#### IV. NUMBER OF STRANDS FOR MINIMUM LOSS

With no constraints on number of strands or strand diameter, the minimum-loss
design will be with a full bobbin. Any design that does not fill the bobbin
can be improved by increasing the number of strands, by a factor s and
decreasing the strand diameter, d_{c} by . This
keeps the dc resistance constant and decreases ac resistance, as shown by (2).
This improvement can be continued until insulation area increases result
in a full bobbin. Thus, the minimum-resistance solution fills the bobbin, and
we can find this solution by analyzing a full bobbin.

For a full bobbin, the outside diameter of the complete litz bundle
is,

(5)

where b_{b} is the breadth of the bobbin, h is the height
allocated for the particular winding under consideration, N is the number of
turns in that winding, and F_{p} is a turn-packing factor for turns
in the winding, expressed relative to perfect square packing
(For F_{p} =1, the litz bundle would occupy of
the window area).

Assuming a factor F_{lp} accounting for serving area, bundle
packing, any filler area, strand packing, and the effect of twist on
diameter, we can find the outside diameter of a single strand

(6)

where n is the number of strands in the overall litz bundle.

The diameter of the copper in a single strand can then be written using (4).

(7)

We now define a total resistance factor

(8)

where F_{dc} is the ratio of dc resistance of the litz wire to
the dc resistance of a single strand winding, using wire with the same
diameter as the litz-wire bundle. Using (6) and (7), we
can show

(9)

Combining (2), (8), and (9) results in

(10)

where

(11)

Equation (10) can now be minimized with respect to n$ to find the optimal
number of strands.

(12)

This will give non-integral numbers of strands; the nearest integral
number of strands can be chosen to minimize ac resistance.

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