Electromagnetic, flow and thermal study of a miniature planar spiral transformer with planar, spiral windings

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1 Electromagnetic, flow and thermal study of a miniature planar spiral transformer with planar, spiral windings J. B. DUMITRU 1, A. M. MOREGA*,1,2, M. MOREGA 1 *Corresponding author 1 POLITEHNICA University of Bucharest, Faculty of Electrical Engineering 313 Splaiul Independentei, , Bucharest, Romania dumitrujeanbogdan@yahoo.com, mihaela@iem.pub.ro 2 Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy Calea 13 Septembrie no. 13, Bucharest, Romania amm@iem.pub.ro* DOI: / S1.7 Abstract: This paper presents mathematical modeling and numerical simulation results for a miniature, planar, spiral transformer (MPST) fabricated in micro-electromechanical MEMS technology. When the MPST is magnetic nanofluid cored, magnetization body forces occur, entraining it into a complex flow. This particular MPST design is then compared with other competing solutions concerning the lumped (circuit) parameters. Finally, the heat transfer problem is solved for different electromagnetic working conditions to assess the thermal loads inside the MPST. Key Words: power transformer, fluid core, magnetic nanofluid, flow, magnetic field, lumped parameters, heat transfer, numerical simulation, finite element. 1. INTRODUCTION Recent advances in the development of micro-power microcontrollers and RF transmitters have led to a growing interest in new wireless devices that use energy harvesting sources (EHS) as an alternative to batteries aimed to scavenge small amounts of energy from artificial light, vibrations, temperature gradients, etc. and to convert it to useful electrical energy [1-5]. A key component of an EHS is the fly-back transformer (called also coupled inductors ), which has to meet certain specifications: small size, low profile, thermal stability, high efficiency, and low cost. A Miniature Planar Spiral Transformer (MPST) with circular windings developed in micro-electromechanical systems (MEMS) may be an alternative to coupled inductors [6]. Usually, the magnetic core of an MPST is made of ferrite. However, recent studies showed that magnetic nanofluids consisting of tiny magnetized iron oxide nanoparticles (magnetite) dispersed in an oil suspension are a sound candidate [6-9] to replace the ferrite. Such magnetic nanofluids, which are becoming a common solution in power transformers as cooling and insulating medium [7-10], also overcome the problem of iron losses in the transformer, especially at high frequencies, due to their near-zero hysteresis, thus enhancing the overall performance of the device., pp ISSN

2 J. B. DUMITRU, A. M. MOREGA, M. MOREGA AN MPST POWER TRANSFORMER An MPST power transformer consists of two circular copper coils built on a ceramic substrate (Al 2 O 3 ). The MEMS technology permits the growth of copper windings with crosssection areas as low as 5050 m 2 and even lower. In the particular design of concern in this study the case and central column of the transformer, which are parts of the magnetic core, are made of 3F3 ferrite. The gap between the windings is filled with magnetic nanofluid [10-13]. If the magnetic field end effects of the windings are neglected, axial symmetry may be assumed for the MPST, which results in a reduced computational effort since the numerical problem may be reduced to a 2D model. Fig. 1 presents a schematic view of this notional MPST with planar windings, and the 2D axial-symmetric computational domain used throughout the numerical simulations. Fig. 1 CAD view of the MPST. Axial symmetry is used to simplify the computational domain. 3.1 THE MAGNETIC FIELD 3. THE MATHEMATICAL MODEL The mathematical model for the magnetic field under steady state conditions is described by the following PDEs, e.g. [13] the windings 1 1 r A J e e, A Ae the ferrite part of the core and ceramic wafers the nanofluid core 0, (1) 1 1 A 0 0 r, (2) 1 0 A M, (3) 0

