Analytical Analysis of Magnetic Levitation Systems with Harmonic Voltage Input

actuators Article Analytical Analysis of Magnetic Levitation Systems with Harmonic Voltage Input Serguei Maximov 1,† , Felipe Gonzalez-Montañez 2,† , Rafael Escarela-Perez 2,*,† , Juan Carlos Olivares-Galvan 2,† and Hector Ascencion-Mestiza 1,† 1 Posgrado en Eléctrica, Tecnológico Nacional de México, Instituto Tecnológico de Morelia, Morelia C.P. 58120, Mexico; [email protected] (S.M.); [email protected] (H.A.-M.) 2 Departamento de Enérgia, Universidad Autónoma Metropolitana Azcapotzalco, Ciudad de México C.P. 02200, Mexico; [email protected] (F.G.-M.); [email protected] (J.C.O.-G.) * Correspondence: [email protected] † These authors contributed equally to this work. Received: 30 July 2020; Accepted: 9 September 2020; Published: 11 September 2020 Abstract: In this paper, a new analytical method using Lagrange equations for the analysis of magnetic levitation (MagLev) systems is proposed, using Thomson’s jumping ring experiment. The method establishes the dependence of the primary and induced currents, and also the equilibrium height of the levitating object on the input voltage through the mutual inductance of the system. The mutual inductance is calculated in two ways: (i) by employing analytical formula; (ii) through an improved semi-empirical formula based on both measurements and analytical results. The obtained MagLev model was analyzed both analytically and numerically. Analytical solutions to the resulting equations were found for the case of a dynamic equilibrium. The numerical results obtained for the dynamical model under transient operation show a close correspondence with the experimental results. The good precision of the analytical and numerical results demonstrates that the developed method can be effectively implemented. Keywords: copper coil; aluminum ring; ferromagnetic material; magnetic levitation; Thomson’s jumping ring; mutual inductance; analytical methods 1. Introduction The principles of magnetic levitation (MagLev) are used in many important applications, such as actuators, power interrupters for eliminating electric arcs, electromagnetic shock absorbers in electrical vehicles, hybrid suspension systems with active control, levitation of superconductor materials, sensors and electromagnetic mass drivers [1]. One of the perspective usages of the MagLev systems in the near future is related to electromagnetic suspension systems and damping shock absorbers. Automobiles and trucks have shock absorbers with similar geometry to that of Thomson’s ring system. These systems are used to damper vibrations generated by imperfections of road surfaces [2–5]. Thus, it is an important challenge to be able mathematically and numerically model, analyze, control and design MagLev devices similar to Thomson’s ring system. There are also highly accurate density measurements and density-based detection methods applied to MagLev systems with similar geometries. In [6–8], an experimental study of a single-ring MagLev system was presented, opening up a wide operational space and enabling object manipulation and density-based measurements. In the literature, there are works focused on the numerical analysis of MagLev systems similar to that of Thomson’s ring. Vilchis et al. [9] presented factors that affect 2D finite element method (FEM) transient simulation accuracy for an axisymmetric model of a Thomson-coil actuator. Yannan Zhou et al. [10] numerically studied a Thomson-coil actuator applied in a circuit breaker in a Actuators 2020, 9, 82; doi:10.3390/act9030082 www.mdpi.com/journal/actuators Actuators 2020, 9, 82 2 of 13 high-voltage direct current system. Tezuka et al. [11] proposed a novel magnetically-levitating motor, able to control five degrees of freedom. In all these systems, analytical or hybrid solutions can be implemented, resulting in accurate and low-demanding computational models. Dynamic models (DM) of the MagLev systems have been previously developed. TJ-Teo et al. [12] presented a new electromagnetic actuator which has two degrees of freedom (linear and rotary motions). Consequently, the developed MagLev system presents two mutual inductances as functions of linear and rotatory positions. Additionally, Nagai et al. [13] realized a compact actuator system by a sensor-free approach. They presented the circuit equations for the solenoid, where the mutual inductance was approximated as a second-order polynomial function. Here, we present a new and rigorous analytical solution to the dynamic equations of the MagLev systems, wherein a mutual inductance, which is the key parameter of electromagnetic force, is calculated analytically. This way, we obtain a method that retains the simplicity of lumped-parameter models plus the accuracy of analytical methods within a field modeling context. Our approach was validated using experimental test results obtained from a Thomson’s jumping ring experiment. 