2-element Narrow Diamond Loop Array for 40 Meters

In my recent survey of 2-element loop arrays for 40 meters I focussed on the potential but not the actuality of real-world designs. In particular, getting these antennas to a 50 + j0 Ω match requires effort I was not willing to undertake without first knowing their performance. That is, I focus first on the pattern and then, and only then, do I consider the match. Never confuse the relative importance of pattern and match: if the pattern doesn't meet your particular operating objectives the match is irrelevant. In other words, a nicely-matched antenna is nice but if that's all you want you can invest in a dummy load or, for that matter, a variety of poorly-performing commercial antennas. Thus the title of this blog.

In this article I take the most promising of the 2-element loop arrays, the one made from narrow diamond loop elements, and turn it into a usable design. The objectives of the design include:

• Achieving an SWR close to 1.0 at the frequency where gain is maximum.
• Favour the CW segment, my favourite.
• Pattern switching, from one broadside direction to the other.
• Mechanically robust.
• Reliable performance.

Of these objectives the last is the most difficult. Vertically-polarized antennas on a metal support (e.g. tower) are prone to interaction with the tower and cabling that runs along the tower. The performance of a directive array is sensitive to any resonances, which can reduce or obliterate the theoretical performance. Unfortunately these interactions are installation dependent, so the best I can do is provide some insight on what to watch for. I cover this topic last.

Now on to the design specifics. I took the "raw" design from the survey article and modified it in the same fashion as I did for the switchable wire yagi antennas. The challenge was to move the array's resonant frequency to coincide with that for maximum gain, and to change the impedance to 50 + j0 Ω. While I did not show it in the survey article, with the maximum gain at 7.000 MHz the resonant frequency is 6.915 MHz (57 + j0 Ω).

The survey antenna and the one described here have an apex height of 15 meters. The supporting structure is assumed to be non-conducting or sufficiently non-resonant to allow it to be absent from the model. This is a matter we'll come back to later, as promised above. The boom length is 7 meters, which is the optimum length discovered in the survey article.

It took close to one hour of adjusting the design on EZNEC to get the frequencies of resonance and maximum gain on the same frequency. The antenna is optimized for CW. For a 2-element parasitic antenna using a reflector element this allow best performance for CW (lower part of the band) while still performing reasonably well higher in the lower part of the SSB segment.

Each loop is 40.46 meters in circumference (10.12 meters per side) when made from 12 AWG insulated copper wire. This is smaller than the 43.9 meters circumference of the survey antenna. As I indicated in the survey article the loop size in a switchable design would be smaller because of the inductive loading of the transmission lines running from each element to the switch box. With an apex height of 15 meters the bottom of the antenna is up 3.94 meters for interior angles of 120° at the top and bottom.

To keep the switch box close to the element feed points I assumed a rigid 7 meters long "boomlet" running between those corners of the elements. This allows for 3.5 meter runs (the minimum possible) of 300Ω ladder line (0.9 velocity factor). A switch box placed anywhere else, such as on the tower, requires longer runs of ladder line, thus smaller loops and poorer performance. The right angles between ladder line and elements also reduces the potential of common-mode currents on these lines. The boomlet not only supports the switch box it also serves to reliably hold the array's shape, as will be described later.

The resulting array has a maximum gain that is only -0.03 db inferior to that of the full-sized elements of the survey antenna, which is negligible. Maximum gain of 3.44 dbi at 10° elevation is set for 7.02 MHz. The maximum F/B is about 60 kHz higher, just as it was in the survey antenna.

The SWR at resonance is 1.1 (56 + j0 Ω) at 7.025 MHz, and slightly lower at 7.015 MHz where it reaches its lowest value. As can be seen the Q of the antenna is higher than that of a single loop antenna, and the SWR rises significantly away from resonance. Since the rise is quite sharp at lower frequencies it is best to set this frequency close to band edge. The SWR (and gain) degradation is more gradual at higher frequencies.

Unlike the 2-element yagi designs there is no need for a beta match or other matching network to match the antenna to 50Ω coax. The naturally high radiation resistance of a full-wave loop combined with the array's high Q takes care of the match for us!

