Numerical Modeling of the Thomson Ring in Stationary Levitation Using FEM-Electrical Network and Newton-Raphson

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array:5 [ 0 => array:3 [ "entidad" => "Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco" "etiqueta" => "a" "identificador" => "aff0005" ] 1 => array:3 [ "entidad" => "Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco" "etiqueta" => "b" "identificador" => "aff0010" ] 2 => array:3 [ "entidad" => "Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco" "etiqueta" => "c" "identificador" => "aff0015" ] 3 => array:3 [ "entidad" => "Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco" "etiqueta" => "d" "identificador" => "aff0020" ] 4 => array:3 [ "entidad" => "Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco" "etiqueta" => "e" "identificador" => "aff0025" ] ] "correspondencia" => array:1 [ 0 => array:3 [ "identificador" => "cor0005" "etiqueta" => "⁎" "correspondencia" => "Corresponding author." ] ] ] ] "titulosAlternativos" => array:1 [ "es" => array:1 [ "titulo" => "Modelación numérica del anillo de Thomson en levitación estacionaria usando circuitos eléctricos, MEF y Newton-Raphson" ] ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0025" "etiqueta" => "Figure 5" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr5.jpeg" "Alto" => 1200 "Ancho" => 1546 "Tamanyo" => 58467 ] ] "descripcion" => array:1 [ "en" => "Average Lorentz force fzav as a function of the distance zs." ] ] ] "textoCompleto" => "IntroductionIn the electric industry is important to have electric systems of immediate breaking and safe. Some of these electric systems consist of power switchers, which use the Thomson ring (Meyer and Rufer, 2006Meyer & Rufer; <a class="elsevierStyleCrossRef" href="#bib0065">Meyer and Rufer, 2006Meyer & Rufer, 2006</a>). Other systems utilize the Thomson ring as actuator to eliminate the electric arcs (<a class="elsevierStyleCrossRef" href="#bib0045">Li, Jeong, Yoon, & Koh, 2010</a>). Other applications of the Thomson ring consist in the levitation of superconductor materials (<a class="elsevierStyleCrossRef" href="#bib0070">Patitsas, 2011</a>). Therefore, it is important the numerical modeling of the Thomson ring. The Thomson ring consists of a coil with ferromagnetic core on which an aluminum ring levitates. The coil is fed by a cosine voltage.The modeling of the electromagnetic field of any electric device (as the Thomson ring) requires of the knowledge of the current density. However, this knowledge cannot be known a priori. It is known a priori the power source voltage instead of current density. In the literature, several methods (<a class="elsevierStyleCrossRefs" href="#bib0020">Belforte et al., 1985; Bissal et al., 2010; Konrad, 1982; Lombard and Meunier, 1992</a>, <a class="elsevierStyleCrossRefs" href="#bib0055">1993; Meunier et al., 1988; Piriou and Razek, 1989</a>) have been developed to calculate the electromagnetic field if the power source voltage is supplied: integro-differential method (<a class="elsevierStyleCrossRef" href="#bib0040">Konrad, 1982</a>); direct methods (<a class="elsevierStyleCrossRef" href="#bib0020">Belforte et al., 1985</a>; Meunier et al., 1988; <a class="elsevierStyleCrossRef" href="#bib0075">Piriou and Razek, 1989</a>); and methods that use electric networks equations (<a class="elsevierStyleCrossRefs" href="#bib0015">Barry and Casey, 1999; Bissal et al., 2010; Lombard and Meunier, 1992, 1993</a>). In this work is supposed that the power source voltage is known and the current density is calculated using electric networks equations.Several studies have analyzed the mathematical and physics models of the Thomson ring. In the work of <a class="elsevierStyleCrossRef" href="#bib0025">Bissal et al. (2010)</a> is modeled the dynamic behavior of the Thomson ring, which consist of a coil without ferromagnetic core. In this work, the coil is fed by a capacitor. <a class="elsevierStyleCrossRef" href="#bib0015">Barry and Casey (1999)</a> obtained analytical solutions of the force acting on the aluminum ring in a stationary levitated position. In the work of <a class="elsevierStyleCrossRef" href="#bib0045">Li et al. (2010)</a> is analyzed the dynamic characteristics of the Thomson ring used as actuator to eliminate the electric arcs. In the work of <a class="elsevierStyleCrossRef" href="#bib0070">Patitsas (2011)</a> is developed a new modality of Thomson ring. This modality consisted in keeping the stable levitation of a superconductor sphere immerse in a magnetic field supplied by a coil.The aim of this work is to analyze the Thomson ring when the aluminum ring is a stationary levitated position. This situation is reached if the coil is fed by a cosine voltage. In the stationary levitation, the state of the electromagnetic field is stable and can be used the phasor equations of the electromagnetic field. These equations are discretized using the Galerkin method. These discretized equations are solved using the COMSOL software (<a class="elsevierStyleCrossRef" href="#bib0030">COMSOL, 2008</a>). It is described the methodology (which uses the Newton-Raphson method) that obtains the separation between the coil and the aluminum ring in stationary levitation (mechanical equilibrium). Also, the separation obtained with this methodology is compared with the experimental data for different values of the power source voltage. It is concluded that the magnetic coupling of the aluminum ring on the coil can be neglected if the source voltage is high.Experimental setupThe Thomson ring used in this work consists of a coil with ferromagnetic core; where an aluminum ring is threaded on the core, as shown in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>. The ferromagnetic core consists in a solid cylinder that is collocated vertically, as is illustrated in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>. In this figure, ZS represents the distance between the coil and the aluminum ring.<elsevierMultimedia ident="fig0005"></elsevierMultimedia>The coil is made of copper wire and consists of 1140 turns (see, <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>). This coil is fed by a cosine voltage given by<elsevierMultimedia ident="eq0005"></elsevierMultimedia><elsevierMultimedia ident="tbl0005"></elsevierMultimedia>where, V0 is the amplitude; ω = 2πf is the angular frequency (f being the natural frequency), as shown in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>. The electric and magnetic characteristics of the materials used in the Thomson ring are indicated in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>. The ferromagnetic core is iron whose relativity permeability is taken from the data base of <a class="elsevierStyleCrossRef" href="#bib0030">COMSOL (2008)</a>.In order to take advantage of axial symmetry, the Thomson ring is represented by means of axisymmetric geometry as depicted in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>. In this Figure, a cylinder coordinate system is chosen so that the r-axis represents the horizontal axis, the z-axis represents the vertical axis. The dimensions of the aluminum ring are: interior radius of 0.031 m, exterior radius of 0.0365 m and height of 0.018 m as illustrated in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>. In this figure, ZS is the separation distance between aluminum ring and copper coil.<elsevierMultimedia ident="fig0010"></elsevierMultimedia>The copper coil forms a toroid with dimensions: interior radius of 0.025 m, exterior radius of 0.039 m, and height of 0.075 m, as shown in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>. The distance between base of ferromagnetic core and base of copper coil is 0.025 m. The copper coil is threaded on a ferromagnetic cylinder (ferromagnetic core). The ferromagnetic core has a radius of 0.0235 m and height of 0.41 m, as depicted in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>.Solution methodologyElectromagnetic field equationsIn this section the equations that describe the electromagnetic field in the Thomson ring are presented. The magnetic field B→=∇×A→ (A→ being the magnetic vector potential) satisfies the Ampere-Maxwell equation<elsevierMultimedia ident="eq0010"></elsevierMultimedia>where, v is the reluctivity, J→ is the current density; and D→=εE→ (¿ being the permittivity) is the electric density. The second term in the Eq. (2) represents the displacement current, which can be dropped if the frequency of the power source is small; in this case, the Eq. (2) is given by<elsevierMultimedia ident="eq0015"></elsevierMultimedia>The current density in this equation depends of the type of region (cupper coil, aluminum ring, air or ferromagnetic core) and is given by:1) Air and ferromagnetic core regionThe current density in air and ferromagnetic core regions is J→=0→; therefore<elsevierMultimedia class="elsevierStyleLink" ident="fx4"></elsevierMultimedia>2) Aluminum ring regionThe current density