2. Model In this work, a system known as the Thomson’s jumping ring is studied, where an aluminum ring is used as a levitating object. The experimental setup is shown in Figure1. The system is composed of a primary copper coil of 1140 turns, and an aluminum ring that moves freely up and down along a laminated ferromagnetic core. /ƌŽŶŽƌĞ Ϭ͘ϬϬϱϯŵ Ϭ͘Ϭϭϴŵ Ϭ͘Ϭϱŵ Ϭ͘ϬϲϮŵ ^ŽƵƌĐĞ sŽůƚĂŐĞ Ϭ͘ϰϭŵ ƵƚŽƚƌĂŶƐĨŽƌŵĞƌ dƌĂŶƐĨŽƌŵĞƌ Ϭ͘Ϭϳϴŵ Figure 1. Experimental setup. The equivalent circuit depicted in Figure2 models this MagLev system. R l l R 1 1 M( z ) 2 2 i i 1 2 V(t) z z Figure 2. Equivalent circuit of the MagLev. G C G C Actuators 2020, 9, 82 3 of 13 The governing differential equations that govern the dynamics of the circuit represented in Figure2 can be obtained from the Lagrangian 2 2 2 l1i1 l2i2 mz˙ L = + + Mi1i2 + − mgz , (1) 2 2 2 |{z} | {z } potential energy kinetic energy where z is the ring height, and i1 and i2 are the electric currents in the primary coil and aluminum ring, respectively. The parameters l1 and l2 are the primary and secondary circuits’ inductances; M is the mutual inductance between the coil and ring; and m and g are the aluminum ring mass and gravitational constant, respectively. Notice that the mutual inductance M(z) is a function of the distance z between the coil and ring [14]. Thus, the Lagrange equations are as follows: d ¶L ¶L − = V(t) − R1i1, (2) dt ¶i1 ¶q1 d ¶L ¶L − = −R2i2, (3) dt ¶i2 ¶q2 d ¶L ¶L − = −bz˙, (4) dt ¶z˙ ¶z where R1 and R2 are the coil and ring resistances; q1 and q2 are the coil and ring electric charges, respectively; and b is the friction coefficient. The right-hand side terms of the Lagrange equations are generalized forces, which are considered as independent terms, associated with the generalized coordinate. Substitution of the Lagrangian (1) into Lagrange equations results in the following system of differential equations (see also [15,16]): di d(M(z)i ) l 1 = V(t) − R i − 2 (5) 1 dt 1 1 dt di d(M(z)i ) l 2 = −R i − 1 , (6) 2 dt 2 2 dt dM(z) mz¨ = i i − bz˙ − mg. (7) 1 2 dz 3. Formula for the Mutual Inductance Mutual inductance between a coil and aluminum ring is a function of the system geometry and the magnetic medium properties between two coils. It can be calculated in two ways: numerically (using FEM) or analytically. In this section, the linear mutual inductance is analytically calculated based on the results of [17], where an infinite isotropic core is considered. According to [17], the mutual inductance can be divided into two parts: M = Mair + Mcore, (8) where Mair is the mutual inductance in air, i.e., in the absence of the core, and Mcore is the contribution of the ferromagnetic core to the whole inductance. The first part of the mutual inductance can be represented in the form (see [17]): a+h b+h 0 z+w p Z a Z b Z Z b 2 Na 2 AB f Mair = m0 dA dB dza dzb × 1 − K( f ) − E( f ) , (9) wahawbhb f 2 a b −wa z Actuators 2020, 9, 82 4 of 13 where s 4r r = a b f 2 2 (10) (zb − za) + (ra + rb) is a dimensionless parameter; K( f ) and E( f ) are complete elliptic integrals of the first and second kind, respectively [18]. Here, (ra, za) and (rb, zb) are pairs of cylindrical coordinates inside the coil and ring (subindices a and b correspond to the coil and aluminum ring, respectively). Namely, ra and rb are radii-vectors and za and zb positions on the core. Integration with respect to these variables signifies that the whole volume of the coil and ring is involved in the calculation of the mutual inductance between them. Parameters wa and wb are the coil and ring widths; ha and hb are their heights; and a and b are their internal radii (see Figure3). Na is the number of turns in the coil. The mutual inductance caused by the core is as follows: ¥ N Z dk = a Mcore 2m0 2 wahawbhb k 0 n o × cos (k(z + wb)) + cos (k(z + wa)) − cos(kz) − cos (k(z + wa + wb)) p n h i × (a + h ) K (k(a + h )) L (k(a + h )) + K (k(a + h )) L (k(a + h )) 2k a 1 a 0 a 0 a 1 a h io p n h − a K (ka)L (ka) + K (ka)L (ka) (b + h ) K (k(b + h )) L (k(b + h )) 1 0 0 1 2k b 1 b 0 b i h io + K0 (k(b + hb)) L1 (k(b + hb)) −b K1(kb)L0(kb) + K0(kb)L1(kb) m kI (bR)I (kR) − bI (bR)I (kR) × r 1 0 0 1 , (11) mrkI1(bR)K0(kR) − bI0(bR)K1(kR) p 2 where b = k + jwms, m = m0mr is the absolute permeability of the core material, mr is its relative permeability and s is core conductance. �! �" ℎ! � � Core ℎ � " Coil Aluminium Ring Figure 3.

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