The chosen design may be too specific to my own needs to be of great interest to others. Since the process of getting there can be employed by others writing about it can be useful. I'll step through my process of selecting a suitable design and how I modified it to meet my objectives.
This is purely a design task, but one that I'm taking seriously. I will go through the steps of selecting a taper schedule and employing EZNEC's SDC (stepped diameter correction). The Leeson correction works well and should lead to zero fuss construction and tuning. Hopefully construction will be in the not too distant future.
For contests the antenna must be low SWR across the entire band -- 21.000 to 21.450 MHz -- have reasonable F/B and as much gain as allowed by the other constraints. Since boom length is one of the two primary determinants of maximum theoretical yagi gain, the 32' boom limits the gain to no better than 11 dbi on 15 meters. In any case the stacking gain (~3 db) is greater than going with an exceptionally large, long boom yagi (~1 to 1.5 db). I can afford to focus on better F/B and match than squeezing out every 0.1 db of additional gain.
This antenna is intended for the 45 meter tall tower I hope to raise later this year. I plan to stack them on that tower, initially fixed then rotatable. The most likely stacking arrangement will have the upper yagi at ~35 meters and the lower one at ~25 meters. They should play well towards Europe and other paths. The heights are chosen to avoid interaction with a 40 meter yagi at the top of the tower and to avoid guy wires. The stacking distance is about 0.7λ, which is near ideal for yagis with a boom length of 0.67λ.
Designing 2 and 3 element yagis is straight-forward. The search space of element lengths and spacing is easily explored. As each additional element is added the challenge to discover an optimum design rapidly increases. Even for just 5 elements the optimization search space is already very large.
Lucky for us that yagi optimization has been going on for decades. Look around and you are sure to find a ready-made design to suit your needs. You should only take care to understand what the word "optimum" means in every instance, since for some this is about gain, F/B, wide-band match, cost or wind load, or some combination of these or other parameters. If the objectives for a design are not stated you ought to be suspicious.
I will therefore begin my search with a "stock" design.
Finding a design template
There have been several generations of designs for optimized yagis. This may seem odd since physics has not evolved. It is modelling software and field measurements that have improved. For yagis we can begin with the NBS (National Bureau of Standards, in the US). For hams the next big step is found in W2PV's seminal work from the early 1980s (see the out of print book "Yagi Antenna Design"). We can do even better today. Two that are of greatest interest to me are those in the ARRL Antenna Book and WA3FET's OWA (optimum wide-band array). There are more if you want to investigate further.
Since W2PV elaborated and improved upon the NBS designs we can start there. From his (pre-NEC) software modelling and optimization he found that making the design complicated yielded only small improvements. He therefore standardized on equal element spacing and equal length directors. This limits the variables to boom length and tuning of the parasites. Unfortunately he didn't have much to say about 5-element designs -- he jumped from 4 to 6 elements -- even though that is ideal for a boom of this length. You cannot simply take a 6-element design and lop off the last director! He also did not say a great deal about match bandwidth, and that matters to me and, I think, most hams.
Modern modelling software made it easier to search the vast space of combinations of element length, placement and boom length to do better. Even so the W2PV designs compare favourably. Even so I opted for a modern design. I did so using the ARRL Antenna Book (22nd edition). Nowhere in there will you find a 5-element 15 meter yagi on a 32' boom. Yet this can be an excellent boom length for such a yagi.
Recall that a yagi's gain is primarily determined by boom length, only requiring enough elements to ensure sufficient mutual coupling to make effective use of the boom length. We then vary element lengths and spacing to get the best combination of gain, F/B and SWR for our needs. A 32' boom on 15 meters is ~0.67λ. Gain better than 10 dbi (free space) is achievable, along with good F/B.
By noting that 32' on 15 meters is the same fraction of λ as 48' on 20 meters I chose the 5-element 20 meter yagi on a 48' boom as my first template. This design shows excellent gain, F/B and broadband match across the 20 meter band. It is an easy matter to scale the design to 15 meters. Parasite tuning is ±6.6% for the reflector and director 3, with the other two directors tapering in length toward the centre frequency.
Adapting the design to the boom
I scaled the element lengths and spacing by multiplying by ⅔ (14 MHz divided by 21 MHz) and adjusting fractionally to squeeze down a further 6" by moving director 3 inward. I used the element taper schedule in the ARRL Antenna Book for a heavy duty 15 meter element.
In the model I built one element for the taper schedule and selected a segment length of 6". The segment length for the element tip will vary somewhat from that value. The centre "tube" for the element-to-boom clamp is one wire of one segment spanning the centre of the element and joining the element halves. I then copied the element for the other four elements and adjusted all element tips to the calculated (scaled) lengths.
It is unsurprising that scaling is insufficient since element diameter and taper affect resonance. By inspecting the modelled gain, F/B and impedance I determined that the performance I want at the 21 MHz frequency is found at 20.6 MHz. All the elements were therefore scaled by 20.6/21.0 by shortening the tip lengths. When scaling antennas always multiply and divide by a scale factor; never add or subtract a fixed amount.