J→=σE→ (being σr the electric conductivity of the aluminum ring) is found using the Faraday law<elsevierMultimedia ident="eq0020"></elsevierMultimedia>Solving this equation for the vector potential A→<elsevierMultimedia ident="eq0025"></elsevierMultimedia>It is observed that this equation does not contain the term of the scalar electric potential gradient (ΔV) due to that there is not a power source in the aluminum region. Substituting Eq. (6) in J→=σrE→<elsevierMultimedia ident="eq0030"></elsevierMultimedia>Substituting Eq. (6) in Eq. (3)<elsevierMultimedia ident="eq0035"></elsevierMultimedia>3) Copper coil regionThe region of the copper coil is modeled as a region that contains N turns where each turn carries the same current iC. In this case, the current density J is uniform with value<elsevierMultimedia ident="eq0040"></elsevierMultimedia>where, SC is the cross section area of the copper coil region. Substituting Eq. (9) in Eq. (3)<elsevierMultimedia ident="eq0045"></elsevierMultimedia>where, Iˆ is a unit vector pointed in direction of the current density.Electrical network equationsIf the current is known, the solution of the Eq. (10) can be realized. However, this current cannot be known a priori. We know a priori the voltage V between the terminals of the coil. An additional equation is required. This equation is obtained using the Kirchhoff voltage law<elsevierMultimedia ident="eq0050"></elsevierMultimedia>where, R is the resistance, ϕ is the magnetic flux that cross all the turns of the coil. The resistance is given by<elsevierMultimedia ident="eq0055"></elsevierMultimedia>where, σC is the electric conductivity of the coil and L is the length of all the turns of the coil. The magnetic flux is given by<elsevierMultimedia ident="eq0060"></elsevierMultimedia>where, the surface SC comprises all the surfaces of the turns of the coil. Using the fact B→=∇×A→ and the Stokes theorem in Eq. (13) we obtain<elsevierMultimedia ident="eq0065"></elsevierMultimedia>where, the trajectory C comprises all the turns of the coil. Substituting Eq. (14) in Eq. (11)<elsevierMultimedia ident="eq0070"></elsevierMultimedia>Phasor equationsThe current in the copper coil is cosine to ensure that the aluminum ring stays in a stationary levitated position. In this situation, the state of the electromagnetic field is stable and the equations of the electromagnetic field can be given in phasor form. In phasor notation, the operator d/dt becomes iω in Eqs. (4), (8) and (10):air and ferromagnetic core regions<elsevierMultimedia ident="eq0075"></elsevierMultimedia>aluminum ring region<elsevierMultimedia ident="eq0080"></elsevierMultimedia>copper coil region<elsevierMultimedia ident="eq0085"></elsevierMultimedia>where, A→ and iC are phasors of the potential A→ and the current iC, respectively.The phasor equation of the electrical network equation Eq. (15) is<elsevierMultimedia ident="eq0090"></elsevierMultimedia>where, V is the phasor of the voltage V.Boundary conditionsIt is observed that Eq. (3) is a second order partial differential equation for the magnetic vector potential A→. The solution of this partial differential equation requires boundary conditions for the vector potential A→. The boundary of the solution domain is chosen so that the vector potential can be dropped (magnetic insulation). The magnetic insulation condition is expressed as<elsevierMultimedia ident="eq0095"></elsevierMultimedia>where, Γ is the boundary of solution domain. In phasor notation, the condition of magnetic insulation is<elsevierMultimedia ident="eq0100"></elsevierMultimedia>DiscretizationUsing the Galerkin method (Lombard and Meunier, 1992Lombard & Meunier; Lombard and Meunier, 1992Lombard and Meunier, 1993Lombard & Meunier; <a class="elsevierStyleCrossRef" href="#bib0055">Lombard and Meunier, 1993Lombard & Meunier, 1993</a>), Eqs. (16)-(19) can be discretized:air and ferromagnetic core regions<elsevierMultimedia ident="eq0105"></elsevierMultimedia>aluminum ring region<elsevierMultimedia ident="eq0110"></elsevierMultimedia>copper coil region<elsevierMultimedia ident="eq0115"></elsevierMultimedia>electrical network equation<elsevierMultimedia ident="eq0120"></elsevierMultimedia>With<elsevierMultimedia ident="eq0125"></elsevierMultimedia>where N represents the number of nodes. The matrices and vectors are defined as<elsevierMultimedia ident="eq0130"></elsevierMultimedia><elsevierMultimedia ident="eq0135"></elsevierMultimedia><elsevierMultimedia ident="eq0140"></elsevierMultimedia><elsevierMultimedia ident="eq0145"></elsevierMultimedia>where, the vector potential A is expanded in the base function βi: A=∑βjAj. The surface Sd is the surface of the solution domain.Mechanical equilibriumThe voltage is a cosine in order to maintain the aluminum ring in a stationary levitated position. This stationary levitation is obtained when the mechanical equilibrium is reached; this is, the Lorentz force averaged in a cycle, fzav equals the gravity force fg. Using the complex notation, the Lorentz force fzav (Barry and Casey, 1999Barry & Casey; Barry and Casey, 1999Hayt and Buck, 2006Hayt & Buck; <a class="elsevierStyleCrossRef" href="#bib0035">Hayt and Buck, 2006Hayt & Buck, 2006</a>) is given by<elsevierMultimedia ident="eq0150"></elsevierMultimedia>with<elsevierMultimedia ident="eq0155"></elsevierMultimedia>where, B→ and J→ are the phasors of magnetic density and current density, J→ respectively. The factor 1/2 in Eq. (31) is due to that the Lorentz force period is half of the magnetic field period (Barry and Casey, 1999Barry & Casey; <a class="elsevierStyleCrossRef" href="#bib0015">Barry and Casey, 1999Barry & Casey, 1999</a>). <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a> shows the flowchart of the obtaining of the average Lorentz force. The steps of this methodology are:<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005">1)Calculate the phasor potential using the phasor equations (Eqs. 22-25) along with boundary condition of magnetic insulation A→⋅nˆ=0 on Γ (see Eq. 20).</li><li class="elsevierStyleListItem" id="lsti0010">2)Determine the phasor magnetic density B→=∇×A→ and phasor current density J→=iωσrA→ (see Eq. 6) in the aluminum ring region.</li><li class="elsevierStyleListItem" id="lsti0015">3)Calculate the average Lorentz force fzav=12∫VrJ→×B→*dV (see Eq. 31).</li></ul><elsevierMultimedia ident="fig0015"></elsevierMultimedia>The space distribution of the electromagnetic field depends of the separation sz between the aluminum ring and the copper coil. Therefore, the average Lorentz force fzav is a function of the separation (fz = fz(sz)). In order to reach the stationary levitation of the aluminum ring, the average Lorentz force fzav equals to the gravity force fg.<elsevierMultimedia ident="eq0160"></elsevierMultimedia>where, zs’ is the separation in stationary levitation and represents the root of Eq. (33). It is observed that Eq. (33) is a transcendental equation. The root of this transcendental equation can be found using a variant of the Newton-Raphson method: secant method (Arfken and Weber, 2005Arfken & Weber; <a class="elsevierStyleCrossRef" href="#bib0010">Arfken and Weber, 2005Arfken & Weber, 2005</a>). The convergence of Newton-Raphson is guaranteed due to that the average Lorentz force fzav(zs) is a function decreasing of the separation zs (see <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>). The secant method is defined by the recurrence relation<elsevierMultimedia ident="eq0165"></elsevierMultimedia><elsevierMultimedia ident="fig0025"></elsevierMultimedia>Experimental validationIn this section we compared the numerical and experimental results for the separation in stationary levitation zs’ as a function of the voltage amplitude in rms, Vrms=V02. The experimental setup was described in the second section. The numerical results are obtained using the proposed methodology in the section above. <a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a> shows the separation zs’ as function of the voltage amplitude Vrms for both experimental and numerical results. The discrepancy between the theoretical and experimental data is at most 12%. This difference can be due to the fact that the numerical modeling does not take into account the temperature effect in the electric conductivity σ.<elsevierMultimedia ident="fig0020"></elsevierMultimedia>Results and discussionIn this section some results obtained by the proposed modeling are studied. The average Lorentz force is examined as a function of the separation distance; the ratio between coil current and ring current, and the spatial distribution of the magnetic field.<a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a> depicts the average Lorentz force fzav as function of the separation zs for a representative voltage amplitude Vrms = 120V. It is observed that the Lorentz force is a decreasing function of the distance zs. This guarantees the convergence of the Newton-Raphson method due to that the derivative dfzavdzS is negative.