To tune the reflector there is either a transmission line stub (again, using 300Ω ladder line) or a tapped coil inside the switch box. The length of the reflector stub, which is shorted at the far end) is 0.8 meters. It will have to be tuned when the antenna is first installed. Since the design, construction and use of the switching and tuning system is, aside from the removal of the beta match, the same as that for the switchable yagi designs I will point to that earlier article rather than repeat the material here.

The gain and F/B performance figures are nearly identical for the survey version of this antenna, right across the band. The plot stops at 7.2 MHz to capture the frequency range where the antenna performs best. To get both the gain and F/B on the same plot the gain is multiplied by 10. As mentioned above, the maximum gain is 3.44 dbi at 7.02 MHz (shown on the chart as 34.4).

Construction

Next, let's turn to the construction of the array. No matter how you look at it a switchable loop array is more structurally complex than a switchable wire yagi. On the other hand it is electrically simpler, mostly due to the ease of matching to a coax feed line. The feed point is also closer to the ground, and can be made closer to the shack by suitable selection of which side of the array to feed. Although transmission line losses are small at 7 MHz there is the expense of a long run to be considered.

The antenna boom is assumed to be tower mounted and trussed in much the same way as was done for the wire yagis. The boom is 1 meter longer and is not cluttered with a feed system. The element bottoms can be secured with a similar type of boom (boomlet) lower down the tower, though trussed from the bottom since the tension is upward.

The two corners of each element are more difficult to secure. While a simple tie-down rope can be used it must be run at an angle shallow enough to apply tension to both the lower and upper quarters of each loop, and to precisely position each corner. Symmetry and parallelism are paramount in a switchable array.

For the side containing the feed and switching system it is preferable to use a boomlet as described above. It can be used for the other side of the array as well. It should be both lightweight and rigid.

Consider the design at right. The boomlet is divided into 3 sections with truss ropes tied at each of the 4 edges. The centre third is aluminum tubing and the outer thirds are telescoping lengths of plastic or fibreglass. The aluminum tube and the rope truss ensure rigidity when the elements are tensioned.

The 4-point tying to a common point, then utilizing a common tie-down rope to a suitable anchor, is similar to that for the diamond vee wire yagi.

The switching and feed system can be mounted on the boomlet as shown at right. The coax can use one of the adjacent truss ropes for support (not shown, so reference the previous drawing).

Other construction details for the boomlet plus switching and feed system, and the tuning procedure, are as earlier described for the switchable wire yagis.

Tower Interaction

By feeding the array at one corner and the use of a common-mode choke in the transmission line there should be negligible interaction with the coax. Horizontally-polarized yagis mounted atop the tower may noticably interact with the 40 meters loop array, but the reverse is typically not a concern. However the tower and cables running along the tower can significantly interact with the vertically-polarized loop array.

How large an interaction we can expect? There is no simple answer since every installation is different. There is not only the resonance of the tower but the tower as loaded by rotatable yagis above the the tower. Even if the antennas are electrically isolated from the tower (e.g. Spiderbeam), the vertically-run feed line, and its capacitive coupling to the tower, can still interact with the loop array.

To gain some insight into the interaction I modelled a tower running up the centre of the array. This is an imperfect model for several reasons:

• The effective diameter of an open lattice tower is not the same as a solid cylinder.
• The tower may be directly connected to ground (lightning protection), but NEC2 does not support this configuration.
• Cables and yagis on the tower will, in general, increase the effective length of a tower since it all looks like capacitive loading.

In my model I made the tower a 60 cm (2') diameter conductor that starts just above ground and is then varied in height. I started at a height of 15 meters (the minimum to support the loop array) and went up to 25 meters. Even if the tower is shorter than 25 meters the capacitive effects of cables and yagis can easily result in a similar effective height.

What I found is that at 15 meters height the interaction was modest. The gain was not affected in the CW segment, gain dropped -0.5 db near 7.2 MHz, F/B was slightly reduced, and the SWR surprisingly improved above 7.15 MHz.

At 25 meters height the interaction was very minor, enough so as to not be a concern.

