Length of wires after they are twisted?

Question

I have looked all over and haven't seen any questions about this topic. This may be more of a mechanical question, but none the less, the dilemma on hand is how does twisting a set of equal length wires affect the end length?

Through some side work, customers have requested a certain twist per inch in their builds. For example, some customers have requested 2 twists of wire per inch of length such that 100" of wire will have 200 twists.

What I'm trying to get an idea of is some kind of equation that can be used to calculate how long of a wire I need to cut and how many twists will be needed to achieve the desired length of wire. I'm considered factors such as AWG, insulation thickness, and how many twists per inch (ranges from 1 twist to 4 twists).

Approaching it from a geometrical standpoint the helical length equation, \$L = \sqrt{H^2+\pi^2D^2} \$. Where L is the length of wire needing to be cut, H is the desired end length, D is the diameter from each wire core center. My trouble comes now trying to figure out how to go from the helical length equation and transforming it what I need... Any suggestions on where to start?

Answer 1

http://inductor.thayerschool.org/papers/stranded.pdf, page 290, equation 4 shows a method of calculating the length factor for twisted wires.

$$ {l_{strand} \over l_{bundle}} = {1 + {{\pi^2 n d_s^2} \over {4 K_a p^2}} } $$

Where:
\$ l_{strand} \$ = actual length of the wire strand
\$ l_{bundle} \$ = length of the twisted bundle
\$ n \$ = number of strands
\$ d_s \$ = diameter of each strand
\$ p \$ = pitch length
\$ K_a \$ = packing factor = \$ A_e / A_b \$
\$ A_e \$ = sum of the cross-sectional areas of all the strands
\$ A_b \$ = overall bundle area

A test I ran on some 25/40 HPN litz wire, my measurements gave a ratio of around 1.0083 to 1.0125. The equation predicted 1.0075. Bundle diameter was taken from the MWS Wire "Litz wire" catalog.

Answer 2

One strategy is to have the twisted wire prepared using a wire twisting machine that works off long spools of the straight wire and then meter out the required length and cut it as needed.

Another strategy may be to approach this from an empirical standpoint and take apart already stranded wire (or prefab your own and then take that apart) and measure the length of the wires when smoothed out.

Lastly via some simple experiments you could determine a small percentage overage and when a customer askes for 200 feet sell them 210 feet and let them trim.