<a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a> shows the spatial distribution of the radial component Br0 of the magnetic density amplitude, for a representative voltage amplitude Vrms = 120V in state of stationary levitation (zs’ = 0.057 m). It also presents the positions of the ferromagnetic core, copper coil and aluminum ring. This <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a> shows that the radial component is higher in regions close to the core, coil and ring edges. In contrast, the radial component Br0 presents small values in positions far away from above edges.<elsevierMultimedia ident="fig0030"></elsevierMultimedia>The total current in the ring ir is realized by means of ir=∫ringJ→⋅dS→; while the total current in the region of the coil is Nic. In <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a> is shown the ratio irNic as function of voltage amplitude Vrms in stationary levitation. It is observed that the highest value (ir / NiC = 0.47) occurs in Vrms = 43.4V corresponding to a separation zs’= 0. The ratio ir / NiC decreases if the voltage amplitude Vrms increases. Also, in a first order approach, the magnetic field originated by any system is proportional to the current of this system. Therefore, the magnetic coupling of the ring on the coil can be neglected for high values of voltage amplitude.<elsevierMultimedia ident="fig0035"></elsevierMultimedia>ConclusionsThe aim of this work was to present a numerical modeling based upon the use of the Galerkin method to simulate the electromagnetic field of the Thomson ring. Also, this modeling is capable of simulating numerically the separation between aluminum ring and copper coil in situation of stationary levitation (the average Lorentz force equals gravity force). This calculation of the separation uses the Newton-Raphson method.The proposed modeling was validated comparing theoretical and experimental results. The compared results were the separation between the aluminum ring and the copper coil (in stationary levitation) for different voltage amplitudes.It is concluded that the magnetic coupling of the aluminum ring on the coil can be neglected if the source voltage is high. Therefore, the coil current can be modeled without taking into account the coupling ring-coil. This means that the coil current is found using a RL (resistance-inductance) circuit; where, the resistance and inductance are parameter of the coil.Citation for this article:Chicago citation styleGuzmán, Juan, Felipe de Jesús González-Montañez, Rafael Escarela-Pérez, Juan Carlos Olivares-Galván, Victor Manuel Jiménez-Mondragon. Numerical modeling of the Thomson ring in stationary levitation using FEM-electrical network and Newton-Raphson. Ingeniería Investigación y Tecnología, XVI, 03 (2015): 431-439.ISO 690 citation styleGuzmán J., González-Montanez F.J., Escarela-Pérez R., Olivares-Galván J.C., Jiménez-Mondragon V.M. Numerical modeling of the Thomson ring in stationary levitation using FEM-electrical network and Newton-Raphson. Ingeniería Investigación y Tecnología, volume XVI (issue 3), july 2015: 431-439." "textoCompletoSecciones" => array:1 [ "secciones" => array:10 [ 0 => array:3 [ "identificador" => "xres528819" "titulo" => "Abstract" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0005" ] ] ] 1 => array:2 [ "identificador" => "xpalclavsec549051" "titulo" => "Keywords" ] 2 => array:3 [ "identificador" => "xres528818" "titulo" => "Resumen" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0010" ] ] ] 3 => array:2 [ "identificador" => "xpalclavsec549052" "titulo" => "Descriptores" ] 4 => array:3 [ "identificador" => "sec0005" "titulo" => "Introduction" "secciones" => array:6 [ 0 => array:2 [ "identificador" => "sec0010" "titulo" => "Experimental setup" ] 1 => array:2 [ "identificador" => "sec0015" "titulo" => "Solution methodology" ] 2 => array:2 [ "identificador" => "sec0020" "titulo" => "Phasor equations" ] 3 => array:2 [ "identificador" => "sec0025" "titulo" => "Boundary conditions" ] 4 => array:2 [ "identificador" => "sec0030" "titulo" => "Discretization" ] 5 => array:2 [ "identificador" => "sec0035" "titulo" => "Mechanical equilibrium" ] ] ] 5 => array:2 [ "identificador" => "sec0040" "titulo" => "Experimental validation" ] 6 => array:2 [ "identificador" => "sec0045" "titulo" => "Results and discussion" ] 7 => array:2 [ "identificador" => "sec0050" "titulo" => "Conclusions" ] 8 => array:1 [ "titulo" => "Further reading" ] 9 => array:1 [ "titulo" => "References" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2014-03-01" "fechaAceptado" => "2014-07-01" "PalabrasClave" => array:2 [ "en" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Keywords" "identificador" => "xpalclavsec549051" "palabras" => array:5 [ 0 => "Thomson ring" 1 => "levitation" 2 => "stationary" 3 => "modeling" 4 => "FEM" ] ] ] "es" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Descriptores" "identificador" => "xpalclavsec549052" "palabras" => array:5 [ 0 => "anillo de Thomson" 1 => "levitación" 2 => "estacionaria" 3 => "modelación" 4 => "MEF" ] ] ] ] "Biografia" => array:5 [ 0 => array:2 [ "identificador" => "vt0005" "biografia" => "Juan Guzmán. Obtained Ph.D. in Energy Engineering from the Universidad Nacional Autónoma de México, México City, Mexico, in 2008. He is currently with the área de ingeniería energética y electromagnética, Departamento de Energía, UAM, Azcapotzalco, México." ] 1 => array:2 [ "identificador" => "vt0010" "biografia" => "Felipe de Jesús González-Montañez. He received the M.Sc. degree in electrical engineering from the Centro de Investigación y de Estudios Avanzados del IPN, México City, Mexico, in 2011. His research interests include the modeling and control of electrical machines." ] 2 => array:2 [ "identificador" => "vt0015" "biografia" => "Rafael Escarela-Pérez. He obtained his B.Sc. in electrical engineering from Universidad Autonoma Metropolitana, Mexico City in 1992 and his Ph.D. from Imperial College, London in 1996. He is interested in the modeling of electrical machines." ] 3 => array:2 [ "identificador" => "vt0020" "biografia" => "Juan Carlos Olivares-Galván. He received the Ph.D. degree in electrical engineering from CINVESTAV, Guadalajara, Mexico, in 2003. His main research interests are related to the experimental and numerical analysis of electromagnetic devices." ] 4 => array:2 [ "identificador" => "vt0025" "biografia" => "Victor Manuel Jiménez-Mondragon. He received the M.Sc. degree in electrical engineering from the Universidad Nacional Autónoma de México, México City, Mexico, in 2012. He is interested in the modeling of electrical machines." ] ] "tieneResumen" => true "resumen" => array:2 [ "en" => array:2 [ "titulo" => "Abstract" "resumen" => "There are a lot of applications of the Thomson ring: levitation of superconductor materials, power interrupters (used as actuator) and elimination of electric arcs. Therefore, it is important the numerical modeling of Thomson ring. The aim of this work is to model the stationary levitation of the Thomson ring. This Thomson ring consists of a copper coil with ferromagnetic core and an aluminum ring threaded in the core. The coil is fed by a cosine voltage to ensure that the aluminum ring is in a stationary levitated position. In this situation, the state of the electromagnetic field is stable and can be used the phasor equations of the electromagnetic field. These equations are discretized using the Galerkin method in the Lagrange base space (finite element method, FEM). These equations are solved using the COMSOL software. A methodology is also described (which uses the Newton-Raphson method) that obtains the separation between coil and aluminum ring. The numerical solutions of this separation are compared with experimental data. The conclusion is that the magnetic coupling of the aluminum ring on the coil can be neglected if the source voltage is high." ] "es" => array:2 [ "titulo" => "Resumen" "resumen" => "Existen una gran cantidad de aplicaciones del anillo de Thomson: levitación de materiales superconductores, interruptores de potencia (usados como actuadores) y eliminación de arcos eléctricos. Por lo tanto, es importante la modelación del anillo de Thomson. El objetivo de este trabajo es modelar la levitación estacionaria del anillo de Thomson. Este anillo de Thomson consiste de una bobina de cobre con núcleo ferromagnético y un anillo de aluminio enhebrado en el núcleo. La bobina se alimenta por un voltaje cosenoidal para asegura el anillo de aluminio en una posición de levitación estacionaria. En esta situación, el campo electromagnético se puede considerar estable y se pueden emplear las ecuaciones fasoriales del campo electromagnético. Estas ecuaciones se discretizan usando el método de Galerkin en el espacio base de Lagrange (método de elementos finitos, FEM). Estas ecuaciones discretizadas se resuelven usando el código COMSOL. Además, se describe una metodología con la cual se puede