IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 16, NO. 2, MARCH 2001
281
Cost-Constrained Selection of Strand Diameter and
Number in a Litz-Wire Transformer Winding
Charles R. Sullivan, Member, IEEE
AbstractDesign of litz-wire windings subject to cost con-
straints is analyzed. An approximation of normalized cost is
combined with analysis of proximity effect losses to find combina-
tions of strand number and diameter that optimally trade off cost
and loss. The relationship between wire size, normalized cost, and
normalized loss is shown to have a general form that applies to a
wide range of designs. A practical design procedure is provided.
Applied to an example design, it leads to less than half the original
loss at lower than the original cost, or, alternatively, under one
fifth the original cost with the same loss as the original design.
Index TermsEddy currents, inductors, litz wire, magnetic de-
vices, optimization costs, power conversion, power transformers,
proximity effect, skin effect.
I. I
NTRODUCTION
L
ITZ WIRE
1
can be used to reduce the severe eddy-current
losses that otherwise limit the performance of high-fre-
quency magnetic components. But litz wire is often avoided by
designers because it can be very expensive. In this paper, we
develop a design methodology considering cost. This approach
enables significant cost reduction with no increase in loss, or
more generally, enables a designer to select the minimum loss
design at any given cost. In a design example, the cost is reduced
by better than a factor of five with no increase in loss, compared
to a design based on a conventional rule of thumb.
Losses in litz-wire transformer windings have been calculated
by many authors [1][6], but relatively little work addresses
the design problem: how to choose the number and diameter of
strands for a particular application. In [7], the optimal stranding
giving minimum loss is calculated. However, this can result in a
very expensive solution with only slightly lower loss than is pos-
sible at considerably lower cost. Although [7] also addresses the
choice of stranding under constraints of minimum strand diam-
eter or maximum number of strands, the real constraint is more
likely to be cost rather than one of these factors.
Analysis of cost is performed at two levels in this paper. First,
a general form for functions describing the cost of litz wire is
hypothesized. This leads to general analytical results describing
the best choice of litz wire for a given transformer winding, in
Manuscript received March 8, 2000; revised November 27, 2000. Recom-
mended by Associate Editor K. Ngo.
The author is with the Thayer School of Engineering, Dartmouth College,
Hanover, NH 03755-8000 USA (e-mail: charles.r.sullivan@dartmouth.edu).
Publisher Item Identifier S 0885-8993(01)02196-2.
1
Sometimes the term litz-wire is reserved for conductors constructed
according to a carefully prescribed pattern, and strands simply twisted together
are called bunched wire. We will use the term litz-wire for any insulated
grouped strands.
Fig. 1.
Types of eddy-current effects in litz wire.
terms of a cost function. At the second level, results that are
less general but are more explicit are obtained through making
the cost function explicit with a polynomial curve fit to man-
ufacturers price quotations. A design methodology, applicable
to the general case, but fleshed out in terms of the specific cost
function, is outlined and illustrated with a design example.
Many analyses of winding loss address only sinusoidal cur-
rent waveforms, but magnetics in high-frequency power con-
verters rarely have waveforms that approximate sinusoids. A
number of authors have developed methods of extending the
analysis of winding loss to nonsinusoidal waveforms [7][13].
Of particular interest is the use of an effective frequency [7],
[10], [11], [13] because that approach allows the use without
modification of optimizations based on sinusoidal waveforms
(including the optimization described here), as explained in the
appendix of [7]. Particularly useful is [13] for a thorough discus-
sion and a compilation of the relevant data for a large number
of common waveforms.
II. L
OSS
M
ODEL
Skin effect and proximity effect in litz-wire windings may be
divided into bundle-level and strand-level effects, as illustrated
in Fig. 1. With properly chosen construction, strand-level prox-
imity effect is the dominant effect that needs to be considered
for choosing the number of strands [7].
