and after that continued with an evolution similar to region I (see Equation (21)).

→ → → F y_model_4 > F y_model_3 > F y_model_2 (21)

However, the thrust force developed in model 1 continued to increase from 8 to 10 m/s, where the thrust force was nearly 294 N (with a slip around 0.5). This value was the highest thrust force obtained in the present work. Region III (High velocities zone): 0 < s < 0.5 10 m < v < 20 m . In this region, • ↔ s moving_part s the evolution of the thrust force decreased in all models. In general, model 1 presented a higher thrust force than others. The behavior of these forces in model 2, model 3, and model 4 were very similar. The thrust forces decreased as the velocity of the secondary part reached the synchronous velocity (20 m/s and s = 0), when model 2 developed the highest thrust force (around 41 N) at slip equal to zero. This phenomenon appeared due to the end-eﬀect and it was not present in asynchronous rotatory machines because the torque obtained at synchronous velocity was equal to zero.

Next, Figure 31 shows the eﬀect of the motion for the levitation force. This ﬁgure gives us the following information:

Model 1, model 2, and model 3 presented two regions of work depending on the selected slip. • The change-point between the levitation zone and the attraction zone varied in the function of the models. So, this change of zone occurred when the values of slips were 0.55 in model 1, 0.6 in model 2, and 0.72 in model 3. However, model 4 did not operate under levitation condition. For all slip values, model 4 developed an attraction force that reached the maximum value near to 568 N with a velocity equal to 12 m/s (slip equal to 0.4). Two regions are shown in Figure 31: Region I: 0.5 < s < 1 0 m < v < 10 m . Inside this • ↔ s moving_part s region, the behavior of the levitation force is given by Equation (22) and followed the same law as in the standstill conditions. Besides, at slip equal to one, model 1, model 2, and model 3 developed the maximum levitation force around 404.04, 204.16, and 65.38 N, respectively:

→ → → → F z_model_4 < F z_model_3 < F z_model_2 < F z_model_1 (22)

It is important to remark the relevance of this force in the TFLIM. The highest values of levitation and attraction forces were obtained in model 1 (around 404.04 N for slip equal to one and 747.63 N, with velocities near to the synchronous velocity or slip equal to zero). Region II: − 0 < s < 0.5 10 m < v < 20 m . In this zone, model 2, model 3, and model 4 continued ↔ s moving_part s the evolution from region I (see Equation (23)). These forces did not continue decreasing with the → slip as occurred with the levitation force from model 1, F z_model_1, that decreased with the slip.

→ → → F z_model_4 < F z_model_3 < F z_model_2 (23) Appl. Sci. 2020, 10, x FOR PEER REVIEW 22 of 26

part reached the synchronous velocity (20 m/s and s = 0), when model 2 developed the highest thrust force (around 41 N) at slip equal to zero. This phenomenon appeared due to the end- effect and it was not present in asynchronous rotatory machines because the torque obtained at synchronous velocity was equal to zero.

Appl. Sci.Next,2020 Figure, 10, 7004 31 shows the effect of the motion for the levitation force. This figure gives us23 ofthe 26 following information:

Curve Levitation Forces vs slip 500

300

100

-100 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

-300

-500

Levitation Force [N] slip -700 Levitation Force Model 1 [N] Levitation Force Model 2 [N] -900 Levitation Force Model 3 [N] Levitation Force Model 4 [N]

Figure 31. Levitation force versus slips.

•Figures Model 32 –1,34 model show the2, and evolution model of3 presented the electrical two current regions for of each work phase. depending These ﬁgureson the selected conﬁrm Appl.the reached Sci. 20202020slip.,, 10 results10 The,, xx FORFOR change-point in PEERPEER standstill REVIEWREVIEW conditions.between the levitation zone and the attraction zone varied 2323in of ofthe 2626 function of the models. So, this change of zone occurred when the values of slips were 0.55 in model 1, 0.6 in model Curve2, and Current0.72 in mode Phasel 3. A However, vs slip model 4 did not operate under levitation9090 condition. For all slip values, model 4 developed an attraction force that reached the maximum value near to 568 N with a velocity equal to 12 m/s (slip equal to 0.4). • 8585 Two regions are shown in Figure 31: Region I: 0.5<<1↔0 ⁄ <_ <10 ⁄. Inside this region, the behavior of the levitation force is given by Equation (22) and followed 8080 the same law as in the standstill conditions. Besides, at slip equal to one, model 1, model 2,

and model7575 3 developed the maximum levitation force around 404.04, 204.16, and 65.38 N, Current [A] Current respectively: [A] Current 7070 (22) __ <__ <__ <__

