Forced Flow Patterns in a Miniature Planar Spiral Transformer with [601400]

Forced Flow Patterns in a Miniature Planar Spiral Transformer with
Ferrofluid Core
J.B. DUMITRU*, A.M. MOREGA*,**, M. MOREGA* and L. PÎSLARU-DĂNESCU***
**Corresponding author
*“POLITEHNICA” University of Bucharest, Fa culty of Electrical Engineering, Bucha rest, Romania
[anonimizat], [anonimizat], [anonimizat], [anonimizat]
**“Gheorghe Mihoc – Caius Iacob” Institute of Statistical Mathematics and Applied Mathematics,
Bucharest, Romanian Academy
***National Institute for Electrical Engineering ICPE-CA, Bucharest, Romania
Abstract: Energy harvesting devices (EHD) utilize s m a l l -scale components with low power losses. Key components, the
electric power transformers (EPTs ), are to convert the voltage/current electromagnetic parameters from the primary, energy
harvesting stage , to the secondary, storage and delivery levels. M agnetic colloidal nanofluids seem to be a sound solution for
building the magnetic circuit of such EPTs, enabling miniaturized constructions , whose implementa tion may benefit of LIGA
fabrication technology. Because the magnetic cores of EPTs are, at least in part, fluid, the occurring magnetization body forces
may result in complex flows that need to be known . Along this line, t his paper presents mathematical m odels and numerical
simulation results for a miniature, planar, spiral EPT. The study is concerned with the magnetically produced flow of the fluid
part of the magnetic core. Several electric powering schemes are envisaged and the pending flows investigate d.
Keywords: Transformer; spiral, planar windings; flow; numerical simulation; finite element; nanofluid.
1. INTRODUCTION
The need for wireless devices that use Energy Harvesting Systems (EHS) as an alternative to batteries [1 -3] motivate
the developing of micro -power microcontrollers and RF transmitters that are used in their electronic conditioning stages.
The EHSs convert artificial light, vibrations, temperature gradients, etc. in useful electrical power [4].
A key component in an EHS is the fly-back power transformer , which must comply to certain specifications
such as small footprint , slim profile, thermal stability, high electromagnetic efficiency, and , not the least, low
cost. Miniature Planar Spiral Transformer s (MPST) in the micro -electromechanical systems (MEMS) class are
sound candidat: [anonimizat] , therefore their study and design is of current concern [ 5-7]. The magnetic core of an MPST is
usually made of ferrite, but recent studies show that magnetic nanofluids (MNF) used as magnetic core (at least
partia lly), may provide for higher power conversion efficiency [5 -7].
MNFs based on iron oxide nanoparticles (magnetite) dispersed in oil suspension are under research for power
transformers as cooling and insulating medi a [7-15], because they may alleviate the di f f i cul t i es r el at ed t o t he
magnetic core losses, especially at high frequencies, due to their almost zero hysteresis.
This study is concerned with a MPST with spiral, circular coils with MNF as part of the magnetic core .
Electric and magnetic field prob lems are formulated and solved to analyze the forced, magnetically produced
flow within the fluid part of the magnetic core. The results indicate that the usage of 2D rather than 3D models
may be of concern, at least in what regards the flow of the fluid m agnetic core.
2. THE MINIATURE PLANAR SPIRAL TRANSFORMER
The MPST concept in this study has two spiral, circular , copper windings, 20 turns for the primary and 40 for
the secondary, that are “grown” in LIGA (Lithographie, Galvanoformung, Abformung [16]) technology on a
ceramic substrate (Al 2O3). The cas e and the central column are made of 3F3 ferrite. The spacing between the
windings is filled with MNF (Fig. 1).
In computing the equivalent, lumped circuit parameters of the MPST, it is customary to consider the planar,
spiral windings as closed loops, circular, concentric turns with no electric terminals. Thus the MPTS becomes an axially symmetric structure, and the study of the electromagnetic field may be conducted using simpler, 2D models rather than full 3D models, Fig. 1.