Between 16 and 23 meters height the interactions range from modest to debilitating. The peak effect (induced current on the tower -- see EZNEC current plot at right) is around a height (or effective height) of 18 to 19 meters. At these heights the impedance is a mess as the following SWR curve demonstrates. Gain is reduced by -1 db or worse and F/B is poor to non-existent across much of the band.

Unfortunately it is difficult to predict what to expect in any particular installation. With luck the effective height of the tower will be sufficiently raised by capacitive cable/yagi interactions to avoid the danger zone.

Update Feb 26: I ought to have mentioned that it is possible to detune a tower with a resonance on one particular band. I skipped over this possibility since I didn't have or know of any good references, and assumed it might be too difficult. One that I since found is this article by W8JI. It's worth a look.

Conclusions

At a height of 15 meters the 2-element narrow diamond loop array has ~1 db better gain than its structurally-closest wire yagi competitor, the inverted vee wire yagi. When account is taken of structural complexity, uncertainty regarding severity of interactions, or the ability to place either antenna at an apex height above 15 meters, the wire yagi may be the better choice.

Where the loop array excels are F/B and feed simplicity, which may sway one's decision. The choice is yours. As they say on the internetz: YMMV. I'll be carefully thinking this over before deciding what if any wire array to build when I am in a position to do so. That time may come later this year.

ARRL DX CW 2014

In my 2013 summary I said that one reason to quickly repair the mast for the multi-band inverted vee was to be ready for the ARRL DX CW contest in February. That weekend event has now come and gone. Although this is not an antenna article it is relevant since, after all, what is the use of antenna if not to make lots of contacts? This article is one ham's perspective on the DX contest possibilities with 5 watts and a few wire antennas at a modest height.

Unlike the CQ Worldwide CW contest in the fall, I decided to get beyond casual and dedicate the weekend to the contest. I took many breaks and got plenty of sleep so this was hardly a hard core effort. That would in any case have made little difference since when conditions were unfavourable to my little pistol station there was no benefit to continue sitting in front of the radio.

My objectives for the contest:

• Top ten in the unassisted QRP all-bands North America category.
• Finally cross the 100 countries mark on 40 meters with my current QRP station.
• Work as many new band-countries as possible.
• Have fun.

To my surprise I may have achieved the first objective, the second I definitely met and I had good success with the third and fourth.

The generally good conditions were a tremendous help of course, though the unsettled geomagnetic conditions from Saturday late afternoon onward negatively impacted my results on 40 and 10 meters. Since at best my signal is marginal on the other side of each QSO even a small increase in ionospheric absorption causes extreme difficulty.

Competition in the QRP category is less than in most others. It is therefore not that noteworthy an achievement to place highly: this speaks to the parable of a big fish in a small pond. That my claimed score in CQ WW CW placed me in the top ten with only a modest effort gave me the confidence to think I could do the same in the ARRL DX CW contest. However it will likely be months until I know. If I go by last year's results I have a reasonable shot.

My choice of contest logging software is N1MM. I like its uncluttered user interface and reliability. However I use only a fraction of its features. For example I do not yet use its programmable CW memories feature, which requires a PC-to-rig interface that I don't have. Instead I used my trusty 30-year old keyer that served me well in contests long past.

The score summary produced by the software is shown adjacent. The claimed score is exaggerated by 2,000 to 4,000 points due to one miscounted multiplier on 10 meters and a small number of suspect QSOs. This is not unusual. The score will be lowered by the log checkers after I submit the log. I had more country errors to correct when I imported the contest log into my old version of HRD 5 (Ham Radio Deluxe).

Unlike CQ WW the ARRL DX is more like a QSO party. For Canadians, that means we cannot work VE and continental US stations. That is, every QSO is DX, and there is no temptation to spend time hunting or running US stations. If your objective is purely DX, this can be an excellent opportunity to add to your band-countries totals. If you live outside of W/VE this is not a real DX contest since you are only allowed to work W/VE. That is why the ARRL DX contests are less popular than CQ WW.

On 40 meters I worked 7 new countries to raise my DXCC count from 95 to 102. That brings me to the bottom rung of DXCC award status with only a little over 4 months of operating. I am pleased with this achievement since it is with a maximum power of 10 watts (5 watts during contests) and a delta loop. The new ones I worked included 3V, FP and ZL, but also more common entities such as OE and a few in the Caribbean region.