It is important6565 to remark the relevance of this force in the TFLIM. The highest values of levitation and11 attraction0.90.9 0.80.8 forces0.70.7 were0.60.6 obtained0.50.5 in0.40.4 model0.30.3 1 (around0.20.2 0.1404.040.1 0N0 for slip equal to slipslip one and -747.63 N, Currentwith velocities Phase A Model near 1 to the synchronousCurrent Phase A Model velocity 2 or slip equal to zero). Region II: 0<<0.5↔10Current Phase A⁄ Model< 3_Current<20 Phase⁄. A In Model this 4 zone, model 2, model 3, and Current Phase A Model 3 Current Phase A Model 4 model 4 continued the evolution from region I (see Equation (23)). These forces did not continue decreasingFigure with 32. the ElectricElectric slip ascurrent current occurred phase phase Awith A versus versus the slips. levitation force from model 1, __, that decreased with the slip. (23) Curve_ Current_ <_ Phase_ < B vs_ slip_

Figures 32–34100100 show the evolution of the electrical current for each phase. These figures confirm the reached results in standstill conditions. 9595

9090

8585 Current [A] Current Current [A] Current 8080

7575 11 0.90.9 0.80.8 0.70.7 0.60.6 0.50.5 0.40.4 0.30.3 0.20.2 0.10.1 00 slipslip Current Phase B Model 1 Current Phase B Model 2 Current Phase B Model 3 Current Phase B Model 4

Figure 33. ElectricElectric current current phase phase B B versus versus slips.

Curve Current Phase C vs slip

9393

8888

8383

7878

7373 Current [A] Current [A]

6868 11 0.90.9 0.80.8 0.70.7 0.60.6 0.50.5 0.40.4 0.30.3 0.20.2 0.10.1 00 slipslip Current Phase C Model 1 Current Phase C Model 2 Current Phase C Model 3 Current Phase C Model 4

Figure 34. Electric current phase C versus slips.

For each model, the relationship between the electrictric currentscurrents followedfollowed EquationEquation (17).(17). Besides,Besides, model 3 was the configuration that presented a low electric current consumed by each phase across thethe rangerange ofof velocities.velocities. So,So, withwith aa slipslip ofof 0.5,0.5, phasephase AA andand phasephase CC inin modelmodel 33 onlyonly consumedconsumed 66.1466.14 and 70.60 A, respectively. Phase B reached its minimum value with a slip value of 0.3 (around to 75.55 A). IfIf wewe comparecompare thethe obtainedobtained resultsresults consideringconsidering thethe wholewhole rangerange ofof slipsslips fromfrom FiguresFigures 30–34,30–34, wewe have available a great amount of information aboutt thethe convenienceconvenience toto includeinclude thethe proposedproposed changeschanges Appl. Sci. 2020, 10, x FOR PEER REVIEW 23 of 26

Curve Current Phase A vs slip 90

75 Current [A] Current

65 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 slip Current Phase A Model 1 Current Phase A Model 2

Current Phase A Model 3 Current Phase A Model 4

Figure 32. Electric current phase A versus slips.

Curve Current Phase B vs slip

100

Current [A] Current 80

75 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 slip Current Phase B Model 1 Current Phase B Model 2 Current Phase B Model 3 Current Phase B Model 4

Appl. Sci. 2020, 10, 7004 24 of 26 Figure 33. Electric current phase B versus slips.

Curve Current Phase C vs slip

73 Current [A]

68 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 slip Current Phase C Model 1 Current Phase C Model 2 Current Phase C Model 3 Current Phase C Model 4

Figure 34. Electric current phase C versus slips.

