#### III. DC RESISTANCE FACTORS

The fraction of the window area occupied by copper in a litz-wire winding will be less than it could be with a solid-wire winding. This leads to higher dc resistance than that of a solid wire of the same outside diameter. A cross section of litz wire is shown in Fig. 1, with the various contributions to cross sectional area marked. In addition to the factors shown in this diagram, the twist of the litz wire also increases the dc resistance. In order to find the optimal number of strands for a litz winding, it is necessary to quantify how the factors affecting dc resistance vary as a function of the number of strands.

###### Fig. 1. Cross sectional area of a litz-wire winding, showing how area is allocated. Area allocated to anything other than copper increases the resistance in a space-limited winding.

Serving
Typically litz bundles are wrapped with textile to protect the thin insulation of the individual strands. This serving adds about 0.06 mm (2.5 mil) to the diameter of the bundle. For a given number of turns filling a bobbin, or a section of a bobbin, the outside diameter of the litz wire must be fixed. The area devoted to serving will then also be fixed, independent of the number of strands.

Strand packing
Simply twisted litz wire comprises a group of strands bunched and twisted into a bundle. More complex constructions begin with this step, and then proceed with grouping and twisting the sub-bundles into higher-level bundles. Particular numbers of strands (one, seven, nineteen, thirty-seven, etc.) pack neatly into concentric circular arrangements. However, with large numbers of strands (e.g., >19), and/or very fine strands (e.g., 44-50 AWG), it is difficult to precisely control the configuration, and the practical packing factor becomes an average number relatively independent of the number of strands. Since the optimal strand diameter is typically much smaller than a skin depth, but the lowest-level bundle can be near a skin depth in diameter, in most cases we can assume that there is a large number of strands in the innermost bundle. Thus, this packing factor is independent of the number of strands.

Bundle packing and filler
The way the strands are divided into bundles and sub-bundles is chosen based on considerations including skin-effect losses, flexibility of the overall bundle, resistance to unraveling, and packing density. In some cases, a non-conducting filler material may be used in the center of a bundle in place of a wire or wire bundle that would, in that position, carry no current because of skin effect.

A typical configuration chosen to avoid significant skin-effect losses should have a carefully designed and potentially complex construction at the large-scale level where bundle diameters are large compared to a skin depth. However, because the optimal strand diameter will be small compared to a skin depth, a simple many-strand twisted bundle may be used at the lowest level. If the overall number of strands is increased, the number of strands in each of these low-level bundles should be increased, but the diameter of each low-level bundle should not be changed, nor should the way they are combined into the higher-level construction be affected. Thus, for our purposes, the bundle packing factor is independent of the number of strands.

Turn packing
The way turns are packed into the overall winding is primarily a function of winding technique, and it is assumed not to vary as a function of the stranding. However, note that loosely twisted litz wires can deform as the winding is constructed, allowing tighter packing. Another option providing tight turn packing is rectangular-cross-section litz wire. In addition to its turn-packing advantage, it has tighter strand and bundle packing, as a result of the mechanical compacting process that forms it into a rectangular cross section.

Twist
The distance traveled by a strand is greater in a twisted bundle than it would be if the strands simply went straight, and so the resistance is greater. An additional effect arises from the fact that a cross section perpendicular to the bundle cuts slightly obliquely across each strand. Thus, the cross section of each strand is slightly elliptical. This reduces the number of strands that fit in a given area, and so effectively increases the resistance. An extreme case of this is illustrated in Fig. 2. The choice of the pitch of the twist ('lay', or length per twist) is not ordinarily affected by the number of strands in the lowest-level bundle, and so, for the purpose of finding the optimal number of strands, we can again assume it is constant.

###### Fig. 2. The cross section of strands becomes elliptical when the bundle is twisted. In this extreme case of lay (length per twist) equal to 4.7 times the bundle diameter, a total of six strands fit where seven would have fit, untwisted.

Strand insulation area
Thinner magnet wire has thinner insulation. However, the thickness of the insulation is not in direct proportion to the wire diameter. Thinner wire has copper in a smaller fraction of the overall cross sectional area, and insulation in a larger fraction.

Of all of the dc-resistance factors considered, this is the only one that varies with the size or number of strands used at the lowest level of the construction. Thus, quantifying this effect on dc resistance gives a good approximation of the total variation in dc resistance as a function of the size or number of strands. The other factors can be lumped into an overall dc resistance multiplying factor which is a constant for the present purposes.

One approach to quantifying the relationship between the insulation area and strand diameter would be to store tables in computers, and use them to find the optimal strand diameter by calculating the losses for different strand diameters until the optimum was found, similar to [6]. However, an analytical description of the variation of insulation thickness with wire size can facilitate an analytical solution for the optimal number of strands.

###### Fig. 3. Insulation build (twice the insulation thickness) for American Wire Gauge (AWG) single-build magnet wire. The dashed curve is the minimum build according to the equation provided by [18]. The lower "staircase" curve is the tabulated data provided by [18] for minimum build' The points marked by 'x' are nominal build from a wire manufacturer¹s catalog, obtained by subtracting tabulated nominal overall diameter and subtracting the exact theoretical nominal wire diameter. The approximation described by (4) comes closest to these points.

An equation describing magnet wire insulation thickness is provided by [18]:

(3)

where B is the minimum insulation build in mils (10-3 in, mil = 25.4 ), X=0.518 for single-build insulation and X= 0.818 for heavy (double) build, and AWG is the American Wire Gauge number1. However, this only applies to wire sizes between 14 and about 30 AWG. For smaller wire sizes, it does not correlate with the tabulated data in [18] (Fig 3). For wire in the range of 30 to 60 AWG, we find a better fit to manufacturers' tabulated nominal insulation build by using

(4)

where dt is the overall diameter, including the insulation thickness, dc is the diameter of the copper only, and dr is an arbitrarily defined reference diameter, used to make the constants unitless. The parameters used for single-build insulation wire were = 0.97 and = 1.12 for dr chosen to be the diameter of AWG 40 wire (0.079 mm). For heavy-build insulation, =0.94 and =1.24. Note that although (4) provides an accurate approximation for wire in the range of 30 to 60 AWG, its asymptotic behavior for large strand diameters is pathological. Insulation thickness goes to zero around 6 AWG, and is negative for larger strands.

1The American wire gauge defines nominal wire diameter in inches as d=0.0050 (92)36-AWG)/39

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