For a design example, we used a 14-turn winding on an RM5 size ferrite
core. The breadth of the bobbin is 4.93 mm, and the breadth of the core
window 6.3 mm. A height of 1.09 mm is allocated to this winding. Based on an
experimental hand-wound packing factor F_{p}=0.85, and litz packing
factor F_{lp}=0.66, unserved, plus a 32
(1.25 mil) thick layer of serving, the above calculation indicates that, for
a frequency of 375 kHz, 130 strands of number 48 wire gives minimum ac
resistance, with a total resistance factor of = 2.35,
ac resistance factor F_{r} = 1.03, and a dc resistance factor
F_{dc} =2.29.

Fig. 4 shows the total calculated resistance factor and its components
as a function of number of strands. The figure and the numbers confirm the
intuition that, because is close to one and the dc
resistance increases only very slowly, the decrease in resistance using finer
strands outweighs the decreased cross sectional area until the ac resistance
factor is brought very close to one. Note that although the factor
F_{dc} is large, only a factor of 1.18 is due to the change in wire
insulation thickness. The remaining factor of 1.95 is due to the the dc
resistance factors that do not vary with number of strands, such as serving
area and strand packing.

for the example discussed in the text at 375 kHz.

Also shown are the ac resistance factor F

and the dc resistance factor F

The minimum total resistance factor is at the point where increases

in F

The optimization leads to choosing a large number of fine strands, which will often mean a high cost, and will sometimes require finer strands than are commercially available. From Fig. 4, one can see that a decrease from the optimum of 130 to about 50 strands entails only a small increase in ac resistance. Consideration of the cost trade-off for a particular application becomes necessary.

Given a sub-optimal number of strands, chosen to reduce costs, a full
bobbin may no longer be best. The problem of choosing the optimal strand
diameter for a fixed number of strands has been addressed by many authors,
though usually just for a single strand [2], [3], [4], [9]. This analysis can be
adapted for more than one strand by simply using the product of the number
of turns and the number of strands Nn in place of the number of turns N. The
result that F_{r opt}=1.5 [2], [3], [4], [9] holds, and

(13)

In many practical cases, cost is a stronger function of the number of strands than of the diameter of the strands. In the range of about 42-46 AWG the additional manufacturing cost of smaller wire approximately offsets the reduced material cost. Thus, designs using the diameter given by (12) often approximate the minimum ac resistance for a given cost.

Fig. 5 shows total resistance factor as a function of the number of
strands for the same example design, but at 1 MHz, where the optimal
stranding is a difficult and expensive 792 strands of AWG 56 wire, and so
analysis of alternatives is more important. The solid line is for a full
bobbin, and the dashed line is for the same number of strands, but with the
diameter chosen for minimum losses, rather than to fill the bobbin. Where
the two lines meet, the optimal diameter just fills the bobbin. Beyond that
point it would not fit, and the line is shown dotted.

The example can be understood more completely by examining contour lines
of total resistance factor F^{'}_{r} as a function of both strand diameter
and number of strands (Fig. 6). The minimum resistance is in the valley
at the upper right (a large number of fine strands). To fit on the bobbin,
designs must be below the dashed diagonal line. Minimum loss designs for
a fixed number of strands can be found by drawing a horizontal line for
the desired number of strands, and finding the point tangent to contour lines.

One could also consider a constraint for minimum wire diameter. Many manufacturers cannot provide litz wire using strands finer than 48 or 50 AWG. On Fig. 6, the minimum resistance for 50 AWG stranding is with a full bobbin, but for 40 AWG wire, the minimum ac resistance can be seen to occur with fewer than the maximum number of strands. This situation can be analyzed by considering (2) with all parameters fixed except for the number of strands, such that

(14)

where is a constant obtained by equating (2) and (14). The total resistance factor is then

(15)

where F_{dc1} is the dc resistance factor with a full bobbin,
for the fixed strand diameter. The value of $n$ that minimizes this
expression is , such that F_{r}=2. This will be
the optimal number of strands, given a fixed minimum strand diameter, unless
this is too many strands to fit in the available window area.