#### V. DESIGN EXAMPLES AND SUB-OPTIMAL STRANDING

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 Fp=0.85, and litz packing factor Flp=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 Fr = 1.03, and a dc resistance factor Fdc =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 Fdc 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.

###### Fig. 4. Total resistance factor, F'r, as a function of number of strands (solid line) for the example discussed in the text at 375 kHz. Also shown are the ac resistance factor Fr (dashed) and the dc resistance factor Fdc (dotted). The minimum total resistance factor is at the point where increases in Fdc balance decreases in Fr with increasing number of strands.

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 Fr 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.

###### Fig. 5. Total resistance factor, F'r, as a function of number of strands for the example discussed in the text at 1 MHz. The solid line indicates resistance factor for a full bobbin. The dashed line shows the lower resistance that is possible by choosing the strand diameter for minimum loss, with the number of strands fixed. Where this optimal diameter results in a full bobbin, the two curves are tangent. For larger numbers of strands, the optimal strand diameter, shown as a dotted line, would over-fill the bobbin, and so is not possible.

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.

###### Fig. 6. Contour lines of total resistance factor, F¹r, as a function of number of strands and diameter of strands, for the example discussed in the text at 1 MHz. The diagonal dashed line indicates a full bobbin. The valley at the upper right is the minimum loss. The minimum loss without over-filling the bobbin is marked by an Œx¹. Contour lines are logarithmically spaced.

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 Fdc1 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 Fr=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.

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