3-element 40 Meter Yagi on a 40' (12 m) Boom

My recent article about long boom yagis reminded me of an omission in my blogging practice. The 15 meter and 20 meter yagis had references to earlier articles or elsewhere so that readers could learn the details of design and performance. I did not do the same for the 40 meter yagis on a 40' (12 meter) boom.

The designs were done some time ago, which is probably why I forgot about them until then. Not long before I wrote that article I sent the EZNEC files to a ham in Europe who was inquiring whether I had a design handy for a 12 meter boom. But in that article I only mentioned them in passing. It is worth writing about them because they do have interesting performance characteristics in comparison to the usual 48' (14.7 meter) boom length commonly used for 3-element 40 meter yagis, including those with a fourth element (coupled resonator, OWA design).

Before diving in it is worthwhile to briefly recap a few points about 3-element yagi design.

• Optimizing for gain only modestly reduces F/B but will significantly reduce the SWR bandwidth. The latter is due to the low radiation resistance associated with maximum gain.
• Frequency of maximum gain is above the usable bandwidth, while F/B tends to peak towards the low end of the usable bandwidth. In practical terms, a 3-element yagi typically has maximum gain on SSB and maximum F/B on CW.
• Designs which maximize SWR bandwidth sacrifice up to 1 db of gain. However F/B can remain good.
• Gain is optimized by bringing the self resonance of the director and reflector closer together. The spread in percentage is a convenient metric, one that I originally learned about in the venerable Yagi Antenna Design book by W2PV. Tuning of the driven element affects the match (and matching network) while having negligible effect on gain, F/B and SWR bandwidth.
• Best performance typically has the driven element offset toward the reflector end of the antenna a small amount.

The addition of a fourth element as the coupled resonator in a high bandwidth OWA design increases the mechanical complexity and load, with the advantages of low SWR from 7.0 to 7.3 MHz and little to no sacrifice of gain or F/B. The worth of the trade off is an individual decision. I include 3-element and 4-element designs for both 15 meter and 12 meter boom lengths in this article.

Overview of the yagis to be compared

The 3-element yagi on a 48' boom in the ARRL Antenna Book is a useful baseline for comparison because of its balance between gain and SWR bandwidth. If that antenna interests you, look there because I won't get into the details here. To summarize, its 2:1 SWR bandwidth is ~200 kHz, gain rises from ~7.8 dbi at 7.0 MHz to ~8.7 at 7.3 MHz, and F/R is 21 db at 7.0 MHz, rises to 27 db at 7.1 MHz then falls to 12 db at 7.3 MHz.

The tuning spread of the parasitic elements is 15%, or ±7.4% of the design midpoint. I will use the latter convention in this article. It is not half of 15% since the exponential mean must be used. You get this with the square root of the ratio between the reflector and director lengths. This is good enough for our purposes though not absolutely precise since the tubing taper schedule has an effect.

For comparison, the 3-element yagi I discussed in an earlier article has  a tuning spread of 6.4% to increase gain by ~0.5 db. F/B continues to be very good, although 2:1 SWR bandwidth is a little less than 200 kHz. With a coupled resonator added the SWR bandwidth is low from 7.0 MHz to 7.3 MHz.

Because reducing the boom length to 40' also reduces achievable gain I further tightened element tuning to 5.3%. This recovers much of the gain while sacrificing SWR bandwidth, as we'll see. The addition of a coupled resonator helps with the latter.

As you likely noticed I have reverted to English units for these yagis. I do this because the tubes and pipes I use are measured in these units and so I model the yagis accordingly. The figures on the left are the distances along the boom the elements are placed and the lengths of each half element. The addition of a couple resonator only affects the length of the driven element. Its length without the coupled resonator is in any case dependent on the matching network: gamma, beta, L-network, etc.

Half element tubing schedule is: 144" of 1.9" pipe; 96" of 1.5" tube; 66" of 1" tube; 66" of ⅞" tube and a variable length of ¾" tube. This is not likely how I would construct the elements. It was an experiment to determine how I could put my stock of aluminum pipe to good use. The idea was spurred by reading Dave Leeson's Physical Design of Yagi Antennas -- it's out of print but the author kindly sent me a copy. For other tapering schedules it is necessary to carefully scale the elements.

For interest I show the 4-element yagi element currents at a higher frequency to highlight one aspect of how the coupled resonator broadens the SWR bandwidth. With the coupled resonator tuned to a higher frequency its current dominates that of the driven element at the high end of the band. In a sense it becomes the driven element despite not being attached to the transmission line. Since it is offset toward the front of the yagi the gain and F/B are slightly affected at higher frequencies.

Performance comparison

In the chart I used the same colour lines for both curves of each antenna -- gain and F/B -- to make it easy to compare yagis. Gain and F/B ought to be easy to distinguish!

I chopped the top of the chart to further improve readability since F/B figures well above 30 db are very difficult to achieve in real antennas due to the precise cancellation of element fields required. It is sufficient to state that F/B is excellent over a portion of the band.

Gains of the 40' boom yagis are about 7.8 dbi at 7.0 MHz and 8.4 dbi at 7.3 MHz. Gains of their 48' boom sisters is a fairly consistent 0.7 db higher. That isn't a large sacrifice to make considering the substantial construction differences for equal robustness.

F/B is significantly better for the 40' boom yagis even though I hadn't planned for that. It may be that further tuning of the 48' boom yagis would erase some or most of the difference. Certainly the yagi in the ARRL Antenna Book does a little better than these gain optimized yagis.

With a fixed matching network the 2:1 SWR bandwidth of the 3-element yagi on a 40' boom is 180 kHz. That's good but not great. With a switchable L-network at the feed point it is possible to achieve a low SWR up to 7.25 MHz. Getting all the way up to 7.3 MHz would require one more switch position. I wouldn't bother though some might want it. At least in the Americas where we have a 300 kHz wide 40 meter band.

The coupled resonator version of the 40' boom yagi can achieve better than 2:1 SWR across the entire band. However I was unsuccessful bringing down the SWR lower than 1.4. Further tuning of the driven element and coupled resonator might do it, but there are no guarantees. My difficulty was exacerbated by NEC2 which has some difficulty correctly modelling the impedance of closely spaced elements even when great care is taken with element segmentation, as I did.
Perhaps in practice it'll do better than the model and can be tweaked once on the tower -- the driven element and coupled resonator are within easy reach. Otherwise it may be necessary to loosen the tuning of the parasitic elements to improve the SWR, at the expense of some gain. I did not explore this path for the present study.

Further work

Despite my best intentions I will not construct and raise a full size 40 meter yagi this year. Maybe not even next year. There is therefore time to play with the models and explore further options. For example, to give up some gain to recover SWR bandwidth. No matter what I do there will be have to be compromises made.

I hope the information in this article spurs a few ideas of your own. These antennas are far more challenging than the 40 meters wire yagis that have been so popular among the readers of this blog.