On 10 meters I managed to squeak out 12 new countries, though nothing rare. On 15 and 20 the additions to my countries worked were less, but enough to put me within reach of 150 on 20 meters.

I had my best QSO productivity on 15 meters. Once the geomagnetic conditions worsened late on Saturday, the signals on 10 meters were only mediocre on Sunday. So I focussed on 15 where I got good results. When I exhausted the possibilities there I searched for new multipliers on 10 and ran up QSO totals on 20. I had poor luck on the north polar path on 20, not managing to work even a single JA.

Unfortunately this time around I worked no new countries. I heard a few but could not get through with my small station. My DXCC total sits at 190, and slowly inching toward 200. This is my total since the beginning of 2013 with my current QRP station, not my all time count which is well over 300.

I've included a picture of my temporary set-up in my not-yet-completed basement shack. The KX3 transceiver is the smallest item! I didn't plan my station around contests so there are some ergonomic problems with the pictured arrangement. In particular the paddles and keyboard are too far apart.

The reason for the two sets of paddles was due to expediency: only my 40-year old Brown Brothers paddles had the proper connecter to plug into my 30-year old keyer. The keyer is homebrew, built from an article in October 1981 QST: The CMOS Super Keyer. I jammed the Brown Brothers paddle between the rig and Bencher paddles since the base is light enough that it slides around a bit on the plastic tabletop.

The antenna switch selects between the TH1vn (favours north & south), the multi-band inverted vee (favours east & west) and the delta loop. I did a lot of switching of antennas while operating 20, 15 and 10 meters to get the best signal on many QSOs. The path often didn't agree with the predicted great circle route, especially around sunrise and after sunset. Although neither high-bands antenna has the performance of a yagi I do have better agility than other stations since I can switch antenna heading instantly.

And with that I'll close off this article. I hope I succeeded in communicating the message that you can have tremendous success as a DXer with QRP and simple wire antennas. You should stick to CW and jump into contests to work lots of DX, including rare ones (stations that will work hard to pull you through for the points!) in a short period of time.

You don't have to be a contester or even enjoy contests. If the wanted DX is in a contest that's where you need to be. Who knows, you might learn to enjoy contests. When DX no longer motivates as it once did, contests may be the stimulus you need to build bigger and better antennas.

Ground Loss and Vertically-polarized Antennas

I made passing references to high ground losses suffered by all of the limited-height, vertically-polarized 40 meters loop antennas in the recent series of articles on DX performance of 40 meters antennas. This is an important topic so I thought it would be interesting to delve more deeply into the subject with the objectives of attempting to quantify the loss contribution and, maybe, see if the loss can be managed.

To do this I will standardize on the 40 meters delta loop antenna I am currently using. Its apex is 15 meters above a typical suburban lossy ground and vertically-polarized by being fed λ/4 down from the apex on one of the delta loop's legs. The adjacent current plot shows how the currents in the two legs are in phase and ideally configured to cancel all but the vertical radiation component. This is our reference for what follows.

To begin we had better be clear on just what we mean when we use the term "ground". Ground is not a magical box to which we connect our electrical supply and equipment for safety, where all the unwanted electrons are shunted off. Of course it can do this if we do it right. It can also help us avoid much of the damage of lightning strikes. Ground also reflects RF radiation and, ominously, absorbs and dissipates RF. Verticals are often said to need extensive radial systems as some sort of talisman to trick the ground into not eating RF.

In reality ground is just a very (very!) big lossy, non-resonant body that conducts, resists and reflects radiation, and it will do so not matter what. That is, ground interacts with all antennas, not just verticals, in both the near field and far field. It just happens that ground has a particular affinity to vertical E-fields and therefore can have a more pronounced impact on vertically-polarized antennas.

Consider the veracity of the following statements. You have probably heard (or spoken) one of more of these yourself.

• Attaching several λ/4-radials to a λ/4 vertical monopole removes much of the ground losses of a ground-mounted or elevated vertical.
• A λ/2 vertical dipole, whether full height or shortened (loaded) not only requires no radials it also has little or no ground loss.
• Horizontal antennas have no ground loss.

