Quarter-wave Transformer Accuracy

I built the experimental delta loop for 20 meters over a few hours this past weekend. It now rests up against the house and mast on which it is to be mounted. As you can see in the picture it stands taller than the eaves of the house! I expect the neighbours will be...impressed when it goes up.

Although large it is actually quite light and can be carried around, lifted and put down without much effort. The heaviest part is the fibreglass mast, consisting of 6 x 4' nesting 1.75" OD tubes. I could get by with lighter-weight fibreglass, except I don't have it and this is good enough for an experiment.

The horizontal arm of the loop is the boom of an ancient Cushcraft 32-19 long-boom yagi for 2 meters, slightly extended. The wire is 12 AWG stranded THHN copper wire. Each leg of the loop has been cut to 7.3 meters, which should resonate at 14.075 MHz when mounted 7 meters above ground.

You should be able to see the feed point (white insulator) on the left arm, 25% of the way up the bottom. This is ¼-wave down from the apex, which is what makes this antenna vertically-polarized and omnidirectional.

What is missing from the antenna is the transmission line. This will consist of a ¼-wave of 75Ω coax, complete with common-mode coax choke. Loops have an impedance that can range anywhere from 100Ω to 200Ω, depending on height, shape and position of the feed point. For this antenna, with the bottom leg at around 7 to 8 meters off the ground, the use of 75Ω for the transformer results in an SWR of about 1.1 at resonance (modelled in EZNEC). The primary line to the rig will be RG-213.

Digging through endless rolls and hunks of coax in my basement I was not successful in finding much in the way of usable 75Ω coax. The only RG-59 was in short patch cord. There is a roll of RG-11 but I know it is in questionable condition. I did eventually find one good section of RG-11 which happened to have connectors and was already in line with a longer length of RG-213. That puzzled me for a moment until I realized it was the transmission line to my old delta loop for 40 meters.

Not wanting to destroy that perfectly good ¼-wave transformer I decided to buy some RG-6. This 75Ω coax is widely available and cheap. It is harder to work with than RG-59 (though about the same diameter) since there is a foil layer underneath the outer braid and the centre insulator is foam. This can be challenging since one has to be certain the foil, the inside of which is the "true" RF conductor, is electrically connected to the pigtail or PL-259 termination. Soldering must be carefully done so as to not melt the foam. The foam also increases the minimum bend radius, which is important when winding the coax choke.

Assuming that all of this is taken care of, the next challenge is calculating the length of the transformer. This requires knowing the velocity factor (VF) of the cable. There are a couple of issues. First, RG-6 from different manufacturers has been found to vary in the insulator construction so that the VF can range from 0.75 to 0.9. This is a wide range! In comparison, cables with solid polyethylene center insulators are pretty consistent in having a VF of 0.66.

The wavelength at 14.150 MHz is 21.19 meters. One quarter of this is 5.3 meters. The transformer will be shorter in proportion to the VF. Foam PE has a nominal VF of 0.83, so that is what I started with in my design. This works out to 4.4 meters. However, because of the VF uncertainty the true length could be anywhere between 3.9 and 4.8 meters.

It is possible to test the cable with suitable equipment to measure the reflection time or impedance transformation to determine the exact VF of any length of coaxial cable. But that's a lot of work, and in any case I don't have the needed equipment.

What I did instead was to do a sensitivity analysis with EZNEC. This is very simple and quick. First I set the transformer length to 4.4 meters and the VF to 0.83 (see above). Next I generated the SWR curve across the band. Here is the result:

The SWR has a minimum value of 1.1 at 14.050 MHz (recall that my major operating interest is CW).

I then repeated the exercise with the lowest and highest VF values I might encounter in typical RG-6 cable. The first is for 0.75 and the second if for 0.9 VF.

The SWR didn't change much but the frequency where it dipped did change. For a VF of 0.75 the SWR minimum is ~14.140 MHz and for a VF of 0.9 it is at or just below bottom of the band.

The single-element delta loop is known as a reasonably broadband antenna. Even in the worst case (VF of 0.9) the SWR stays below 2 even near the top of the band.

This sensitivity analysis tells me I have little to worry about in regard to VF of precision measurement of the transformer length. This might not be true for antennas with a smaller bandwidth, which includes any directive array and antennas for 160 and 80 meters.

I will almost certainly have to adjust the antenna length in any case due to environmental factors that are not in the model. The affect of VF is just one more factor, and one that I could not distinguish from any other. Further, should the experiment succeed and I go ahead and add more bands the interactions will affect both resonance and impedance.

This is just a lengthy way of saying that the VF of the coax is not much of a concern: the SWR should end up being good enough. Regardless of the impedance match, the pattern will be unaffected.

Now if I can only find the time to get to prepare the transmission line and raise the antenna. It may have to wait until at least next week.