Fig. 1. The MPST with planar, spiral windings, and fe rrite – magnetic nanofluid core, and the equivalent, simplified , 2D axial model [17].
While this approach may provide for satisfactorily accurate results, as will be shown, it may be less so when
the flow of the fluid core , produced by magnetization body forces forced, is of concern. In that case , the spiral
windings cannot be replaced with equivalent, closed, concentric turns, and the electric terminals should be
represented too. We present mathematical models for the magnetic field in the MPST and the forced flow field
within the fluid part of the magnetic core. Both 2D (axial) and 3D models are utilized to evaluate the magnetic,
lumped circuit parameters of the MPST and the flow field within the MNF core of the MPST.
3. THE MATHEMATICAL MODEL
The magnetic field for steady state conditions is described by the mathematical model
inside the coils
, (1)
inside the ferrite part of the magnetic core and ceramic holders
, (2)
inside the magnetic nanofluid core
, (3)
where, A [T·m] is the magnetic vector potential, J [ A / m2] is the electric current density, µ0 = 4π×10-7 H / m i s t h e
magnetic permeability of vacuum , µr i s t h e r e l a t i v e p e r m e a b i l i t y f o r d i f f e r e n t other parts of the transformer , and
M [A/m] is the magnetization of the MNF , model led through [17]
. (4)
Here, H [A/m] is the magnetic field strength, and α, β are empiric constants, which are selected to fit the
magnetization curve , e.g., [6]. The magnetic problem (1) -(4) is closed by a magnetic insulation boundary condition

on the outer surface of the case , n×A = 0, where n is the outward pointing normal to the computational domain.
The magnetic field, produced by the current flowing through the MPST windings, generates magnetic body
forces within the MNF , which are responsible for the forced flow of the fluid core. The mathematical model that
describes the flow in steady state conditions is given by
momentum conservation (Navier -Stokes )
, (5)
mass conservation law
, (6)
where, u [m/s2] is the velocity, p [N/m2] is the pressure, ρ [kg/m3] is the mass density, η [N⋅s/m2] is the dynamic
viscosity, and [ N / m3] is the magnetization body force. Body forces of thermal nature are not
significant here [17].
The boundary conditions that close t h e flow problem are a s f ollows : no slip (zero velocity) at the outer
boundaries of the MNF domain .
The self-inductances of the MPST are calcula ted with the energy method [17]
, (7)
where i = 1 subscript denotes the primary winding, i = 2 the secondary wind ing, Ii [A] are the electric currents ,
wm,1 [ J / m3] is the magnetic energy density inside the MPST when the primary winding powered a n d t h e
secondary winding is open c i r c u i t , wm,2 [ J / m3] is the magnetic energy density when the secondary winding is
powered a nd the secondary is open circuit, and V is the MPST volume.
The mutual inductance is computed using Neumann method using
, (8)
where Ф21 [Wb] is the total flux magnetic produced by the current in the primary winding when the seconda ry
coil is o pen circuit. The coupling factor, defined through
, (9)
measures the magnetic efficiency of the design of the MPST.
3. NUMERICAL SIMULATION RESULTS AND DISCUSSION
First, 2D axial models were considered. The numerical sim ulation results for two types of the MNFs
presented here are: the “1000 Gs” magnetic fluid, with [eq. (4)] α = 4.71 ×104A/m, and β = 2.08 ×10-5m/A, and
the “100 Gs” magnetic fluid, with α = 6.2 ×103A/m, and β = 1.79 ×10-4m/A.
Three different working conditions : both windings power on, additional magnetic fluxes; both windings on,
differential magnetic fluxes; primary on and secondary off; primary on and secondary on, were analyzed. Fig. 2
shows the magnetic field in these cases, obtained through numerical simu lation . The magnetic field is seen
through its field lines (contour lines of magnetic vector potential ), whose color is proportional with the local
value of the magnetic flux density . The upper winding is the primary and the lower winding is the secondary.

