#### 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, dc 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 bb 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 Fp is a turn-packing factor for turns in the winding, expressed relative to perfect square packing (For Fp =1, the litz bundle would occupy of the window area).

Assuming a factor Flp 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 Fdc 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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