All the above statements are false. Let's look at them one by one.

The currents in the elevated ground plane peak at the feed point just like in any vertical or horizontal dipole. The antenna is a λ/2 in length, with half of that vertical and half of that horizontal, though split 4 ways: the current in each radial is ¼ the current in the monopole element. It is the right angle between the antenna halves which give the ground plane antenna its low feed point impedance of 30Ω, not the ground.

With the antenna raised to 3 meters above ground (about the same height as the bottom of our reference 40 meters delta loop) the EZNEC-modelled ground loss is -5.5 db. This is almost identical to that for the delta loop and, for that matter, all the other close-to-the-ground loops and verticals I've modelled. As for its DX performance, it is awful: the gain at 10° elevation is -1.9 dbi, worse than every antenna in the recent summary article. The only remarkable thing here is that a vertically-polarized antenna such as a delta loop can be so competitive versus dipoles and inverted vees even though most of the radiated energy heats the earthworms.

Just to show that ground is not in play with the ground plane antenna I raised its height to 100 meters above ground (easy to do with software, though not in real life). The impedance dropped a few ohms, ground loss improved to -2.5 db, and the pattern remained omnidirectional, but gained several vertical lobes as happens with any high antenna. The ground plane antenna even works fine with just one radial, except that it becomes somewhat directive. Using 4 radials makes the pattern omnidirectional. A ground plane is similar to an inverted vee tipped on its side.

As I said, this loss of -5.5 db is also true of every low vertical antenna I modelled, which include full-size and shortened verticals. So that takes care of the second statement as well.

As for the final statement, horizontal antennas do indeed have loss. It's just that the loss is typically small, and therefore not often worth discussion. To give two examples, the reference 40 meters dipole and inverted vee reference antennas in the previous antenna comparison have losses of -0.6 and -1.2 db, respectively, at an apex height of 15 meters. The loss rises at lower heights and drops at greater heights.

In short, the factors which affect ground loss for any antenna at a particular frequency are:

• Polarization: vertical is worse than horizontal polarization, often by 5 or more db.
• Ground type: loss rises with lower conductivity and higher dielectric constant.
• Height: the farther an antenna is from ground the lower the loss.

To gain an insight to the magnitude of the loss I created an EZNEC model with two grounds. This is a nice feature of the software that allows the study of artificial ground treatments that affect antenna performance.

Each ground has its own set of characteristics: conductance, dielectric constant and height. They must be concentric, each with a fixed inner and outer radius. The outermost ground extends to infinity. I set the heights to zero so that they are coplanar. The test antenna shares the same origin as ground so the delta loop is centered above the ground plane.

The outer ground is the same medium ground I've used all along. For the inner ground I made some rough estimates of the characteristics of a metallic ground plane made of something like open-mesh galvanized steel chicken wire. Although I would never pave my yard with this stuff I wanted to make the experiment somewhat plausible. I chose a conductivity of 100 S/m (better than seawater but much worse than a solid metal sheet) and a dielectric constant of 10. Since the mesh is open the field associated with currents on the mesh penetrates the ground, which will therefore donate some of its dielectric attributes to the ground plane. The specific numbers are not critical, which I tested with a range of values for these ground characteristics, so don't worry too much if you disagree with my numerical choices.

I should point out that a set of radials under the antenna is not equivalent to the mesh described above. I tested several of these antenna with a ground plane made up of a small number of short and long radials. It takes a lot of these non-resonant radials before an appreciable effect is seen. Unless you are committed to laying 100 or more of these on or in the ground (as done by AM broadcasters and some 160 meters enthusiasts) it won't help much. It takes a lot of metal to make an effective ground plane.

I then ran the model through EZNEC, measuring the loss, and the gain and elevation angle of peak far-field radiation, varying the ground plane radius from 0 meters (no ground plane) out to 100 meters (2.5λ). Here is a plot of the results.

As you can see the loss is acutely dependent on the extent of the metal ground plane, improving from -5.9 db to -1.0 db. The affect of the local ground on the antenna's net efficiency is plain to see.

