Electromagnetic, flow and thermal study of a miniature ......61 Electromagnetic, flow and thermal study of a miniature planar spiral transformer with planar, spiral windings INCAS

INCAS BULLETIN, Volume 6, Special Issue 1/ 2014, pp. 59 – 66 ISSN 2066 – 8201

Electromagnetic, flow and thermal study of a miniature

planar spiral transformer with planar, spiral windings

J. B. DUMITRU1, A. M. MOREGA*

,1,2, M. MOREGA

1

*Corresponding author 1“POLITEHNICA” University of Bucharest, Faculty of Electrical Engineering

313 Splaiul Independentei, 060042, Bucharest, Romania

[email protected], [email protected] 2“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and

Applied Mathematics of the Romanian Academy

Calea 13 Septembrie no. 13, 050711 Bucharest, Romania

[email protected]*

DOI: 10.13111/2066-8201.2014.6.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.

J. B. DUMITRU, A. M. MOREGA, M. MOREGA 60

INCAS BULLETIN, Volume 6, Special Issue 1/ 2014

2. AN MPST POWER TRANSFORMER

An MPST power transformer consists of two circular copper coils built on a ceramic

substrate (Al2O3). The MEMS technology permits the growth of copper windings with cross-

section areas as low as 5050 m2 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. THE MATHEMATICAL MODEL

3.1 THE MAGNETIC FIELD

The mathematical model for the magnetic field under steady state conditions is described by

the following PDEs, e.g. [13]

the windings

eAeA AJ er ,11

0 , (1)

the ferrite part of the core and ceramic wafers

0110 Ar , (2)

the nanofluid core

010 MA , (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, Je [A/m2] 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]

HM arctan , (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 = 1.510-5 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

mgp fuuu 2 , (5)

mass conservation law

0 u , (6)

where u [m/s] is the velocity field, p [N/m2] is the pressure field, ρ [kg/m3] is the mass

density, η [Ns/m2] is the dynamic viscosity, and HMf mg [N/m3] 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

QTkTCp )(u , (7)

where ρ [kg/m3] is the mass density, T [K] is the temperature field, k [W/m·K] is the thermal

conductivity, and Q [W/m3] 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,

qconv h (T Tamb ) (h = 2 W/m2·K, natural convection).

The ambient temperature is set Tamb = 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]



d2

,2

V

im

i

ii wI

L , (8)

where Ii [A] are the currents in the primary (i = 1) and secondary (i = 2) windings, wm,1

[J/m3] is the magnetic energy density when the primary winding is fed by the current I1 and

the secondary winding is open (I2 = 0), while wm,2 [J/m3] is the magnetic energy density

when the secondary winding is fed by I2, (I1 = 0); Ω [m3] is the volume of the MPST.

The mutual inductance between the windings is computed using von Neumann method

[13],

1

212,11,2

iLLM

, (9)

where Ф21 [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 L11L22 .

4. 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.



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(10-9m/s).

Fig. 3 The flow through streamlines and arrows (left). Velocities are of order O(10-9m/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).


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).


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.



a. Nanofluid magnetic core. Differential magnetic

fluxes. Maximum temperature is 346.747 K and

minimum temperature is 346.613 K.

b. Ferrite magnetic core. Differential magnetic fluxes.

Maximum temperature is 346.709 K and minimum

temperature is 346.631 K.

c. Nanofluid magnetic core. The primary is powered

secondary is open. Maximum temperature is 319.945

K and minimum temperature is 319.884 K.

d. Ferrite magnetic core. The primary is powered and the

secondary is open. Maximum temperature is 319.92 K and

minimum temperature is 319.89 K.

e. Nanofluid magnetic core. The primary is powered

and the secondary is open. Maximum temperature is

320.045 K and minimum temperature is 319.839 K.

f. Ferrite core. The primary is powered and the secondary is

open. Maximum temperature is 319.734 K and minimum

temperature is 319.673 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 0.1oC) in this small sized MPST, which justifies the assumption that

the thermal forces are neglected.