Half element taper schedule:
- 3" length of 3" diameter, to account for the 3" x 6" element-to-boom clamp
- 27" length of ⅞" tubing, half of a continuous 54" length for the centre section of the full element
- 36" length of ¾" tubing, which is half of the visible portion of a continuous 112" length that is inserted through the ⅞" tube
- ½" tubing for the element tip, adjusted to the required element length, not including the portion hidden within the ¾" tube
Element half-lengths and spacing from the rear of the yagi are as follows. The tip length is the half-element length minus 66" (the rest of the half-element) plus 3" for overlap.
- Reflector: 144.11"; 0"
- Driven element: 135.7"; 48"
- Director 1: 133.8"; 106.5"
- Director 2: 131.75"; 239"
- Director 3: 126.77; 378"
The hairpin is a shorted ~400 Ω stub 11.5" long, made from ½" tubing spaced 4" centre-to-centre. The stub should be several inches longer to allow for adjustment with a shorting bar. The feed point will out of reach when mounted on the tower, but that's a topic for another day.
With different stub parameters the length will change. The inductance of a short stub is proportional to the length and inversely proportional to the nominal impedance. This handy approximation only applies to transmission line lengths that are short relative to wavelength.
The antenna is remarkably consistent across the band. Unsurprisingly it behaves very much like the 5-element 20 meter yagi it is based on, the performance of which is shown above in the YW screen capture (YW is yagi design software included with the ARRL Antenna Book). A example of the azimuth pattern is shown at right.
The ARRL optimized design has parasite tuning of ±6.6%, measured for the reflector and director 3. The other directors fall in between, and the director spacing is quite wide. This is an antenna that will require wind load and weight compensation so that it is mechanically balanced.
Radiation resistance has a steep drop at the high end of the range where gain reaches a maximum. The antenna is tuned to ensure that this point occurs above 21.45 MHz. Doing so gives us the best conditions for a broadband match with a simple network. R and X components of the feed point impedance vary little across the band.
The completed yagi model has a forward gain that gradually rises from 10.2 dbi at 21 MHz to 10.45 dbi at 21.45 MHz. F/B stayed in a range between 22 and 26 db, which is excellent. F/B was measured at the maximum side (back) lobe rather than the exact rearward direction since the side lobes change position with frequency. I believe this gives the best idea of its QRM and QRN rejection potential.
Plots of gain and F/B appear further below, compared to a variation of the design. These performance figures agree well with the 20 meter yagi this antenna is based on (shown above in the YW screen capture).
The hairpin (beta) match was calculated at the end of the process, once the yagi was completely designed. As can be seen from the EZNEC plot below it does very well indeed without the contortions I went through with the 40 meter 3-element yagi I recently discussed.
The SWR is below 1.3 across the band, which is ideal for broadband transmitters and amplifiers. With a smidgen of transmission line loss the SWR in the shack will be even lower. There is no need for an OWA design and the added load and cost of a couple resonator. However we do want to use low loss coax to ensure that our design and construction effort is not wasted!
It is tempting to push the design further to try and get closer to the theoretical maximum gain of ~11 dbi. We can't push too much or we'll lose the excellent broadband match and F/B. High gain is associated with high Q. It is also inadvisable to play around too much with element spacing and tuning since manual design of this nature is not conducive to getting good result. This antenna is already heavily optimized and very sensitive to small changes. I know, I tried!
We do have some room for adjustment since the match is so good; some increase in antenna Q will still result in an acceptable broadband match. To ease into this I tightened the parasite tuning from ±6.6% to ±5.5% by shortening the reflector to 143.38" and lengthening director 3 to 128.68". This includes shifting the entire yagi (adjusting all elements) down by ~100 kHz in order to centre its best performance within the band. Adjusted directors 1 and 2 are 134.47" and 133.07", respectively.
A further tightening of the tuning to ±5% degraded gain and F/B. Small changes to the lengths and spacing of the other directors degraded gain and F/B. I doubt we can do better even with serious optimization modelling.
The match requires a driven element length of 135.8" and a stub length of 9.8". The SWR is little changed, except for the beginning of a sharp increase at the high end of the band. It is still a very good 1.6.
Comparing the gain and F/B of the two yagis is instructive. While I was able to improve the gain as much as 0.2 db in the CW segment that advantage disappears for SSB, where both yagis are roughly equivalent. F/B is similar for CW and the lower SSB segment and then becomes as much as 4 db worse higher in the band. However the difference is quite small.
In my opinion the differences do not clearly favour one design over the other, and both are excellent. The ±6.6% yagi is more consistent across the band and may be acceptable for that reason alone. Luckily I am in no rush to finalize my choice.
When I get to the point of building these antennas I will have more to say with regard to construction, tuning, installation and stacking. I am looking forward to building this antenna.