The elevation angle where the broadside lobe peaks gets lower -- better for DX -- as the ground plane grows. A bump centred at a radius of 10 meters might be surprising but is easily explained, as I'll do further along in this article.

The plotted gain is for the elevation angle peak radiation not the 10° I've used in the analyses until now. I made this choice to give a sense of how the pattern shape changes with the increasing ground plane radius. The gain at 10° is also increasing but is not plotted. At a radius of 85 meters (coincidentally, 2λ) the peak is at 10°, and the gain is far superior to even the previously modelled 2-element yagis up 15 meters. Unfortunately this requires several acres of property, all covered with metal mesh! It's cheaper and easier to put up a dipole or yagi on a taller tower.

Let's look at the mechanics of how these performance figures are generated. It has much to do with the construction of the far-field gain.

The far field is the vector sum of the sky wave and ground-reflected wave. If the reflected wave does not undergo a phase shift, at low angles the two waves (with the same final elevation angle) have approximately the same phase and reinforce each other. As you may know there is an angle, the Brewster angle, determined by ground characteristics, below which the reflected wave is phase-inverted. When this happens the two cancel. That is why antennas, vertical and horizontal, always fail as the elevation angle approaches 0°.

The elevation angle where the lobe peaks also drops as ground conductance rises. If you can't pave the planet in metal your next best choice is an island in the ocean. That's why so may DXpeditions to islands, such as the recent FT5ZM operation, can have unbelievably great signals even on the low bands. With just 10 watts and a delta loop I was able to work them on 40 meters.

When the ground reflection at a particular angle is off the metal ground plane the loss and phase shift are small. You can see this effect in the adjacent elevation far-field plot. In this instance the ground plane radius is 85 meters, which is where the space wave from the top of the delta loop at 10° is reflected.

Notice the sharp drop in gain below 10°. Those involve reflections from the beyond the metal ground plane.

As hinted above this also explains the elevation angle bump at a radius of 10 meters. With no ground plane (radius of 0) the ground is consistently medium and behaves as we have come to expect. As the ground plane is extended outward the first low-loss reflection is at a steep incident angle from the bottom of the antenna. As the radius continues to be extended the lower incidence angle reflections from more of the antenna are off the ground plane. At 85 meters the entire antenna can illuminate the ground plane at a downward angle of 10°. It's straight-forward geometry.

I suppose one possible recommendation is to measure off 85 meters from your antenna in the direction that matters most for your DXing activity. Determine which neighbour that is, knock on their door and ask if you can pave their yard with chicken wire. For free! People like free stuff.

To close off let's pretend that we are the ones on an island DXpedition with a delta loop. We edit the model by reversing the two grounds, with the inner one becoming a medium ground (or worse if the island is rocky) while the outer one is seawater. I put the antenna in the centre of a circular island that is 200 meters across (100 meters radius) and 5 meters above sea level.

As you can see the elevation plot is noticably odd. There is an excellent but narrow peak at 2° elevation but with a node at 10° and lots of radiation at high angles where the poor island ground comes into play. Loss is also substantial, not much different from not having the ocean there at all; the shore is too far from the antenna to counter most of the ground loss.

A simple vertical would likely do better since it would not have as much high-angle radiation as the delta loop. The loss, however, is not so easily defeated. Unless you buy the island and pave it with metal.

Height & Gain: 40 Meters Wire Antenna Summary

For those who have followed along I have written a number of articles over the past several weeks on one and two-element wire antennas for 40 meters. My objectives are particularly aimed at DX (low angle) performance and a single support for the antenna apex. Some of these antennas were only intended as references -- the dipole and inverted vee -- for the purpose of comparison.

[The TL;DR version of this long article is to just read the chart below. I at least want the long version to document my observations for future use. Others might not.]

This article is an opportunity to distill a few performance aspects of all these antennas in one place. The result is one (very) busy chart plus several discussion points. To introduce the chart I will make a few points, most of which have appeared earlier.

