Coax impedance
The characteristic impedance of a coaxial line depends only on the ratio of its two diameters and the dielectric between them, not on its length:
C = 55.6 × εr ÷ ln(D ÷ d) pF per metre
D is the inside diameter of the outer conductor (the outside of the dielectric) and d the diameter of the inner conductor. A 2.0 mm inner conductor in 7.0 mm of solid polyethylene, a ratio of 3.5, works out at 50 Ω and about 100 pF per metre, which is close to RG-213. Use the diameter of the dielectric for the outer conductor. A stranded inner conductor behaves like a solid one slightly thinner than its overall diameter, so measured stranded cables come out a few ohms low.
The calculator also gives the frequency where the cable stops behaving as simple coax because a second mode can travel in it. Well below that frequency is fine: for RG-213 it's well above 10 GHz, but for large hardline it can fall to a few GHz.
Twin line and ladder line
For two parallel round wires, the impedance depends on the ratio of their centre spacing to their diameter:
The exact arcosh form is used here, which matters when the wires are close together. Real ladder line has plastic spacers and insulation, so its effective permittivity is a little above 1. Window line sold as 450 Ω typically measures 360 to 400 Ω with a velocity factor of about 0.9. Open wire line with spreaders every metre or so is close to the air value.
Making your own line
Home-made lines are useful for matching sections, phasing lines and Q-sections. Copper tube inside copper pipe with PTFE spacers makes excellent low-loss air-spaced line for VHF and UHF, and two wires held apart by spreaders make open wire feeder for HF. After working out the dimensions, check the impedance with a NanoVNA: terminate the far end with a resistor of the expected impedance, and the input should read close to that value across the band. The coax stub calculator works out the length of a quarter-wave matching section.
Questions
How do I work out the impedance of coax?
Measure the inner conductor diameter and the diameter of the dielectric. Z0 = 138 ÷ √εr × log10(D ÷ d). For solid polyethylene, εr is 2.25.
What D/d ratio gives 50 Ω?
About 3.5 with solid polyethylene and 2.3 with air. For 75 Ω, about 6.5 with polyethylene and 3.5 with air.
What impedance is 450 Ω window line really?
Usually 360 to 400 Ω when measured, with a velocity factor of about 0.9, because the plastic web raises the effective permittivity.
Why is coax 50 Ω?
It's a compromise. Air-spaced coax has its lowest loss at about 77 Ω and handles the most power at about 30 Ω. 50 Ω balances the two.