Enlightenment sought with respect to phasing harnesses. Given that in the general case, in order to achieve 90 degrees difference of phase, you employ two lengths of coax, one measuring an odd multiple of 1/4 lambda, and the other an even multiple. The above 1/4-lambda requirement for both, so I understand, because of employing the coax itself to transform impedance along said 1/4-lamda sections, even and odd. This you do so as to bring the apparent impedance up to 100 Ohms where the even and odd sections meet. That way you create a point where you can connect a 50 Ohms feedline such that the 1:2 ratios of Z balance out. That for this to work, the 1/4 and 1/2 lambda phase feedlines must themselves be 75 Ohms. All this I understand. But what if it were NOT the case, that you needed to be transforming impedance? Suppose, for instance, that instead of two 50 Ohm, you were wanting to phase together a pair of 100 Ohm balanced antennas. And further suppose that you created 100 Ohm balanced feedline from parallel runs of 50 Ohm coax by tying their shields tied together at both ends. You then would have 100 Ohms Z all the way down. Refer, if you please, to the diagram I have attached. Obviously all would be well if all three sets of shielded, balanced 100 Ohms Z feedline were odd multiples of 1/4 lambda. This, I think can't help but work. And here is my question: Seeing as how no impedance transformation needs to take place, might one get by with ONLY the single 90-degree phasing length bing an exact 1/4 lambda? That is to say, is there any mathematical reasoning (as is the case with power dividers providing 180 degrees) that the other two lengths (those connecting to the paired antennas) might be of any random length convenient ... so long as each exactly equaled the other? Or put more succinctly, if not needing to transform impedance, might one get away with having only a 1/4-lambda difference between the lengths? Note on the diagram: I have drawn only the minimum so as to simply for the question. The point of having a box would be for relays to switch the 90 degrees lag from one antenna to the other, reversing its favored direction. The box proclaiming 1:1 is meant to represent a balun. Please kindly answer the question as asked. I know that other ways do exist, and that everyone has their favorite. I'm only asking about just this one simple thing. So I'd immensely grateful comments not going off on every possible tangent.

If I understood you properly: You got two runs of Z=100 line. The length difference of them is ΒΌ WL. The lines are loaded 100 Ohm each, no reactance in the loads. Yes you will have 50 Ohm input in the point where the lines meet, and you have 90 dg difference between the loads. Another matter that we cannot usually have such perfect loads in an antenna, the loads (the loops or the elements) usually have very different impedance, even if they are perfectly equal in physical size. They are different because of the coupling. The difference can be about 6 times or so

The 90 degrees, then, is the main issue? Because what put me onto this track of thinking was reading about power dividers from an article that used wave guides in its examples. Cited very specifically was that, with such an arrangement, the antenna leads need only be equal, not necessarily multiples of any particular wavelength fraction. Of course, those same examples cited 180 degrees for the phasing they sought. Still, the article was comparing against wavelength fractions of coax feedline to do the same thing. The article was very big on that being a principle advantage of power dividers. And reading this, I could not help but wonder.

Your simple phasing arrangement is commonly used for circular polarization, where two antennas are at right angles to each other with minimal coupling.