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I use coaxial transmission line transformers for simple power splitting and combining networks. This is for connecting multiple resonant antennas in array configurations. It's a standard technique that uses a 1/4 wavelength section of coax to match end loads/sources of different impedance. It's become really common in the ham radio designs I'm looking at. Here's a brief note on that: physics behind the impedance matching performed in this case Here is a practical example of one implementation: https://www.qsl.net/dk7zb/PVC-Yagis/2x25_ohm.htm, using 5/4 wavelength cables.

It's so well known and understood (although my theoretical understanding of it is limited), that I wouldn't have a question, except - many designs suggest that odd multiples of 1/4 wavelength should also work, e.g. 3/4, 5/4, 7/4, etc.

I find that intuitively hard to accept for the following reasons. I am somewhat confused and want to know the correct way to implement this. It seems logical to me, that at the end of a 1/4 wavelength transformer, the transformed impedance appears, and so it should be connected only to a coax of the target impedance, not the transformer impedance.

For example, to transform 25 ohm to 100 ohm, I can use 1/4 wavelength of sqrt(25*100) = 50 ohms cable. Call 25 ohms the source, if I want to continue the cable run, I should attach any length of 100 ohms cable to the transformer. That makes perfect sense to me, although I don't fully understand the principle of operation of the transformer itself and just accept that it is true.

But now I'm being told that I could just as well use for example, a 3/4 wavelength of 50 ohms (matching) cable. My conceptual problem is this: I could view that broken down into a series connection of 1/4 wavelength 50 ohms and 1/2 wavelength of 50 ohms. In the first case, there can be a 1/2 wavelength of 100 ohms cable plus any length in addition at the end of the transformer. In the second, there is an actual 1/2 wavelength of 50 ohms cable plus any length of 100 ohms after that.

Questions: How can both cases be valid, or is the theory of odd quarter wavelengths just wrong? If I use lengths like 7/4 wavelengths transformer, what are the consequences? Apparently people are using this without experiencing return losses, but what about signal loss due to the mis-match?

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  • \$\begingroup\$ Clearly the longer ones have more ohmic loss so they are not identical. However they may be "close enough". \$\endgroup\$ Mar 6, 2021 at 19:37

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A transmission line terminated in a mismatched load of a given impedance will repeat that impedance every λ/2 (electrical length) back from the termination (e.g. here). You can think of your odd multiples of λ/4 transformer as consisting of a single λ/4 impedance transformer (using the calculations you've already described) plus an arbitrary number of λ/2 sections which repeat the mismatched impedance back to the λ/4 section.

You can use this to make a longer matching section. So your example of DK7ZB's stacked Yagi array is using a 5λ/4 transformer to allow larger vertical spacing of the individual Yagi antennas.

There are trade-offs of course. A longer transformer means higher losses. Multiple sections magnify variations in wavelength so a longer transformer will have a narrower bandwidth. You may be able to achieve operation on harmonically related bands - a λ/4 transformer on 2 m might be usable as a 3λ/4 transformer on 70 cm, if you're lucky.

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