obtener la separación entre la bobina y el anillo de aluminio. Esta metodología usa el método de Newton-Rapson. Las soluciones numéricas de esta separación se comparan con datos experimentales. Se concluye que el acoplamiento magnético entre el anillo de aluminio sobre la bobina se puede despreciar si el voltaje de alimentación es alto." ] ] "multimedia" => array:41 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 1591 "Ancho" => 1571 "Tamanyo" => 113855 ] ] "descripcion" => array:1 [ "en" => "Thomson ring setup." ] ] 1 => array:7 [ "identificador" => "fig0010" "etiqueta" => "Figure 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 1288 "Ancho" => 2115 "Tamanyo" => 177997 ] ] "descripcion" => array:1 [ "en" => "a) axisymmetric representation of the Thomson ring, b) experimental setup." ] ] 2 => array:7 [ "identificador" => "fig0015" "etiqueta" => "Figure 3" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr3.jpeg" "Alto" => 1335 "Ancho" => 1377 "Tamanyo" => 125133 ] ] "descripcion" => array:1 [ "en" => "Flowchart of the obtaining of fzav." ] ] 3 => array:7 [ "identificador" => "fig0020" "etiqueta" => "Figure 4" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr4.jpeg" "Alto" => 1196 "Ancho" => 1574 "Tamanyo" => 84821 ] ] "descripcion" => array:1 [ "en" => "zs’ vs V0 for theoretical and experimental data." ] ] 4 => array:7 [ "identificador" => "fig0025" "etiqueta" => "Figure 5" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr5.jpeg" "Alto" => 1200 "Ancho" => 1546 "Tamanyo" => 58467 ] ] "descripcion" => array:1 [ "en" => "Average Lorentz force fzav as a function of the distance zs." ] ] 5 => array:7 [ "identificador" => "fig0030" "etiqueta" => "Figure 6" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr6.jpeg" "Alto" => 1665 "Ancho" => 2123 "Tamanyo" => 159509 ] ] "descripcion" => array:1 [ "en" => "Spatial distribution of the radial component Br0(T)." ] ] 6 => array:7 [ "identificador" => "fig0035" "etiqueta" => "Figure 7" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr7.jpeg" "Alto" => 1128 "Ancho" => 1478 "Tamanyo" => 70837 ] ] "descripcion" => array:1 [ "en" => "irNic as function of Vrms." ] ] 7 => array:7 [ "identificador" => "tbl0005" "etiqueta" => "Table 1" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="" valign="top" scope="col" style="border-bottom: 2px solid black"> \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">Aluminum ring \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">Copper coil \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="left" valign="top" scope="col" style="border-bottom: 2px solid black">Ferromagnetic core \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">Relativity permittivity (¿r) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">Relativity permeability (μr) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">4000 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">Electric conductivity (σ) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">3.77 × 107 S/m \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">5.99 × 107 S/m \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1.12 × 107 S/m \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">Coil turn (N) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="" valign="top"> \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">1140 turns \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="" valign="top"> \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">Natural frequency (f) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="" valign="top"> \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="left" valign="top">60 Hz \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="" valign="top"> \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab852183.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Parameters used in the Thomson ring." ] ] 8 => array:6 [ "identificador" => "eq0005" "etiqueta" => "(1)" "tipo" => 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Modelación numérica del anillo de Thomson en levitación estacionaria usando circuitos eléctricos, MEF y Newton-Raphson

Guzmán Juana,

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maestro_juan_rafael@hotmail.com

Corresponding author.

, González-Montañez Felipe de Jesúsb, Escarela-Pérez Rafaelc, Olivares-Galván Juan Carlosd, Jiménez-Mondragon Victor Manuele

a Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco

b Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco

c Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco

d Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco

e Departamento de Energía Universidad Autónoma Metropolitana (UAM) Unidad Azcapotzalco

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    "titulo" => "Numerical Modeling of the Thomson Ring in Stationary Levitation Using FEM-Electrical Network and Newton-Raphson"
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        "titulo" => "Modelaci&#243;n num&#233;rica del anillo de Thomson en levitaci&#243;n estacionaria usando circuitos el&#233;ctricos&#44; MEF y Newton-Raphson"
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          "en" => "<p id="spar0040" class="elsevierStyleSimplePara elsevierViewall">Average Lorentz force <span class="elsevierStyleItalic">f<span class="elsevierStyleInf">zav</span></span> as a function of the distance <span class="elsevierStyleItalic">z<span class="elsevierStyleInf">s</span></span>&#46;</p>"
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    "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0025">Introduction</span><p id="par0005" class="elsevierStylePara elsevierViewall">In the electric industry is important to have electric systems of immediate breaking and safe&#46; Some of these electric systems consist of power switchers&#44; which use the Thomson ring &#40;Meyer and Rufer&#44; 2006Meyer &#38; Rufer&#59; <a class="elsevierStyleCrossRef" href="#bib0065">Meyer and Rufer&#44; 2006Meyer &#38; Rufer&#44; 2006</a>&#41;&#46; Other systems utilize the Thomson ring as actuator to eliminate the electric arcs &#40;<a class="elsevierStyleCrossRef" href="#bib0045">Li&#44; Jeong&#44; Yoon&#44; &#38; Koh&#44; 2010</a>&#41;&#46; Other applications of the Thomson ring consist in the levitation of superconductor materials &#40;<a class="elsevierStyleCrossRef" href="#bib0070">Patitsas&#44; 2011</a>&#41;&#46; Therefore&#44; it is important the numerical modeling of the Thomson ring&#46; The Thomson ring consists of a coil with ferromagnetic core on which an aluminum ring levitates&#46; The coil is fed by a cosine voltage&#46;</p><p id="par0010" class="elsevierStylePara elsevierViewall">The modeling of the electromagnetic field of any electric device &#40;as the Thomson ring&#41; requires of the knowledge of the current density&#46; However&#44; this knowledge cannot be known a priori&#46; It is known a priori the power source voltage instead of current density&#46; In the literature&#44; several methods &#40;<a class="elsevierStyleCrossRefs" href="#bib0020">Belforte <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1985&#59; Bissal <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2010&#59; Konrad&#44; 1982&#59; Lombard and Meunier&#44; 1992</a>&#44; <a class="elsevierStyleCrossRefs" href="#bib0055">1993&#59; Meunier <span class="elsevierStyleItalic">et al&#46;</span>&#44; 1988&#59; Piriou and Razek&#44; 1989</a>&#41; have been developed to calculate the electromagnetic field if the power source voltage is supplied&#58; integro-differential method &#40;<a class="elsevierStyleCrossRef" href="#bib0040">Konrad&#44; 1982</a>&#41;&#59; direct methods &#40;<a class="elsevierStyleCrossRef" href="#bib0020">Belforte <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1985</a>&#59; Meunier <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1988&#59; <a class="elsevierStyleCrossRef" href="#bib0075">Piriou and Razek&#44; 1989</a>&#41;&#59; and methods that use electric networks equations &#40;<a class="elsevierStyleCrossRefs" href="#bib0015">Barry and Casey&#44; 1999&#59; Bissal <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2010&#59; Lombard and Meunier&#44; 1992&#44; 1993</a>&#41;&#46; In this work is supposed that the power source voltage is known and the current density is calculated using electric networks equations&#46;</p><p id="par0015" class="elsevierStylePara elsevierViewall">Several studies have analyzed the mathematical and physics models of the Thomson ring&#46; In the work of <a class="elsevierStyleCrossRef" href="#bib0025">Bissal <span class="elsevierStyleItalic">et al</span>&#46; &#40;2010&#41;</a> is modeled the dynamic behavior of the Thomson ring&#44; which consist of a coil without ferromagnetic core&#46; In this work&#44; the coil is fed by a capacitor&#46; <a class="elsevierStyleCrossRef" href="#bib0015">Barry and Casey &#40;1999&#41;</a> obtained analytical solutions of the force acting on the aluminum ring in a stationary levitated