Table 1 – The MPST lumped parameters.

primary self-inductance, L11 [H] 1.661810-4

secondary self-inductance, L22 [H] 6.647110-4

mutual inductance, M [H] 3.271810-4

coupling factor, k 0.9844

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 2007-2013 of the Romanian Ministry of Labor, Family and

Social Protection through the Financial Agreement POSDRU/107/1.5/S/76903. The work

was conducted in the Laboratory for Multyphysics Modeling at UPB.



REFERENCES

[1] D. Lee, Energy harvesting chip and the chip based power supply development for a wireless sensor, Sensors

2008, vol. 8, pp. 7690-7714; DOI: 10.3390/s8127690.

[2] M. Bakkali, R Carmona-Galan, A Rodriguez-Vazquez, A prototype node for wireless vision sensor network

applications development, IEEE eXplore publication, ISBN: 9781424459964, DOI:

10.1109/ISVC.2010.5656420.

[3] O. Brand, G. K. Fedder, Eds., Advanced Micro & Nanosystems, vol. 2, CMOS-MEMS, Wiley-VCH Verlag,

GmBH & Co. KGaA, 2005, ISBN: 3-257-31080-0.

[4] J. Hogerheiden, M. Ciminera, G. Jue, Improved planar spiral transformer theory applied to a miniature

lumped element quadrature hybrid, IEEE Trans. on Microwave Theory and Techniques, vol. 45, 4, pp.

543-545, 1997.

[5] A. M. Morega, J. C. Ordonez, P. A. Negoias, M. Morega, R. Hovsapian, Optimal electrical design of spherical

photovoltaic cells”, COMSOL Conference, Congress Center of the Czech Technical University, Prague,

October 26, 2006.

[6] T. H. Tsai, L. S. Kuo, P. H. Chen, D. S. Lee, C. T. Yang, Applications of ferro-nanofluid on a micro-

transformer, Sensors 2010, vol. 10, pp. 8161-8172, DOI:10.3390/s100908161.

[7] K. Raj, R. Moskovitz, Ferrofluid, Cooled electromagnetic device and improved cooling method, 1993, US

Patent No. 5,462,685.

[8] L. Beitelman, V. Segal, C. P. Curran, Cooling electromagnetic stirrers, 2005, US Patent No. 6,927,510 B1.

[9] V. Segal, A. Rabinovich, D. Nattrass, K. Raj, A. Nunes, Experimental study of magnetic colloidal fluids

behavior in power transformers, J. Magn. and Magn. Mater., vol. 215-216, pp. 513-515, 2000.

[10] A. M. Morega, M. Morega, L. Pâslaru-Dănescu, V. L. Stoica, F. Nouraş, F. D. Stoian, A Novel, Ferrofluid-

Cooled Transformer. Electromagnetic Field and Heat Transfer by Numerical Simulation, OPTIM 2010,

12th Int. Conference on Optimization of Electrical and Electronic Equipment, 22-24 May 2010, DOI

10.1109/OPTIM.2010.5510528, Print ISBN 978-1-4244-7019-8, pp. 401-406, Brasov, Romania.

[11] A. M. Morega, L. Pâslaru-Dănescu, M. Morega, A novel nanoparticles magnetic fluid microactuator device,

OPTIM 2012, 13th Int. Conference on Optimization of Electrical and Electronic Equipment, 24-26 May

2012, Brasov, Romania, paper RD-006602, IEEE-Xplore.

[12] A. M. Morega, M. Morega, J. B. Dumitru, L. Pîslaru-Dănescu and V. Stoica, Magnetic and electric sizing of

a miniature planar spiral transformer, in The 7th Int. Conf. Electrical and Power Engineering, EPE 2012,

Iaşi, Romania, Oct. 25–27, 2012.

[13] A. M. Morega, J. B. Dumitru, M. Morega, L. Pîslaru- Dănescu, Flow patterns in the magnetic nanofluid core

of a miniature planar spiral transformer, Environmental Engineering and Management J., vol. 12, 6, pp.

1171-1177, June 2013.

[14] *** COMSOL A.B. v. 3.5a (2010), 4.2a (2011).

Documents

Electromagnetic, flow and thermal study of a miniature ......61 Electromagnetic, flow and thermal study of a miniature planar spiral transformer with planar, spiral windings INCAS