3 61 Electromagnetic, flow and thermal study of a miniature planar spiral transformer with planar, spiral windings where A [T m] is the magnetic vector potential, J e [A/m 2 ] is the external electric current density, 0 = 4π10-7 H/m is the magnetic permeability of free space, and r is the relative permeability. The magnetization, M [A/m], is approximated by the analytic formula [10, 11] M arctan H, (4) where H [A/m] is the magnetic field strength, and α, β are empiric constants selected to accurately fit the magnetization curve of the nanofluid (α = 3050 A/m, β = m/a, [12]). Magnetic insulation boundary condition (na = 0, where n is the outward pointing normal) on the outer surface of the computational domain (the outer surface of the ferrite casing, Fig. 1) closes the problem. 3.2 THE FORCED FLOW IN THE MAGNETIC NANOFLUID CORE The magnetic field produced by the currents in the MPST windings results in magnetic body forces that entrain the fluid core into a forced motion. Assuming that the nanofluid is Newtonian and its flow is laminar, incompressible, the mathematical model that describes it under steady state conditions is provided by momentum balance mass conservation law u u p u fmg 2, (5) u 0, (6) where u [m/s] is the velocity field, p [N/m 2 ] is the pressure field, ρ [kg/m 3 ] is the mass density, η [Ns/m 2 ] is the dynamic viscosity, and fmg M H [N/m 3 ] is the magnetization body force. The magnetic field flow coupling is one way: magnetization body forces produce the flow, whereas the very slow motion of the magnetic nanofluid (as will be seen later) does not perturb the magnetic field. Thermal forces are neglected because the device is too small for the gravity to influence the flow as will be seen later, the system is almost isothermal. The boundary conditions for the flow problem are no slip (zero velocity) at the outer boundaries of the magnetic fluid core computational domain. 3.3 HEAT TRANSFER PROBLEM The thermal field inside the MPST is analyzed by solving the energy equation C p T ( kt Q u ), (7) where ρ [kg/m 3 ] is the mass density, T [K] is the temperature field, k [W/m K] is the thermal conductivity, and Q [W/m 3 ] is the heat rate generation in the windings by Joule effect. The boundary condition for the top and lateral walls of the MSPT is convective heat flux, q conv h(t T amb ) (h = 2 W/m 2 K, natural convection). The ambient temperature is set T amb = 300 K. Symmetry is assumed at the symmetry axis. The bottom (the mounting part) is thermally insulated. 3.4 CALCULATING THE LUMPED PARAMETERS OF THE MPST The self-inductances are calculated by using the energy method [13]

4 J. B. DUMITRU, A. M. MOREGA, M. MOREGA 62 2 L ii w, d 2 m i, (8) I i V where I i [A] are the currents in the primary (i = 1) and secondary (i = 2) windings, w m,1 [J/m 3 ] is the magnetic energy density when the primary winding is fed by the current I 1 and the secondary winding is open (I 2 = 0), while w m,2 [J/m 3 ] is the magnetic energy density when the secondary winding is fed by I 2, (I 1 = 0); Ω [m 3 ] is the volume of the MPST. The mutual inductance between the windings is computed using von Neumann method [13], M 21 L2,1 L1,2, (9) i1 where Ф 21 [Wb] is the total magnetic flux produced by the primary current when the secondary winding is open. A key indicator in the design of the MPST is the (magnetic) coupling factor, k M L 11 L NUMERICAL SIMULATION RESULTS The mathematical model (1)-(9) was solved for numerically, in the finite element (FEM) technique, as implemented by [14]. First, the magnetic field is solved and the magnetic body forces are computed. Then the flow problem in the fluid core is addressed. Finally, the heat transfer is solved for. Fig. 2 shows the magnetic field in the MPST with nanofluid core through field lines and arrows of magnetic flux density. Three working conditions are considered. In the first case both windings are powered such that the electrical currents have opposite directions the nominal working condition. In the second case the primary is powered and the secondary is open. In the third case the secondary is powered and the primary is open. The magnetic field computed in the second and third cases is used to calculate the lumped circuit magnetic parameters of the MPST. a. Differential fluxes. Both windings are powered and the currents have the opposite signs. b. The primary is on and the secondary is off. c. The primary is on and the secondary is off. Fig. 2 Magnetic flux density spectra. The upper wing is the primary and the lower winding is the secondary. Fig. 3 presents the velocity field, through arrows and streamlines for nominal working conditions, and the pressure in the flow field.

