At the suggestion of a local antenna elmer, I started experimenting with simple yagis, but rapidly bogged down, as I was having trouble keeping track of the effects of making various changes and also having things converge, so to speak. But software is my profession, and so I wrote a Perl front-end to talk to NEC2, that prepared NEC input 'decks' [yes, like me, it goes back to the old FORTRAN punched card era (barely in both cases)], condensed the output, and with the aid of some C code, plotted those results to a .gif file (or as a .cgi).
Then I started by trying a simple experiment. Now, we know roughly how a quad loop behaves, and how a folded dipole acts, but what about things in between? I remember an article about stretched quads, so I thought my first experiment would be to examine what happens to the impedance and the pattern from a folded dipole, to a conventional quad loop, and then beyond into structures that perhaps don't have names because they aren't very useful. The results were that the folded dipole impedance looked plausible, the conventional quad was only close to what I had expected, but the impedance formed a nice curve (which I have yet to actually plot). But it also showed that the impedance got close to 50 ohms at about a 3:7 aspect ratio. So, I went looking for that article, but it turned out to be for a different structure. But there was also a recent QST article (reproduced in the current ARRL Handbook for 10 meters), and it was indeed saying roughly that my experiment predicted, that is, that at about a 1:2 aspect ratio, the impedance was about 50 ohms and it had a decent pattern as well. So, already, this approach got me some interesting results.
So the next step was to examine the three element yagi. Now, this examination seems to usually be done at HF frequencies, so I decided to pick 2 meters instead, and fairly narrow element diameter, (e.g. 1/8", since that's what I had lying around). But, then, how do I tune it? It occurred to me that resonance usually happens roughly where the imaginary component of the complex impedance crosses zero (or else at the maximum transmitted power). Now, I could compute the frequency response, and adjust the length of the driven element to attain resonance. But, since I am doing this in software, I may as well scale the whole antenna. But I'd rather not change the boom length, so I scale in the same manner as scaling for element diameter. So, I look for the resonance and scale length of all of the elements by the ratio of the predicted vs. the desired frequency. That gets me into the right ballpark, so to speak, and so I have made this part of my customary experimentation. To summarize, I construct the antenna in software, scale element lengths to resonance, and look at the results.
In the first experiment, I used the approximate dimensions given in ARRL Antenna Book [16th Edition] as a starting point, and looked at what happened when I moved the driven element with respect to the reflector and director, as suggested by Figure 16 of the chapter on Yagis, but with a fixed director length and an overall length of 0.4 wavelengths instead of 0.3 wavelengths, since there is little reason to keep things that short at 2 meters. Indeed, it behaved as noted, with the impedance peaking in a similar place. But 25 ohms does not present a good match to 50 ohm coax, and since, unlike HF, there's little reason to keep the boom length short, let's look at the boom length. (Note that not all graphs are included here in the interest of disk space, network bandwidth and readability; for example, here's the original for the next experiment).
So, in the second experiment, we see that the gain peaks with a boom length of somewhere between 0.4 and 0.5 wavelengths, and impedance increases with boom length, at least, within the range that produces plausible gain. At 0.6 wavelengths, the impedance is about 50 ohms, and at a cost of about 0.5 dB, which I might be willing to give up for sake of simplicity. So, let's rerun the first experiment at that boom length.
So, the third experiment, we see that there is a trade-off between gain and front-to-back ratio (F/B), and that the impedance peaks at what looks like a possible compromise between gain and F/B ratio. And 50 ohms is between that and the peak in gain, and is perhaps a better choice.
Let's look at the effects of changing single elements. Changing the reflector spacing shows a trade-off between gain and impedance, with F/B ratio increasing in the same manner as the gain, with the best gain (and F/B ratio) at about 8 ohms. This raise the question of using a folded dipole or other means of impedance transform, but that is beyond the scope of this discussion so far. Changing the reflector length does not have much effort on the impedance, but seems to produce a trade-off between gain and F/B ratio.
Changing the director spacing has a large effect on F/B ratio, and shows a trade-off between F/B ratio and impedance. It also seems not to have much effect on the forward gain. Changing the director length also does not have much effect on the impedance, and with a mild trade-off between gain and F/B ratio. That change seems to mostly affect the F/B ratio. Changing both lengths in opposite directions tends to minimize effects on the resonant frequency, but we're not worrying about that here. That also has a minor effect on gain, with F/B ratio seeming to be most directly affected.
Now, we've been able to vary things in a number of ways, but have not come across which makes a very large improvement on the ARRL Antenna Book suggestion. It also appears now, from looking at varying other parameters than boom length, that in trying to match directly, we'll be giving up about 1 dB of gain and that's a fairly large price for drive simplicity. Now, if we're going to make two yagis, and either stack them or make them circularly polarized, then 25 ohms might be another value to try for. But, as a post-script, we can examine what happens if we use a folded-dipole as a driven element. This show more promise, but then, how do we adjust it? We can't just make the driven element a little long and trim it to obtain resonance or good SWR in the band. So, it's intriguing, but perhaps best left to the experts, who probably have better ideas. It looks like more experiments are in order.