• The elevation angle selected for comparison is 10°. This is the median value of measured elevation angles for medium to long path lengths at 7 MHz, which range from 0° to 20°. The peak gain is almost always higher than 10° but that is of no interest to me. The vast majority of antenna articles I've read focus on the peak gain at whatever elevation angle it occurs, but that is often not indicative of DX performance on the low bands where antennas are, for most hams, at low heights.
• Ground, topography and "obstacles" in the environment of the antenna can have a strong effect. On this I had to compromise if the comparison is to have any meaning. I therefore assume a medium, real ground, as modelled in EZNEC, which is approximately what many suburban hams will have. I also assume that the ground is flat out to the horizon and that there are no obstacles that would shadow or couple to the antenna. This includes a tower or metal support for the antenna that might be resonant on 40 meters. In the real world where we all live these interactions are inevitably present. As a general rule, the lower the antenna the greater the interactions. Interactions are more than not reduce antenna performance, especially for arrays with two or more elements.
• The loops and loop arrays are all fed for vertical polarization. When fed for horizontal polarization their DX performance (gain on the chart) would be substantially worse.
• The interior apex angle of the inverted vees, narrow diamond loop and chevron loop is 120°. The explanation for this choice is in the articles of the associated antenna articles. In general, a smaller interior apex angle for these antennas result in one or more of reduced bandwidth (higher Q), lower gain at 10° elevation, and reduced F/B (for the 2-element antennas).
• The gain shown in the chart for the 2-element antennas is at the frequency where it is maximum. It will be lower at other frequencies. You'll need to read the individual antenna articles for how the gain varies with frequency. 
• All wires in the model are 12 AWG insulated copper, except for the chevron which uses thicker 10 AWG. Choice of wire affects element length and resistive loss. After decades of playing with antennas I have come to prefer insulated wires to keep corrosion at bay. This not only makes adjustments easier in the future (no sanding/cleaning) but also avoids resistive loss due to oxidation.
• Links to the antenna articles: 2-element yagi with wire dipoles; 2-element yagi with inverted vees; 2-element yagi with diamond vees; delta loop; chevron loop; diamond loops; 2-element delta loop and 2-element narrow diamond loop.

Yes, it's a busy chart, but it isn't too difficult to unpack if you note that the dipole and inverted vee start from the bottom left and are roughly parallel, and the three 2-element yagis (dipole elements, inverted vee elements and diamond vee elements) start at the centre left and are also mostly parallel. All the loops and loop arrays cease to increase in gain as the height increases.

There are a few conclusions to be taken from this chart, which I'll list. Just keep in mind that there are a number of factors that could reduce the performance you might see in actual practice.

Gain Performance

If you must have gain, at any height, you should put up a wire yagi. While it may seem an expensive waste, a rotatable 2-element short yagi only 15 meters high will do very well. As you turn down the element ends, as you must with a fixed (or switchable) wire yagi, make the angle as shallow as possible. Alternatively, or below about 18 meters, a 2-element narrow diamond loop array will give about the same gain and superior F/B. But first test for tower resonance which would impact performance.

If you want to keep it simple and use a single element a vertically-polarized loop is the superior choice at heights below 18 meters. The choice of loop configuration is determined by how much F/S and low-SWR bandwidth you can live with, suitable tie-down points and interactions with the tower, a high-bands yagi above the loop and the local environment. The chevron loop offers the best gain but requires the greatest amount of compromise. The delta loop offers the best omnidirectionality but with unexceptional gain.

Copper Losses

Voltage Divider

Wire antennas are made from conductors that are narrower than found in the typical rotatable yagi made from aluminum alloy tubes. This results in resistive (I²R) losses that are higher than in the rotatable yagis despite copper being a better conductor than aluminum. Aluminum oxide, which readily coats the surfaces of yagi elements is a good enough conductor to not change this relationship.

Oxidized bare copper wires (cuprous oxide) have higher resistance, and therefore loss. The impact on low HF is usually not a large concern since the oxidation doesn't usually penetrate too deep and the skin depth is greater. Stranded bare copper is worse than solid bare wire since there is more surface area that will oxidize. I use stranded copper wire in most of my antennas, but I use insulated wire to avoid corrosion. There is dielectric loss in the plastic insulation, but it is small at low frequencies.