5 63 Electromagnetic, flow and thermal study of a miniature planar spiral transformer with planar, spiral windings Apparently the flow depends on the structure of the magnetic field and it consists of two recirculation cells with opposite flows and of low velocity of the order O(10-9 m/s). Fig. 3 The flow through streamlines and arrows (left). Velocities are of order O(10-9 m/s). Pressure (contours) and magnetic body forces (arrows, right) red represents higher local pressure. Nominal working conditions. When the primary winding is powered and the secondary is open the flow in the fluid core consists of one recirculation zone in the fluid core volume, as Fig. 4 shows. The motion in the nanofluid intensifies near the middle section (by the symmetry axis) where it reaches higher velocities, of the order O(10-5 m/s). Fig. 4 The flow through streamlines and arrows (left). Velocities are of order O(10-5 m/s). Pressure (contours) and magnetic body forces (arrows, right) red represents higher pressures. Primary is on, secondary is off. In the third case, when the secondary is powered and the primary is off, the flow structure is similar to the one in the previous case (primary on, secondary off), except that the flow is now in reversed direction (see Fig. 5). Fig. 5 The flow through streamlines and arrows (left). Velocities are of order O(10-5 m/s). Pressure (contours) and magnetic body forces (arrows, right) red represents higher pressures. Secondary is on, primary is off. The thermal study was concerned with all three powering schemes. Two types of magnetic core, nanofluid and ferrite, were considered to compare the classical design with the novel solution. The resulting temperature distributions and heat flux are shown in Fig. 6.

6 J. B. DUMITRU, A. M. MOREGA, M. MOREGA 64 a. Nanofluid magnetic core. Differential magnetic fluxes. Maximum temperature is K and minimum temperature is K. b. Ferrite magnetic core. Differential magnetic fluxes. Maximum temperature is K and minimum temperature is K. c. Nanofluid magnetic core. The primary is powered secondary is open. Maximum temperature is K and minimum temperature is K. d. Ferrite magnetic core. The primary is powered and the secondary is open. Maximum temperature is K and minimum temperature is K. e. Nanofluid magnetic core. The primary is powered and the secondary is open. Maximum temperature is K and minimum temperature is K. f. Ferrite core. The primary is powered and the secondary is open. Maximum temperature is K and minimum temperature is K. Fig. 6 Temperature (color map) and heat flux (streamlines) within the MPST. As the results show, there are no notable differences between the nanofluid core and the ferrite core from the heat transfer point of view, although it is expected that under transient situations the nanofluid core may perform better than the ferrite [12]. The forced convection heat transfer inside the nanofluid core has little influence in the thermal balance as indicated by the negligibly small velocity. The temperature gradients are insignificant (less than 0.1 o C) in this small sized MPST, which justifies the assumption that the thermal forces are neglected.

7 65 Electromagnetic, flow and thermal study of a miniature planar spiral transformer with planar, spiral windings Table 1 The MPST lumped parameters. primary self-inductance, L 11 [H] secondary self-inductance, L 22 [H] mutual inductance, M [H] coupling factor, k The lumped parameters of the transformer where calculated out of the numerical simulation results (8), (9). Table I summarizes these findings. It may be conjectured that the excellent coupling factor is due to the central ferrite column that provides a negligibly small magnetic reluctance path, which directly couples the MPST windings. 5. CONCLUSIONS This paper presents mathematical modeling and numerical simulation results for a MPST concept proposed for equipping the fly-back converter in EHD devices. Numerical 2D simulations were conducted, under steady state conditions, to compute the magnetic field, the forced flow in the magnetic nanofluid core, the thermal field within the apparatus, and the lumped circuit magnetic parameters of the MPST. The results show that the MPST does not reach magnetic saturation levels during normal working condition. The magnetic nanofluid forced flow in the core is due to magnetic body forces. The three powering schemes that were considered indicate that the flow patterns and the pressure gradients are sensitive to the structure of the magnetic field. When both windings are powered the resulting flow consists of two opposite recirculation cells of low velocity, of the order O(10-9 m/s). When the MPST windings are powered in turn (one is powered the other one is off) the flow consist in a single recirculation zone with velocity of the order O(10-5 m/s), whose rotation depends on the powering scheme. The thermal field was analyzed for both magnetic nanofluid and ferrite cores. The numerical results show that the power transformer is almost isothermal and that the thermal gradients are less than 0.1 degrees. Therefore the flow due to thermal forces is negligibly small. There are no significant discrepancies between the magnetic and ferrite cored designs in what concerns the heat load. They are both below the thermal failure limit. The lumped circuit magnetic parameters of the transformer, self and mutual inductances, and the coupling coefficient, k, which is very close to one (the ideal transformer limit) suggest that the electromechanical design is efficient. ACKNOWLEDGMENTS J. B. Dumitru acknowledges the support offered by the Sectorial Operational Programme Human Resources Development of the Romanian Ministry of Labor, Family and Social Protection through the Financial Agreement POSDRU/107/1.5/S/ The work was conducted in the Laboratory for Multyphysics Modeling at UPB.

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