For each model, the relationship between the electricelectric currents followed Equation (17). Besides, model 3 waswas thethe configurationconfiguration that that presented presented a lowa low electric electric current current consumed consumed by by each each phase phase across across the therange range of velocities. of velocities. So, withSo, with a slip a slip of 0.5, of 0.5, phase phase A and A and phase phase C in C model in model 3 only 3 only consumed consumed 66.14 66.14 and and70.60 70.60 A, respectively. A, respectively. Phase Phase B reached B reached its minimum its minimum value value with with a slip a slip value value of 0.3 of 0.3 (around (around to 75.55 to 75.55 A). A). If we compare the obtained results considering the whole range of slips from Figures 30–34, we haveIf we available compare athe great obtained amount results of information considering about the whole theconvenience range of slips to from include Figures the 30–34, proposed we havechanges available into the a great geometry amount of the of information TFLIM, in order abou tot the achieve convenience a more etoffi includecient motor. the proposed We can conclude changes that with slip values above 0.5, model 4 developed the highest thrust force. It supposes that the changes included in the geometric parameters are useful to increase the thrust force inside the range of velocities between 0 and 10 m/s. However, model 4 always operated under attraction force condition, so an increment in the thrust force implied a reduction in the levitation force. In this way, the thrust force developed by model 3 was near to model 4 and it operated inside the levitation zone for low velocities. Therefore, we can say that model 3 is an optimum model to operate with velocities below 10 m/s. However, for slip values below 0.5, model 1 presents the best behavior. In conclusion, the proposed changes are irrelevant for velocities from 10 to 20 m/s and model 3 is the most efficient configuration for all range of velocities in comparison with model 1, that presents in phase B, the highest electric current consumed, around 95 A at the synchronous velocity.

5. Conclusions The main goal of this paper was to develop a new configuration for a more efficient LIM, based on a prototype model called model 1. To this end, we proposed three changes to the geometric parameters of the fixed and moving parts and we defined a new model for each proposed change. The first change was incorporated into model 2, including three different regions into the aluminum layer through two slots of dielectric, in order to separate the loops of the induced currents. Thus, the thrust force generated over the lateral teeth was developed in the desired direction. The second change was proposed in model 3, where we included a ferromagnetic yoke in the primary part to produce a longitudinal magnetic flux. Finally, the third change was included in model 4 and implied the optimization of the width of the transversal slots. In standstill conditions (slip = 1), model 4 achieved the maximum thrust force of all models, around 75% with respect to the prototype model (model 1). However, this model presented a great disadvantage because it produces a loss of levitation force which begins to operate inside an attraction zone. With motion and velocities below 10 m/s (slip > 0.5), model 4 also developed the maximum thrust force. However, for velocities between 10 and 20 m/s (slip < 0.5), the changes added into the geometry did not contribute to increasing the thrust force. In this case, model 1 developed the maximum thrust force. Besides, for the whole range of analyzed velocities, model 4 operated inside the attraction zone. Appl. Sci. 2020, 10, 7004 25 of 26

Hence, for velocities below 10 m/s (slip > 0.5), model 3 presented a similar thrust force as model 4 and operated inside the levitation zone for velocities below 6 m/s (slip > 0.7). Regarding the consumed electric currents, model 3 presented the most efficient configuration for all slip values because it achieved two main magnetic circuits operating inside the motor—transversal and longitudinal. In brief, for velocities below 10 m/s, model 3 represents an improvement in respect to model 1, due to the high thrust force developed and the low electric current consumed. To simulate all proposed models with FEM-3D, we presented a method that allows one to analyze the magnetic field density along the airgap in an LIM. The method is composed of two steps. First, according to an analytical method, the first harmonic of the magnetic field density in the airgap was computed. This is the starting point of the simulations with FEM-3D. Second, with an iterative process, the external circuit coupling to supply the LIM was fixed. To this end, we used two electrical parameters: the number of turns of the coils per phase, Nphase, and the level of voltage in an AC three-phase system, VLINE. Using FEM-3D, the coupling between the primary winding of the LIM and the voltage sources has been defined completely. In addition, the selected transitory state in FEM-3D gives us valuable information to design a control strategy of a LIM. In the transitory state, the use of FEM-3D allows us to analyze the unbalanced system originated by the longitudinal end-effect. It is a phenomenon that cannot be neglected and presents the same distribution for all models and values of slip: IPhase_a < IPhase_c < IPhase_b. This behavior implies a great advantage of using FEM-3D for two reasons. Firstly, the highest phase (Ib) will be a reference signal to design the insulated scheme and the electrical protection of the LIM. Secondly, the unbalanced electric system supposes high variations in the forces during the transitory state. This effect implies that it is necessary to carry out an analysis of the vibration modes using FEM when we want to design the structural support of an LIM in mechanical engineering.

Author Contributions: All authors have collaborated in the writing of the paper. J.A.D.H. has made the design and simulation of all experiments and prepared the original draft presentation. N.D.C. and E.G.V. collaborated in the supervision of the work and in the review and editing of the draft. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported in part by the Spanish Ministry of Economy and Competitiveness under the Projects CICY DPI2017-84259-C2-2-R, PID2019-108377RB-C32, and the group of educational innovation project GID2016-6. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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