The resistive losses in the single-element wire antennas is typically no worse than -0.1 db, and so can be considered of no significance. The chevron loop losses are more significant, though still no more than -0.5 db, due to its low radiation resistance. The conductor resistance is in series with the radiation resistance, so as with any voltage divider circuit the lower the radiation resistance the greater the current and therefore the power dissipated in the conductor resistance.

In the 2-element wire antennas the impact of resistive losses is more pronounced since the radiation resistance is lower. This results in greater current flow and thus greater loss in the conductor resistance. This is not a large effect but is still significant. The typical loss figure for the 2-element antennas in this article is -0.3 db.

W8JI has a description of radiation resistance, height and loss you should consider reading if you'd like to learn more on this important topic. It also applies to the next topic.

Element Length vs. Gain

I think everyone knows that the free-space broadside gain of a λ/2 dipole is ~2.1 dbi. When the length is shorter the gain declines. One way to understand this is to imagine a dipole getting shorter and shorter until it becomes a point source. A point source is an isotropic radiator with a gain of 0 dbi. That we cannot actually build this antenna is unimportant. It is enough to understand that as an antenna element is shortened the gain declines.

It is important to understand that this loss of gain for short antenna elements is not due to losses. What actually happens is that the pattern broadens such that less power goes toward the broadside direction. However it is also true that short antennas have lower impedances which require matching networks, and those matching networks can introduce noticable loss.

The 2-element wire yagis that I modelled have elements that are shorter than λ/2. I did note this at the time as an advantage since it allows the antenna to fit a smaller space. The reason the elements are shorter (0.41λ) is due to the inductive loading of the ladder line running from each element to the switch box. In comparison to a wire yagi with full λ/2 elements the gain of these switchable yagis is -0.3 db.

Thus we have -0.3 db due to I²R loss and -0.3 db due to element shortening, which sums to -0.6 db. The gain of a full-size 2-element rotatable yagi would therefore be 0.6 db greater than that for the 2-element wire dipole yagi on the chart.

It is interesting to briefly look at how these figures compare to the expected gain from a short rotatable yagi such as those made by Cushcraft and Force 12. The I²R losses are largely eliminated but the even shorter elements (~0.32λ and ~0.28λ, respectively) increase that gain reduction by about another 0.1 to 0.2 db, plus a small loss due to the loading coils. So if you were to put up one of these small commercial yagis you would see a gain (at the frequency it peaks) indistinguishable from the modelled 2-element wire dipole yagi (top line in the chart above) and only 0.6 db worse than a 2-element full-size rotatable yagi.

The 2-element loop arrays I modelled have not been through the "shortening" process to make them switchable. When I get around to doing so expect that the gain will be similarly reduced from that shown in the chart above.

Antenna Height and Tuning

You should expect that the impedance and resonant frequency of all the single-element antennas will change with height above ground. The rate of change is greatest at the lowest height. Loops and inverted vees are particularly sensitive to height since for a given apex height they get closest to the ground.

Proximity to buildings wires (and other conductors) and utility wires will result in interactions that will skew the antenna pattern and shift resonance and impedance from the modelled values. Since these interactions are difficult to predict some tuning is required once the antenna is installed. These interactions are typically greatest when the antenna is low and the other conductors are within either broadside lobe.

Tuning of 2-element yagis and loop arrays is less sensitive to height. Once the 40 meter beam's lowest point is higher than about 8 meters the impedance and resonant frequency hardly shift at all with further increase of height. The same applies to the frequencies of maximum gain and F/B. While this is a topic all its own I can summarize the reason for this useful attribute, one that applies to all yagis, quads and similar beam antennas.

As the gain of an antenna rises the solid angle of the main lobe(s) covers a smaller area of the far field. That is, the antenna "sees" less of the ground and other conductors that are not within the free-space lobe(s). With a F/B or F/S over 10 db anything behind or beside the antenna, respectively, farther than ~λ/4 is pretty much invisible to the antenna, and therefore does not appreciably interact. The same is true of the ground, though more so for the yagis than the higher-beam width loop arrays.

This is why yagi manufacturers can confidently state the exact dimensions of their antennas regardless of where it is to be installed. Further tuning just isn't required. This is especially true of the higher bands where the distance to ground and other conductors is farther in terms of wavelength.