We represent winding losses by
(1)
where
is a factor relating dc resistance to an ac resistance
which accounts for all winding losses, given a sinusoidal current
with rms amplitude
. As discussed in Appendix A, internal
08858993/01$10.00 � 2001 IEEE 282
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 16, NO. 2, MARCH 2001
and external strand-level proximity effect loss can be accounted
for with the approximate expression
(2)
where
radian frequency of a sinusoidal current;
number of strands;
number of turns;
diameter of the copper in each strand;
resistivity of the copper conductor;
breadth of the window area of the core;
factor defined in Appendix A, accounting for field dis-
tribution in multiwinding transformers, normally equal
to one.
For waveforms with a dc component, and for some nonsinu-
soidal waveforms, it is possible to derive a single equivalent fre-
quency that may be used in this analysis [7]. In an inductor, the
field in the winding area depends on the gapping configuration,
and this analysis is not directly applicable [14].
III. C
OST
A
NALYSIS
Attempting to quantify cost for academic analysis is problem-
atic; prices change with volume, manufacturer, time, and negoti-
ation. However, many important results depend only on the gen-
eral form of the cost function. In particular, the general solutions
derived in the Appendix for optimal cost/loss tradeoff designs
depend only only the assumption that the cost of a length of litz
wire can be approximately described by
(3)
where
base cost per unit length associated with the
bundling and serving operations;
cost basis function proportional to the additional
cost per unit mass for a given strand diameter
;
number of strands;
length of the wire.
Since we have not specified a form for
, the only loss
of generality in assuming this form (3) is in the assumption that
depends only on
, and not on . Examination of pricing
from litz-wire manufactures indicates that this assumption is a
valid approximation. Note that for the purpose of optimization
with a fixed winding length, we can ignore
, and consider
only the cost variation which is proportional to
.
In order to gain intuition about the variation of cost, and to
provide specific numerical results, it is useful to find an approx-
imate expression for
. From manufacturers pricing, we
find that the following function, normalized to a value of one for
large-diameter wire, is a good approximation for a wide range
of values of
and
(4)
where
is in meters,
m , and
m . This function, proportional to cost per unit mass,
Fig. 2.
Normalized cost per unit mass and normalized cost per unit length,
as modeled by (4). Both are normalized such that the minimum values are
one, for the purpose of display in this graph. The cost per unit mass increases
monotonically, reflecting the cost of drawing a given quantity of copper into
finer and finer strands. The cost per unit length is found by multiplying cost per
unit mass by mass per unit length, as described in the text. Below 44 AWG, the
decreasing mass dominates the trend, making the cost per unit length decrease
as the wire gets smaller. Above 44 AWG the cost per unit mass increases
rapidly enough that the increased manufacturing cost dominates the decreased
material cost, and cost per unit length increases. Both 38 AWG and 48 AWG
cost about twice as much as 44 AWG. For 38 AWG, this cost increase is a
result of the larger mass of copper required. For 48 AWG, the cost increase is
due to the expense of forming the wire into very fine strands.
is shown in Fig. 2, along with the normalized cost per unit
length,
.
is approximately constant for large di-
ameters, but by around 40 AWG it has started rising signifi-
cantly. 44 AWG is notable as the size at which the cost per unit
length is a minimum. At 48 AWG, cost per unit length has in-
creased significantly and cost per unit mass has increased dra-
matically. Few manufacturers will provide constructions using
finer strands than this, and though (4) is not based on data be-
yond this point, it does appropriately rise very rapidly. Although
(4) represents a smooth function, wire based on standard sizes
is cheaper than arbitrary choices, and the actual cost function
has significant ripples because of this. In particular, even-num-
bered sizes are generally cheaper and more readily available
than odd-numbered sizes. The extent of this variation is highly
sensitive to volumeat sufficiently high volumes, there would
be no penalty for using odd, or even custom sizes. Thus, such
variations are omitted from this analysis; we assume the cost is
described by the smooth function shown.
IV. C
HOOSING
N
UMBER AND
D
IAMETER OF
S
TRANDS
The design choice of number and diameter of strands can
be conceptualized and illustrated as a two-dimensional (2-D)
space. In the case of a full bobbin, the choices in this space form
a line, and the tradeoff between cost and loss becomes a simple
matter of evaluating both cost and loss along this line, which
can be described by using calculations in [7]. However, with
cost constraints, a full bob