Zo of two wire line

I saw a recent discussion where the blind were leading the blind on the dimensions of a twisted two wire line for Zo=50Ω for use in a balun.

The poster had used an online calculator which used the well known log function for estimating Zo of an air spaced two wire line… the calculator, like most quotations of the formula do not state clearly that it is only an approximation of limited validity, and the calculator returned results for ridiculous inputs (like negative spacing).

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The graph above (Duffy 2008) shows the log approximation, and the underlying acosh based estimate. I say estimate because the acosh function does not account for proximity effect which becomes significant at the very closest spacings, and internal inductance which becomes significant at lower frequencies. Proximity effect depends on more than just the spacing/diameter ratio and so cannot be shown on the above graph.

So how did our poster find dimensions for wires for Zo=50Ω when the log graph above shows that as the wire centre to centre spacing approaches the wire diameter, it the wires approach touching, Zo approaches 83Ω?

Easy, he specified a centre to centre spacing less than the wire diameter, ie the wires were not just touching but significantly shared the same volume… and the silly calculator came up with 50Ω.

Based on measurement experience and (Duffy 2011, Duffy 2001), a pair of 1.6mm diameter wires with 0.05mm enamel (ie heavy build) spaced 1.7mm (ie in contact) should have Zo around 35Ω… but you are unlikely to achieve that spacing consistently when winding onto a ferrite toroid.

In this day of computing, calculating the acosh function is not difficult and it is simply slackness that leads to a calculator that isn’t accurate below about Zo=200Ω, it is just a trap for the gullible.

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