A. The 1000 Gs magnetic fluid core.

B. The 100 Gs magnetic fluid core.
a. Both winding s are on and the
fascicular fluxes are opposite . b. Both windings are on, additional
fascicular magnetic fluxes . c. The primary winding on and
the secondary winding off. d. The secondary winding is off
and the primary winding is o n.
Fig. 2. Magnetic flux density for the 1000 Gs and 100 Gs magnetic fluid cores. Values are in Tesla [18].
Fig. 3 shows spectra of the forced flow within the fluid magnetic core for the cases presented in Fig. 2.

A. The 1000 Gs magnetic fluid. Opposite magnetic fluxes. Ve locities are of order O(10-8m/s).

B. The 100 Gs magnetic fluid. Opposite magnetic fluxes. Velocities are of order O(10-8m/s).

C. The 1000 Gs magnetic fluid. Additional magnetic fluxes. Velocities are of order O(10-4m/s).

D. The 100 Gs mag netic fluid. Additional magnetic fluxes. Velocities are of order O(10-4m/s).

E. The 1000 Gs magnetic fluid. The primary on and the secondary off. Velocities are of order O(10-4m/s).

F. The 100 Gs magnetic fluid. The primary is on and the second ary is off. Velocities are of order O(10-4m/s).

G. The 1000 Gs magnetic fluid. The primary is off and the secondary is on. Velocities are of order O(10-2m/s).

H. The 100 Gs magnetic fluid. The primary is off and the secondary is on. Velocities are of order O(10-4m/s).
Fig. 3. Velocity field (left), pressure and magnetic forces (right) inside the fluid magnetic core.
The MNF flow depends on the particular powering scheme of the MPST, and may be structured either into
two recirculation regions of low velocity, in opposite directions (Fig. 3, A), or single, larger flow cells, of higher
velocity (Fig. 3, B,C).
Things are changing substantially when seen through the 3D model. Fig. 4 shows the voltage surface map (DC
powering), the magnetic field and the magnetic , and the fluid flow in the MPST with the 1000 Gs magnetic fluid
when both windings are on, additional fascicular magnetic fluxes .

A. The electric potential on the windings. Values are in [V]. B. The magnetic flux density. Values are in [T]

C. The flow – 3D perspective. Values are in [m/s]. D. The flow – frontal view . Values are in [m/s].
Fig. 4. The electric field, magnetic field, and the magnetic fluid core flow in the MPST with 1000 Gs fluid . Both win dings are on hence
the magn etic fluxes are additional.
It may be concluded that, although too slow either to pose concerns or to help in the heat transfer (the MPST

seems to reach a stationary state that is isothermal), the MNF flow is complex, fully 3D, with no symmetry, unlike
the flow predicted by the 2D models. Further more, Fig. 5 presents , comparatively, the magnetic field and the fluid
core flow for the MPST with 100 Gs and 1000 Gs magnetic fluid, for three of the powering schemes analyzed
using the 2D axial model. The electr ic terminals , not included in the 2D model, and the different modes crossings
produce different magnetic field spectra in those (terminals ) regions.

A. Both windings are on, additional
magnetic fluxes – 1000 Gs magnetic fluid. B. Primary winding is on – 1000 Gs
magnetic fluid. C. Secondary winding is on – 1000 Gs
magnetic fluid.

A. Both windings are on, additional
magnetic fluxes – 100 Gs magnetic fluid. B. Primary winding is on – 100 Gs
magnetic fluid. C. Secondary winding is on – 100 Gs
magnetic fluid.
Fig. 5. The magnetic flux density for several powering schemes and two magnetic fluid cores [18].
The forced flows inside the MNF core for the different powering schemes obtained using the 3D model show
off very different flows, of swirl ty pe, as compared with those obtained using the 2D axial models.

A. Both windings are on, additional
magnetic fluxes – 1000 Gs magnetic fluid. B. Primary winding is on – 1000 Gs
magnetic fluid. C. Secondary winding is on – 1000 Gs
magnetic fluid.