position&#46; In the work of <a class="elsevierStyleCrossRef" href="#bib0045">Li <span class="elsevierStyleItalic">et al</span>&#46; &#40;2010&#41;</a> is analyzed the dynamic characteristics of the Thomson ring used as actuator to eliminate the electric arcs&#46; In the work of <a class="elsevierStyleCrossRef" href="#bib0070">Patitsas &#40;2011&#41;</a> is developed a new modality of Thomson ring&#46; This modality consisted in keeping the stable levitation of a superconductor sphere immerse in a magnetic field supplied by a coil&#46;</p><p id="par0020" class="elsevierStylePara elsevierViewall">The aim of this work is to analyze the Thomson ring when the aluminum ring is a stationary levitated position&#46; This situation is reached if the coil is fed by a cosine voltage&#46; In the stationary levitation&#44; the state of the electromagnetic field is stable and can be used the phasor equations of the electromagnetic field&#46; These equations are discretized using the Galerkin method&#46; These discretized equations are solved using the COMSOL software &#40;<a class="elsevierStyleCrossRef" href="#bib0030">COMSOL&#44; 2008</a>&#41;&#46; It is described the methodology &#40;which uses the Newton-Raphson method&#41; that obtains the separation between the coil and the aluminum ring in stationary levitation &#40;mechanical equilibrium&#41;&#46; Also&#44; the separation obtained with this methodology is compared with the experimental data for different values of the power source voltage&#46; It is concluded that the magnetic coupling of the aluminum ring on the coil can be neglected if the source voltage is high&#46;</p><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0030">Experimental setup</span><p id="par0025" class="elsevierStylePara elsevierViewall">The Thomson ring used in this work consists of a coil with ferromagnetic core&#59; where an aluminum ring is threaded on the core&#44; as shown in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>&#46; The ferromagnetic core consists in a solid cylinder that is collocated vertically&#44; as is illustrated in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>&#46; In this figure&#44; <span class="elsevierStyleItalic">Z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">S</span></span> represents the distance between the coil and the aluminum ring&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia><p id="par0030" class="elsevierStylePara elsevierViewall">The coil is made of copper wire and consists of 1140 turns &#40;see&#44; <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#41;&#46; This coil is fed by a cosine voltage given by<elsevierMultimedia ident="eq0005"></elsevierMultimedia></p><elsevierMultimedia ident="tbl0005"></elsevierMultimedia><p id="par0035" class="elsevierStylePara elsevierViewall">where&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf">0</span> is the amplitude&#59; <span class="elsevierStyleItalic">&#969;</span> &#61; 2<span class="elsevierStyleItalic">&#960;f</span> is the angular frequency &#40;<span class="elsevierStyleItalic">f</span> being the natural frequency&#41;&#44; as shown in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#46; The electric and magnetic characteristics of the materials used in the Thomson ring are indicated in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#46; The ferromagnetic core is iron whose relativity permeability is taken from the data base of <a class="elsevierStyleCrossRef" href="#bib0030">COMSOL &#40;2008&#41;</a>&#46;</p><p id="par0040" class="elsevierStylePara elsevierViewall">In order to take advantage of axial symmetry&#44; the Thomson ring is represented by means of axisymmetric geometry as depicted in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>&#46; In this Figure&#44; a cylinder coordinate system is chosen so that the <span class="elsevierStyleItalic">r</span>-axis represents the horizontal axis&#44; the <span class="elsevierStyleItalic">z</span>-axis represents the vertical axis&#46; The dimensions of the aluminum ring are&#58; interior radius of 0&#46;031 m&#44; exterior radius of 0&#46;0365 m and height of 0&#46;018 m as illustrated in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>&#46; In this figure&#44; <span class="elsevierStyleItalic">Z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">S</span></span> is the separation distance between aluminum ring and copper coil&#46;</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia><p id="par0045" class="elsevierStylePara elsevierViewall">The copper coil forms a toroid with dimensions&#58; interior radius of 0&#46;025 m&#44; exterior radius of 0&#46;039 m&#44; and height of 0&#46;075 m&#44; as shown in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>&#46; The distance between base of ferromagnetic core and base of copper coil is 0&#46;025 m&#46; The copper coil is threaded on a ferromagnetic cylinder &#40;ferromagnetic core&#41;&#46; The ferromagnetic core has a radius of 0&#46;0235 m and height of 0&#46;41 m&#44; as depicted in <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>&#46;</p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0035">Solution methodology</span><p id="par0050" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic"><span class="elsevierStyleBold">Electromagnetic field equations</span></span></p><p id="par0055" class="elsevierStylePara elsevierViewall">In this section the equations that describe the electromagnetic field in the Thomson ring are presented&#46; The magnetic field B&#8594;&#61;&#8711;&#215;A&#8594; &#40;A&#8594; being the magnetic vector potential&#41; satisfies the Ampere-Maxwell equation<elsevierMultimedia ident="eq0010"></elsevierMultimedia></p><p id="par0060" class="elsevierStylePara elsevierViewall">where&#44; <span class="elsevierStyleItalic">v</span> is the reluctivity&#44; J&#8594; is the current density&#59; and D&#8594;&#61;&#949;E&#8594; &#40;<span class="elsevierStyleItalic">¿</span> being the permittivity&#41; is the electric density&#46; The second term in the Eq&#46; &#40;2&#41; represents the displacement current&#44; which can be dropped if the frequency of the power source is small&#59; in this case&#44; the Eq&#46; &#40;2&#41; is given by<elsevierMultimedia ident="eq0015"></elsevierMultimedia></p><p id="par0065" class="elsevierStylePara elsevierViewall">The current density in this equation depends of the type of region &#40;cupper coil&#44; aluminum ring&#44; air or ferromagnetic core&#41; and is given by&#58;</p><p id="par0070" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">1&#41; Air and ferromagnetic core region</span></p><p id="par0075" class="elsevierStylePara elsevierViewall">The current density in air and ferromagnetic core regions is J&#8594;&#61;0&#8594;&#59; therefore</p><p id="par0080" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleInlineFigure"><elsevierMultimedia class="elsevierStyleLink" ident="fx4"></elsevierMultimedia></span></p><p id="par0085" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">2&#41; Aluminum ring region</span></p><p id="par0090" class="elsevierStylePara elsevierViewall">The current density J&#8594;&#61;&#963;E&#8594; &#40;being &#963;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">r</span></span> the electric conductivity of the aluminum ring&#41; is found using the Faraday law<elsevierMultimedia ident="eq0020"></elsevierMultimedia></p><p id="par0095" class="elsevierStylePara elsevierViewall">Solving this equation for the vector potential A&#8594;<elsevierMultimedia ident="eq0025"></elsevierMultimedia></p><p id="par0100" class="elsevierStylePara elsevierViewall">It is observed that this equation does not contain the term of the scalar electric potential gradient &#40;&#916;<span class="elsevierStyleItalic">V</span>&#41; due to that there is not a power source in the aluminum region&#46; Substituting Eq&#46; &#40;6&#41; in J&#8594;&#61;&#963;rE&#8594;<elsevierMultimedia ident="eq0030"></elsevierMultimedia></p><p id="par0105" class="elsevierStylePara elsevierViewall">Substituting Eq&#46; &#40;6&#41; in Eq&#46; &#40;3&#41;<elsevierMultimedia ident="eq0035"></elsevierMultimedia></p><p id="par0110" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">3&#41; Copper coil region</span></p><p id="par0115" class="elsevierStylePara elsevierViewall">The region of the copper coil is modeled as a region that contains <span class="elsevierStyleItalic">N</span> turns where each turn carries the same current <span class="elsevierStyleItalic">i</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">C</span></span>&#46; In this case&#44; the current density <span class="elsevierStyleItalic">J</span> is uniform with value<elsevierMultimedia ident="eq0040"></elsevierMultimedia></p><p id="par0120" class="elsevierStylePara elsevierViewall">where&#44; <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">C</span></span> is the cross section area of the copper coil region&#46; Substituting Eq&#46; &#40;9&#41; in