Coaxial Cable Twelfth-wave Transformer calculation
Updated 11/07/2019
A solution for matching two coax lines of 75 and 50 ohms is based on a web article by Darrel Emerson, AA7FV. The article is now located at https://www.cv.nrao.edu/~demerson/twelfth/twelfth.htm. AA7FV stated that his article is a summary of the article published in QST for June 1997. He referenced an original article published in 1961, "A Convenient Transformer for Matching Coaxial Lines" in Electronic Engineering, Vol. 33, pp 42-44.
To avoid having to do the math, which involves arctangents and such, I developed a spreadsheet which is downloadable from this website. The spreadsheet includes a diagram of how the matching sections are placed in the feed line, so that the person doing the calculation entries have something to refer to. Note that in the case of a dipole with an approximate feedpoint impedance of 70 to 75 ohms, fed with RG-6 or other 75 ohm coax, only one 75-to-50 ohm 12th-wave transformation is needed, at the shack entry end of the feedline. If the antenna feedpoint is 50 ohms (such as a gamma match of a beam antenna) using 75 ohm coax as the main part of the feedline, two 12-wave transformers will be required: one for 50 to 75 ohms at the antenna, and one for 75 to 50 ohms at the shack end of the feedline.
For those who use 0.405" 75 ohm coax for the feedline, RG-11U or equivalent is available. For those who want to save money on the feedline, low loss RG-6 CATV or satellite coax is good. RG-6 typically uses F connectors. For a good reference for RF connectors and adapters is Max Gain Systems at https://mgs4u.com. To simplify the 12th-wave transformer coax cables and their connectors, let's let's assume that a good quality 50 ohm RG-8X and a good quality 75 ohm RG-59 are used for the transformer sections. The main reasons are (1) these are each is available with at least 95% copper shield braid, and (2) both are able to use UG-176 adapters for PL-259 UHF solder-type male connectors. Short double-male UHF adapters are used for joining the transformer sections. The 75 ohm RG-6 feedline with its aluminum shield braid could use a single F-to-PL-259 male adapter; or two, if one is used to connect RG-6 to a UHF female dipole center. An F male connector for RG-6 such as a Belden Snap & Seal compression type could be used there. An inexpensive crimp RG-6 F connector might work at the shack end if the feedline is not under stress at the point of the transformation. Otherwise, a compression type could be used. One source of RG-59 having 95% coverage copper braid is The Wireman CQ135-RG59 B/U. For those who prefer 0.405" coax, UHF and N connectors and adapters are available.