A. Both windings are on, additional
magnetic fluxes – 100 Gs magnetic fluid. B. Primary winding is on – 100 Gs
magnetic fluid. C. Secondary winding is on – 100 Gs
magnetic fluid.
Fig. 6. The forced flow in the magnetic fluid core for several powering schemes and two magnetic fluid cores [18] .
The main reasons for this behavior are the spiral current paths (rather than closed, circular, concentric turns),
and the electric terminals that are present in the 3D model and not in the 2D ones, which explain the differences
in the magnetic field spectra. The magnetic body forces produced through the MNF core magnetization result
then in different types of 3D flows as compared with their 2D counterparts. F i g . 6 s h o w s c l e a r l y t h e f l o w
patterns that are accompan ying the different powering schemes of the MPST.

A. Both windings are on, additional
magnetic fluxes – 1000 Gs magnetic fluid. B. Primary winding is on – 1000 Gs
magnetic fluid. C. Secondary winding is on – 1000 Gs
magnetic fluid.

A. Both w indings are on, additional
magnetic fluxes – 100 Gs magnetic fluid. B. Primary winding is on – 100 Gs
magnetic fluid. C. Secondary winding is on – 100 Gs
magnetic fluid.
Fig. 6. The forced flow in the MNF core for several powering schemes and two magneti c fluid cores [18] .
For opposite amperturns (both windings on, opposite fascicular magnetic fluxes) the flow is still 3D, but
horizontally stratified into two circular, horizontal cells, unlike the flow produced by additional amperturns
(Fig. 4), when such a stratification does not exist.
Although negligibly small for heat transfer purposes, these flows may help in preventing the magnetic fluid
sedimentation.

Table 1 summarizes the results concerning the lumped circuit parameters of the MPST with 1000 Gs
magnetic fluid, which is used in the prototype construction.
Table 1 – The lumped circuit magnetic parameters of the MPST for the MPST with 1000 Gs magnetic fluid core [18].
Lumped circuit magnetic parameter 2D model 3D model
L
11 [H] – primary self -inductanc e 3.41×10-5 3.7×10-4
L
22 [H] – secondary self -inductance 1.37×10-3 1.33×10-3
M [H] – mutual inductance 6.8×10-4 1.41×10-3
k – coupling factor 0.99 0.989
In what concern the lumped circuit magnetic parameters, the discrepancies between the results obtai ned based on
the 2D and 3D models using the energy method (7) and Neumann formula (8) is less important, as shown in Table 1.
3. CONCLUSIONS
Mathematical modeling and numerical simulation may be used to analyze, complex structures such as MPSTs, with
the a im to optimize their design. This study was concerned with a monophased MPST for EHD devices. Different
powering schemes that provide steady state operational conditions were considered. 2D and 3D models were used to
compute the magnetic field and the magn etic fluid core flow and lumped circuit magnetic parameters of the MPST.
The 2D models lead to similar results for the magnetic parameters of the MPST. On the other hand, the flows
obtained using the two approaches are different: the 2D flows present verti cally organized recirculation regions,
with no azimuthal component, while the 3D flows, swirle like show off, basically, horizontally organized cells.
Depending the powering scheme there may be one or more cells. In general, in the 2D models, powering both windings
produces two recirculation zones with low velocities, of the order O(10-8 m/s), and powering one of the windigns only
may result in flows that consist of a single recirculation region with velocities of order O(10-2…10-4 m/s).
The lumped circuit m a g n e t i c parameters of the MPST , self and mutual inductances, and the coupling
coefficient, k, very close to one (ideal transformer) – o b t a i n e d f o r b o t h m o d e l s – i n d i c a t e that the proposed
electromechanical design for the MPST is efficient.
ACKNOWLEDGEMENTS
The work was conducted in the Laboratory for Multiphysics Modeling, at UPB, within the framework of the PNCDI
– II research grant ASEMEMS -HARVEST 63/2013. The magn etic prope rties of the magnetic nanofluids used here were
measured at the University POLIT EHNICA of Timi șoara and Timi șoara Branch of the Romani an Academy .
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