Eq&#46; &#40;3&#41;<elsevierMultimedia ident="eq0045"></elsevierMultimedia></p><p id="par0125" class="elsevierStylePara elsevierViewall">where&#44; I&#710; is a unit vector pointed in direction of the current density&#46;</p><p id="par0130" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic"><span class="elsevierStyleBold">Electrical network equations</span></span></p><p id="par0135" class="elsevierStylePara elsevierViewall">If the current is known&#44; the solution of the Eq&#46; &#40;10&#41; can be realized&#46; However&#44; this current cannot be known a priori&#46; We know a priori the voltage <span class="elsevierStyleItalic">V</span> between the terminals of the coil&#46; An additional equation is required&#46; This equation is obtained using the Kirchhoff voltage law<elsevierMultimedia ident="eq0050"></elsevierMultimedia></p><p id="par0140" class="elsevierStylePara elsevierViewall">where&#44; <span class="elsevierStyleItalic">R</span> is the resistance&#44; <span class="elsevierStyleItalic">&#981;</span> is the magnetic flux that cross all the turns of the coil&#46; The resistance is given by<elsevierMultimedia ident="eq0055"></elsevierMultimedia></p><p id="par0145" class="elsevierStylePara elsevierViewall">where&#44; <span class="elsevierStyleItalic">&#963;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">C</span></span> is the electric conductivity of the coil and <span class="elsevierStyleItalic">L</span> is the length of all the turns of the coil&#46; The magnetic flux is given by<elsevierMultimedia ident="eq0060"></elsevierMultimedia></p><p id="par0150" class="elsevierStylePara elsevierViewall">where&#44; the surface <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">C</span></span> comprises all the surfaces of the turns of the coil&#46; Using the fact B&#8594;&#61;&#8711;&#215;A&#8594; and the Stokes theorem in Eq&#46; &#40;13&#41; we obtain<elsevierMultimedia ident="eq0065"></elsevierMultimedia></p><p id="par0155" class="elsevierStylePara elsevierViewall">where&#44; the trajectory <span class="elsevierStyleItalic">C</span> comprises all the turns of the coil&#46; Substituting Eq&#46; &#40;14&#41; in Eq&#46; &#40;11&#41;<elsevierMultimedia ident="eq0070"></elsevierMultimedia></p></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0040">Phasor equations</span><p id="par0160" class="elsevierStylePara elsevierViewall">The current in the copper coil is cosine to ensure that the aluminum ring stays in a stationary levitated position&#46; In this situation&#44; the state of the electromagnetic field is stable and the equations of the electromagnetic field can be given in phasor form&#46; In phasor notation&#44; the operator <span class="elsevierStyleItalic">d</span>&#47;<span class="elsevierStyleItalic">dt</span> becomes <span class="elsevierStyleItalic">i&#969;</span> in Eqs&#46; &#40;4&#41;&#44; &#40;8&#41; and &#40;10&#41;&#58;</p><p id="par0165" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">air and ferromagnetic core regions</span><elsevierMultimedia ident="eq0075"></elsevierMultimedia></p><p id="par0170" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">aluminum ring region</span><elsevierMultimedia ident="eq0080"></elsevierMultimedia></p><p id="par0175" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">copper coil region</span><elsevierMultimedia ident="eq0085"></elsevierMultimedia></p><p id="par0180" class="elsevierStylePara elsevierViewall">where&#44; A&#8594; and <span class="elsevierStyleItalic"><span class="elsevierStyleBold">i</span></span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">C</span></span> are phasors of the potential A&#8594; and the current <span class="elsevierStyleItalic"><span class="elsevierStyleBold">i</span></span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">C</span></span>&#44; respectively&#46;</p><p id="par0185" class="elsevierStylePara elsevierViewall">The phasor equation of the electrical network equation Eq&#46; &#40;15&#41; is<elsevierMultimedia ident="eq0090"></elsevierMultimedia></p><p id="par0190" class="elsevierStylePara elsevierViewall">where&#44; <span class="elsevierStyleItalic"><span class="elsevierStyleBold">V</span></span> is the phasor of the voltage <span class="elsevierStyleItalic">V</span>&#46;</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">Boundary conditions</span><p id="par0195" class="elsevierStylePara elsevierViewall">It is observed that Eq&#46; &#40;3&#41; is a second order partial differential equation for the magnetic vector potential A&#8594;&#46; The solution of this partial differential equation requires boundary conditions for the vector potential A&#8594;&#46; The boundary of the solution domain is chosen so that the vector potential can be dropped &#40;magnetic insulation&#41;&#46; The magnetic insulation condition is expressed as<elsevierMultimedia ident="eq0095"></elsevierMultimedia></p><p id="par0200" class="elsevierStylePara elsevierViewall">where&#44; &#915; is the boundary of solution domain&#46; In phasor notation&#44; the condition of magnetic insulation is<elsevierMultimedia ident="eq0100"></elsevierMultimedia></p></span><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0050">Discretization</span><p id="par0205" class="elsevierStylePara elsevierViewall">Using the Galerkin method &#40;Lombard and Meunier&#44; 1992Lombard &#38; Meunier&#59; Lombard and Meunier&#44; 1992Lombard and Meunier&#44; 1993Lombard &#38; Meunier&#59; <a class="elsevierStyleCrossRef" href="#bib0055">Lombard and Meunier&#44; 1993Lombard &#38; Meunier&#44; 1993</a>&#41;&#44; Eqs&#46; &#40;16&#41;-&#40;19&#41; can be discretized&#58;</p><p id="par0210" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">air and ferromagnetic core regions</span><elsevierMultimedia ident="eq0105"></elsevierMultimedia></p><p id="par0215" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">aluminum ring region</span><elsevierMultimedia ident="eq0110"></elsevierMultimedia></p><p id="par0220" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">copper coil region</span><elsevierMultimedia ident="eq0115"></elsevierMultimedia></p><p id="par0225" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">electrical network equation</span><elsevierMultimedia ident="eq0120"></elsevierMultimedia></p><p id="par0230" class="elsevierStylePara elsevierViewall">With<elsevierMultimedia ident="eq0125"></elsevierMultimedia></p><p id="par0235" class="elsevierStylePara elsevierViewall">where <span class="elsevierStyleItalic">N</span> represents the number of nodes&#46; The matrices and vectors are defined as<elsevierMultimedia ident="eq0130"></elsevierMultimedia><elsevierMultimedia ident="eq0135"></elsevierMultimedia><elsevierMultimedia ident="eq0140"></elsevierMultimedia><elsevierMultimedia ident="eq0145"></elsevierMultimedia></p><p id="par0240" class="elsevierStylePara elsevierViewall">where&#44; the vector potential <span class="elsevierStyleItalic">A</span> is expanded in the base function <span class="elsevierStyleItalic">&#946;</span><span class="elsevierStyleInf">i</span>&#58; <span class="elsevierStyleItalic">A</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#8721;<span class="elsevierStyleItalic">&#946;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span><span class="elsevierStyleItalic">A</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span>&#46; The surface <span class="elsevierStyleItalic">S</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">d</span></span> is the surface of the solution domain&#46;</p></span><span id="sec0035" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0055">Mechanical equilibrium</span><p id="par0245" class="elsevierStylePara elsevierViewall">The voltage is a cosine in order to maintain the aluminum ring in a stationary levitated position&#46; This stationary levitation is obtained when the mechanical equilibrium is reached&#59; this is&#44; the Lorentz force averaged in a cycle&#44; <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">zav</span></span> equals the gravity force <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span>&#46; Using the complex notation&#44; the Lorentz force <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">zav</span></span> &#40;Barry and Casey&#44; 1999Barry &#38; Casey&#59; Barry and Casey&#44; 1999Hayt and Buck&#44; 2006Hayt &#38; Buck&#59; <a class="elsevierStyleCrossRef" href="#bib0035">Hayt and Buck&#44; 2006Hayt &#38; Buck&#44; 2006</a>&#41; is given by<elsevierMultimedia ident="eq0150"></elsevierMultimedia></p><p id="par0250" class="elsevierStylePara elsevierViewall">with<elsevierMultimedia ident="eq0155"></elsevierMultimedia></p><p id="par0255" class="elsevierStylePara elsevierViewall">where&#44; B&#8594; and J&#8594; are the phasors of magnetic density and current density&#44; J&#8594; respectively&#46; The factor 1&#47;2 in Eq&#46; &#40;31&#41; is due to that the Lorentz force period is half of the magnetic field period &#40;Barry and Casey&#44; 1999Barry &#38; Casey&#59; <a