oss Calculation with Multiple Windings, Arbitrary Waveforms, and Two-Dimensional Field Geometry
w1
w2
w3
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Abstract--
Charles R. Sullivan
A. Review of Previous Analytical Approaches
Winding Loss Calculation with Multiple Windings,
Arbitrary Waveforms, and Two-Dimensional Field Geometry
IEEE Industry Applications Society
Annual Meeting
Phoenix, AZ, 3-9 October, 1999
NTRODUCTION
Fig. 1. Three-winding transformer (w1-w3) in which two-dimensional
field geometry is important.
A method for calculating eddy-current (proximity-
effect) losses in transformer and inductor windings is introduced.
The new method is capable of analyzing losses due to two- and
three-dimensional field effects in multiple windings with arbitrary
waveforms in each winding. It uses a simple set of numerical mag-
netostatic field calculations to derive a matrix describing the trans-
former or inductor. This is combined with a second matrix calcu-
lated from derivatives of winding currents to computer total ac loss.
Experiments show the method is accurate for coils that are not in
or close to self-resonance.
Thayer School of Engineering
8000 Cummings Hall, Dartmouth College, Hanover, NH 03755-8000
charles.r.sullivan@dartmouth.edu
603-643-2477
http://engineering.dartmouth.edu/inductor
I. I
Magnetic component performance is essential for
high-frequency power conversion; often the magnetic
components are the most expensive and largest parts of
a system, and can pose some of the most severe loss and
thermal problems. Circuit designs are often predicated on
minimizing requirements for magnetic components. De-
spite the importance of magnetic components, the state
of the art in magnetic component design leaves much to
be desired. In particular, standard methods of analyzing
winding loss [1-18] ([16] gives a useful review) assume
a one-dimensional field for analyzing eddy-current ef-
fects in windings. But two-dimensional effects are impor-
tant in any magnetic component that includes a discrete
air gap. In this paper, we introduce a new method that
includes the effects of two- or three-dimensional fields,
while taking into account multiple windings and non-
sinusoidal waveforms that may be different in each wind-
ing. The method applies to round-wire windings, includ-
ing litz-wire windings.
Although different descriptions can be used, most ex-
isting analytical calculations of high-frequency winding
loss are fundamentally equivalent to one of three analy-
ses. The most rigorous approach uses an exact calculation
of losses in a cylindrical conductor with a known current,
subjected to a uniform external field, combined with an
expression for the field as a function of one-dimensional
position in the winding area [6, 18]. Perhaps the most
commonly cited analysis [7] uses equivalent rectangu-
lar conductors to approximate round wires, and then pro-
ceeds with an exact one-dimensional solution. Finally,
one may use only the first terms of a series expansion of
these solutions, e.g. [15, 19, 20].
For designs in which one-dimensional field analysis is
accurate, and where wire strands are small compared to
a skin-depth, these various methods are approximately
equivalent [6], despite one small discrepancy explained
in [21]. Although the basic analysis is usually based on
sinusoidal waveforms, a number of authors have devel-
oped methods of extending this analysis to non-sinusoidal
waveforms through Fourier analysis or other methods
[11, 13, 19, 20, 21, 22, 23].
A major limitation of all of this work is that it does not
apply to components in which the field geometry is not
one-dimensional. This includes nearly all inductors and
gapped transformers, in which the two-dimensional field
geometry due to the gap significantly affects losses [24].
Standard one-dimensional analysis is also unable to ana-
lyze transformers with two-dimensional winding layout,
such as the one shown in Fig. 1. In [25] an analytical ap-
proach is developed for two-dimensional fields in gapped
single-winding inductors. Although this is an important
accomplishment, the results are too complex for routine
design work, they are specific to one geometry, and they
do not account for multiple windings or arbitrary wave-
forms. 

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B. Limitations of General Purpose Electromagnetic
Analysis
Scale Problems
Optimization
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Work in computational electromagnetics has produced
general-purpose field analysis methods, and commercial
software is available for two and three-dimensional so-
lutions of arbitrary problems, including analysis of eddy
currents. However, there are two major limitations of this
approach for high-frequency magnetics in power elec-
tronics applications.
The first limitation is a scale problem. Transform-
ers and inductors often require many turns of fine
wire, or may use stranded wire such as litz wire to
reduce eddy current losses. The wire stands may be
as small as 30-50
m in diameter (44-48 AWG),
while the overall dimensions may be tens of cen-
timeters. Thus, the length scales involved can vary
over two to four orders of magnitude, and there
may be as many as 10,000 or more strands of wire.
Even when larger wire is used, the skin depth in
the wire can be small, for example 100 m at 400
kHz, which creates the same problem. This leads
to a need for a large number of elements in finite
element analysis, and thus slow simulations and
large memory requirements. To circumvent this
problem, software vendors recommend modeling
a stranded winding as a region of uniform current
density. While this is helpful for analyzing field
distributions, it provides no information on losses
in the stranded winding.
With existing field analysis, optimization must be
done by trial and error. Particularly when each iter-
ation takes hours to analyze via finite element anal-
ysis, true optimization is not practical, except in a
few academic experiments, which then provide in-
formation about only one particular design.
II. N
A
M
In order to circumvent both the limitations of one-
dimensional analytical methods, and the limitations of
existing numerical methods, we use a combination of nu-
merical calculation of the overall field geometry, with
analytical calculation of its interaction with the winding
strands. This avoids the scale problem, but allows ap-
plying the power of modern computers to quickly obtain
a much more accurate solution than would be available
through one-dimensional analysis. A similar approach
was reported in [26] for gapped single-winding induc-
tors with sinusoidal waveforms. The method we report
here is more powerful, in that it is capable of analyzing
multi-winding transformers with different non-sinusoidal
waveforms in each winding.
We start with the calculation of loss in a conducting
cylinder in a uniform field, perpendicular to the axis of
the cylinder, with the assumption that the field remains
constant inside the conductor, equivalent to the assump-
tion that the diameter is small compared to a skin depth.
This results in instantaneous power dissipation
in a
wire of length [19]
(1)
where
is the flux density,
is the resistivity of the
wire, and
is its diameter. The average loss depends
on the time average of the squared derivative of the field,
. We can also use the spatial average of this quan-
tity to calculate the time average of total ac loss in a wind-
ing
(2)
where
is the number of turns in winding ,
is
the average length of a turn,
indicates a spatial
average over the region of winding , and
indicates
a time average. For a litz-wire winding, the same ex-
pression may be used to calculate strand-level proximity-
effect loss. In this case,
must represent the product
of the number of turns and the number of strands in each
turn (i.e.,
is the total number of strands in the wind-
ing), and
is the diameter of the individual strands. The
length of a turn may also need to be adjusted to account
for the increased distance that a strand travels on account
of twisting. This calculation neglects bundle-level ef-
fects, but this is usually valid because with proper bundle
construction, they can be made negligible [21].
In a given winding, the field,
may be expressed as
the superposition of fields due to currents in each wind-
ing. We can then express the loss in winding
of a two-
winding transformer as
(3)
where
and
is the field due to current
in winding . We can express this in terms of the currents
as
(4)
where
is a constant relating current and loss, to
be calculated in the next section. The total ac loss in all
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RANSFORMER USED FOR VERIFICATION
IELD
ALCULATIONS
ERIFICATION
TABLE I
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Windings
Number of turns
162:162
Number of layers in