class="elsevierStyleCrossRef" href="#bib0015">Barry and Casey&#44; 1999Barry &#38; Casey&#44; 1999</a>&#41;&#46; <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a> shows the flowchart of the obtaining of the average Lorentz force&#46; The steps of this methodology are&#58;<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005"><span class="elsevierStyleLabel">1&#41;</span><p id="par0260" class="elsevierStylePara elsevierViewall">Calculate the phasor potential using the phasor equations &#40;Eqs&#46; 22-25&#41; along with boundary condition of magnetic insulation A&#8594;&#8901;n&#710;&#61;0 on &#915; &#40;see Eq&#46; 20&#41;&#46;</p></li><li class="elsevierStyleListItem" id="lsti0010"><span class="elsevierStyleLabel">2&#41;</span><p id="par0265" class="elsevierStylePara elsevierViewall">Determine the phasor magnetic density B&#8594;&#61;&#8711;&#215;A&#8594; and phasor current density J&#8594;&#61;i&#969;&#963;rA&#8594; &#40;see Eq&#46; 6&#41; in the aluminum ring region&#46;</p></li><li class="elsevierStyleListItem" id="lsti0015"><span class="elsevierStyleLabel">3&#41;</span><p id="par0270" class="elsevierStylePara elsevierViewall">Calculate the average Lorentz force fzav&#61;12&#8747;VrJ&#8594;&#215;B&#8594;&#42;dV &#40;see Eq&#46; 31&#41;&#46;</p></li></ul></p><elsevierMultimedia ident="fig0015"></elsevierMultimedia><p id="par0275" class="elsevierStylePara elsevierViewall">The space distribution of the electromagnetic field depends of the separation <span class="elsevierStyleItalic">s</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">z</span></span> between the aluminum ring and the copper coil&#46; Therefore&#44; the average Lorentz force <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">zav</span></span> is a function of the separation &#40;<span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">z</span></span> &#61; <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">z</span></span>&#40;<span class="elsevierStyleItalic">s</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">z</span></span>&#41;&#41;&#46; In order to reach the stationary levitation of the aluminum ring&#44; the average Lorentz force <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">zav</span></span> equals to the gravity force <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span>&#46;<elsevierMultimedia ident="eq0160"></elsevierMultimedia></p><p id="par0280" class="elsevierStylePara elsevierViewall">where&#44; <span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span>&#8217; is the separation in stationary levitation and represents the root of Eq&#46; &#40;33&#41;&#46; It is observed that Eq&#46; &#40;33&#41; is a transcendental equation&#46; The root of this transcendental equation can be found using a variant of the Newton-Raphson method&#58; secant method &#40;Arfken and Weber&#44; 2005Arfken &#38; Weber&#59; <a class="elsevierStyleCrossRef" href="#bib0010">Arfken and Weber&#44; 2005Arfken &#38; Weber&#44; 2005</a>&#41;&#46; The convergence of Newton-Raphson is guaranteed due to that the average Lorentz force <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">zav</span></span>&#40;<span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span>&#41; is a function decreasing of the separation <span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span> &#40;see <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>&#41;&#46; The secant method is defined by the recurrence relation<elsevierMultimedia ident="eq0165"></elsevierMultimedia></p><elsevierMultimedia ident="fig0025"></elsevierMultimedia></span></span><span id="sec0040" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0060">Experimental validation</span><p id="par0285" class="elsevierStylePara elsevierViewall">In this section we compared the numerical and experimental results for the separation in stationary levitation <span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span>&#8217; as a function of the voltage amplitude in rms&#44; Vrms&#61;V02&#46; The experimental setup was described in the second section&#46; The numerical results are obtained using the proposed methodology in the section above&#46; <a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a> shows the separation <span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span>&#8217; as function of the voltage amplitude <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">rms</span></span> for both experimental and numerical results&#46; The discrepancy between the theoretical and experimental data is at most 12&#37;&#46; This difference can be due to the fact that the numerical modeling does not take into account the temperature effect in the electric conductivity <span class="elsevierStyleItalic">&#963;</span>&#46;</p><elsevierMultimedia ident="fig0020"></elsevierMultimedia></span><span id="sec0045" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0065">Results and discussion</span><p id="par0290" class="elsevierStylePara elsevierViewall">In this section some results obtained by the proposed modeling are studied&#46; The average Lorentz force is examined as a function of the separation distance&#59; the ratio between coil current and ring current&#44; and the spatial distribution of the magnetic field&#46;</p><p id="par0295" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a> depicts the average Lorentz force <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">zav</span></span> as function of the separation <span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span> for a representative voltage amplitude <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">rms</span></span> &#61; 120<span class="elsevierStyleHsp" style=""></span>V&#46; It is observed that the Lorentz force is a decreasing function of the distance <span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span>&#46; This guarantees the convergence of the Newton-Raphson method due to that the derivative dfzavdzS is negative&#46;</p><p id="par0300" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a> shows the spatial distribution of the radial component <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">r0</span></span> of the magnetic density amplitude&#44; for a representative voltage amplitude <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">rms</span></span> &#61; 120<span class="elsevierStyleHsp" style=""></span>V in state of stationary levitation &#40;<span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span>&#8217; &#61; 0&#46;057 m&#41;&#46; It also presents the positions of the ferromagnetic core&#44; copper coil and aluminum ring&#46; This <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a> shows that the radial component is higher in regions close to the core&#44; coil and ring edges&#46; In contrast&#44; the radial component <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">r0</span></span> presents small values in positions far away from above edges&#46;</p><elsevierMultimedia ident="fig0030"></elsevierMultimedia><p id="par0305" class="elsevierStylePara elsevierViewall">The total current in the ring <span class="elsevierStyleItalic">i</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">r</span></span> is realized by means of ir&#61;&#8747;ringJ&#8594;&#8901;dS&#8594;&#59; while the total current in the region of the coil is <span class="elsevierStyleItalic">Ni</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span></span>&#46; In <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a> is shown the ratio irNic as function of voltage amplitude <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">rms</span></span> in stationary levitation&#46; It is observed that the highest value &#40;<span class="elsevierStyleItalic">i</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">r</span></span> &#47; <span class="elsevierStyleItalic">Ni</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">C</span></span> &#61; 0&#46;47&#41; occurs in <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">rms</span></span> &#61; 43&#46;4<span class="elsevierStyleHsp" style=""></span>V corresponding to a separation <span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span></span>&#8217;&#61; 0&#46; The ratio <span class="elsevierStyleItalic">i</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">r</span></span> &#47; <span class="elsevierStyleItalic">Ni</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">C</span></span> decreases if the voltage amplitude <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">rms</span></span> increases&#46; Also&#44; in a first order approach&#44; the magnetic field originated by any system is proportional to the current of this system&#46; Therefore&#44; the magnetic coupling of the ring on the coil can be neglected for high values of voltage amplitude&#46;</p><elsevierMultimedia ident="fig0035"></elsevierMultimedia></span><span id="sec0050" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0070">Conclusions</span><p id="par0310" class="elsevierStylePara elsevierViewall">The aim of this work was to present a numerical modeling based upon the use of the Galerkin method to simulate the electromagnetic field of the Thomson ring&#46; Also&#44; this modeling is capable of simulating numerically the separation between aluminum ring and copper coil in situation of stationary levitation &#40;the average Lorentz force equals gravity force&#41;&#46; This calculation of the separation uses the Newton-Raphson method&#46;</p><p id="par0315" class="elsevierStylePara elsevierViewall">The proposed modeling was validated comparing theoretical and experimental results&#46; The compared results were the separation between the aluminum ring and the copper coil &#40;in stationary levitation&#41; for different voltage amplitudes&#46;</p><p id="par0320" class="elsevierStylePara elsevierViewall">It is concluded that the magnetic coupling of the aluminum ring on the coil can be neglected if the source voltage is high&#46; Therefore&#44; the coil current can be modeled without taking into account the coupling ring-coil&#46; This means that the coil current is found using a RL &#40;resistance-inductance&#41; circuit&#59; where&#44; the resistance and inductance are parameter of the coil&#46;</p><p id="par0325" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Citation for this article&#58;</span></p><p id="par0330" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Chicago citation style</span></p><p id="par0335" class="elsevierStylePara elsevierViewall">Guzm&#225;n&#44; Juan&#44; Felipe de Jes&#250;s Gonz&#225;lez-Monta&#241;ez&#44; Rafael Escarela-P&#233;rez&#44; Juan Carlos Olivares-Galv&#225;n&#44; Victor Manuel Jim&#233;nez-Mondragon&#46; Numerical modeling of the Thomson ring in stationary levitation using FEM-electrical network and Newton-Raphson&#46; <span class="elsevierStyleItalic"><span class="elsevierStyleBold">Ingenier&#237;a Investigaci&#243;n y Tecnolog&#237;a</span></span>&#44; XVI&#44; 03 &#40;2015&#41;&#58; 431-439&#46;</p><p id="par0340" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">ISO 690 citation style</span></p><p id="par0345" class="elsevierStylePara elsevierViewall">Guzm&#225;n J&#46;&#44; Gonz&#225;lez-Montanez F&#46;J&#46;&#44; Escarela-P&#233;rez R&#46;&#44; Olivares-Galv&#225;n J&#46;C&#46;&#44; Jim&#233;nez-Mondragon V&#46;M&#46; Numerical modeling of the Thomson ring in stationary levitation using FEM-electrical network and Newton-Raphson&#46; <span class="elsevierStyleItalic"><span class="elsevierStyleBold">Ingenier&#237;a Investigaci&#243;n y Tecnolog&#237;a</span></span>&#44; volume XVI &#40;issue 3&#41;&#44; july 2015&#58; 431-439&#46;</p></span></span>"
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          "titulo" => "Introduction"
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              "titulo" => "Phasor equations"
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            2 => "stationary"
            3 => "modeling"
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        "biografia" => "<p id="spar0015" class="elsevierStyleSimplePara elsevierViewall"><span class="elsevierStyleItalic">Juan Guzm&#225;n&#46;</span> Obtained Ph&#46;D&#46; in Energy Engineering from the Universidad Nacional Aut&#243;noma de M&#233;xico&#44; M&#233;xico City&#44; Mexico&#44; in 2008&#46; He is currently with the &#225;rea de ingenier&#237;a energ&#233;tica y electromagn&#233;tica&#44; Departamento de Energ&#237;a&#44; UAM&#44; Azcapotzalco&#44; M&#233;xico&#46;</p>"
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        "biografia" => "<p id="spar0060" class="elsevierStyleSimplePara elsevierViewall"><span class="elsevierStyleItalic">Felipe de Jes&#250;s Gonz&#225;lez-Monta&#241;ez&#46;</span> He received the M&#46;Sc&#46; degree in electrical engineering from the Centro de Investigaci&#243;n y de Estudios Avanzados del IPN&#44; M&#233;xico City&#44; Mexico&#44; in 2011&#46; His research interests include the modeling and control of electrical machines&#46;</p>"
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        "biografia" => "<p id="spar0065" class="elsevierStyleSimplePara elsevierViewall"><span class="elsevierStyleItalic">Rafael Escarela-P&#233;rez&#46;</span> He obtained his B&#46;Sc&#46; in electrical engineering from Universidad Autonoma Metropolitana&#44; Mexico City in 1992 and his Ph&#46;D&#46; from Imperial College&#44; London in 1996&#46; He is interested in the modeling of electrical machines&#46;</p>"
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        "biografia" => "<p id="spar0070" class="elsevierStyleSimplePara elsevierViewall"><span class="elsevierStyleItalic">Juan Carlos Olivares-Galv&#225;n&#46;</span> He received the Ph&#46;D&#46; degree in electrical engineering from CINVESTAV&#44; Guadalajara&#44; Mexico&#44; in 2003&#46; His main research interests are related to the experimental and numerical analysis of electromagnetic devices&#46;</p>"
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        "biografia" => "<p id="spar0075" class="elsevierStyleSimplePara elsevierViewall"><span class="elsevierStyleItalic">Victor Manuel Jim&#233;nez-Mondragon&#46;</span> He received the M&#46;Sc&#46; degree in electrical engineering from the Universidad Nacional Aut&#243;noma de M&#233;xico&#44; M&#233;xico City&#44; Mexico&#44; in 2012&#46; He is interested in the modeling of electrical machines&#46;</p>"
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        "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">There are a lot of applications of the Thomson ring&#58; levitation of superconductor materials&#44; power interrupters &#40;used as actuator&#41; and elimination of electric arcs&#46; Therefore&#44; it is important the numerical modeling of Thomson ring&#46; The aim of this work is to model the stationary levitation of the Thomson ring&#46; This Thomson ring consists of a copper coil with ferromagnetic core and an aluminum ring threaded in the core&#46; The coil is fed by a cosine voltage to ensure that the aluminum ring is in a stationary levitated position&#46; In this situation&#44; the state of the electromagnetic field is stable and can be used the phasor equations of the electromagnetic field&#46; These equations are discretized using the Galerkin method in the Lagrange base space &#40;<span class="elsevierStyleItalic">finite element method</span>&#44; FEM&#41;&#46; These equations are solved using the COMSOL software&#46; A methodology is also described &#40;which uses the Newton-Raphson method&#41; that obtains the separation between coil and aluminum ring&#46; The numerical solutions of this separation are compared with experimental data&#46; The conclusion is that the magnetic coupling of the aluminum ring on the coil can be neglected if the source voltage is high&#46;</p></span>"
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        "resumen" => "<span id="abst0010" class="elsevierStyleSection elsevierViewall"><p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">Existen una gran cantidad de aplicaciones del anillo de Thomson&#58; levitaci&#243;n de materiales superconductores&#44; interruptores de potencia &#40;usados como actuadores&#41; y eliminaci&#243;n de arcos el&#233;ctricos&#46; Por lo tanto&#44; es importante la modelaci&#243;n del anillo de Thomson&#46; El objetivo de este trabajo es modelar la levitaci&#243;n estacionaria del anillo de Thomson&#46; Este anillo de Thomson consiste de una bobina de cobre con n&#250;cleo ferromagn&#233;tico y un anillo de aluminio enhebrado en el n&#250;cleo&#46; La bobina se alimenta por un voltaje cosenoidal para asegura el anillo de aluminio en una posici&#243;n de levitaci&#243;n estacionaria&#46; En esta situaci&#243;n&#44; el campo electromagn&#233;tico se puede considerar estable y se pueden emplear las ecuaciones fasoriales del campo electromagn&#233;tico&#46; Estas ecuaciones se discretizan usando el m&#233;todo de Galerkin en el espacio base de Lagrange &#40;m&#233;todo de elementos finitos&#44; FEM&#41;&#46; Estas ecuaciones discretizadas se resuelven usando el c&#243;digo COMSOL&#46; Adem&#225;s&#44; se describe una metodolog&#237;a con la cual se puede obtener la separaci&#243;n entre la bobina y el anillo de aluminio&#46; Esta metodolog&#237;a usa el m&#233;todo de Newton-Rapson&#46; Las soluciones num&#233;ricas de esta separaci&#243;n se comparan con datos experimentales&#46; Se concluye que el acoplamiento magn&#233;tico entre el anillo de aluminio sobre la bobina se puede despreciar si el voltaje de alimentaci&#243;n es alto&#46;</p></span>"
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Article information

ISSN: 14057743
Original language: English

DOI:10.1016/j.riit.2015.05.005

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2021 May	33	13	46
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2020 December	42	7	49
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