Sunday, December 28, 2014

Stacking HF Yagis - The Basics

Stacking of two identical yagis can provide an additional 3 db gain (doubling). That is, under suitable conditions. There is lot of fantastic information around about the realities of stacking, balanced against a lot of good but incomplete information. There is also lots of nonsense here and there. It all makes for a big mess.

This article is my attempt to add a bit of light to the basics of stacking HF yagis. I figure, why not, winter and the holidays make for a good time to do a little dreaming about big antennas. Although I have never had stacked yagis (for the same band) in my various stations over the years I have used many and helped others design them.

So sit back and join me in a bit of dreaming about big antennas. Keep in mind this is not a construction article, nor is it about choosing the optimum design or a critique of others' designs.

If you already thoroughly know the basics you'll learn nothing new here.

Pieces of the puzzle

The diagram at right includes the main components of a two-yagi stack, as often deployed. It is obviously not to scale or even "anatomically correct". My drafting skills are poor! Hopefully I can get the idea across with my minimum outlay of effort.

The upper yagi is on top of the tower and rotatable. The lower yagi may be fixed in one direction or rotatable with a ring rotor or a hinged arm. The latter would have a dead spot somewhere since rotation would be less than 360°. Then there are those that have rotatable towers.

Each of the labelled components will be covered in the discussion below, in no particular order.

Additive gain from subtractive arrays

Moxon has a nice discussion of these distinct classes of gain (directive) antenna arrays. Here are my paraphrased definitions of both:
  • Subtractive: Arrangement of driven and parasitic elements within the antenna's near field designed to reduce or cancel the net field in most directions to achieve directivity; that is, gain and F/B. Its characteristics include: relatively compact size; narrow bandwidth; and low radiation resistance (for maximum gain). The yagi is the usual and perhaps best example.
  • Additive: Two or more antennas with low mutual impedance have their patterns combine in the far field to produce directivity. When suitably designed the antennas in the additive array do not "see" each other (small mutual impedance), therefore the characteristics of each antenna in the array are as if they were solitary. A power divider allocates the power among antennas, along with any required phase shifting.
When stacking yagis, the ideal is additive combination of the separate antenna patterns without so much mutual impedance between them that the net pattern is degraded. In the real world the ideal is rarely achieved. The confounding factors are getting the antennas far enough apart and far enough from ground that addition is the dominant factor in pattern determination. A lesser objective is to either accept some degradation or to modify the design of the individual antennas to leverage the mutual impedance to equal or even exceed the ideal (3 db) gain.

It's really just that simple. I have noticed that some (most?) of the literature on this topic does not clearly differentiate how the two array types each affect the outcome in their own way. This omission can cause confusion.

Example: Stacked 3-element Tri-band Yagis

An example will help to elucidate the discussion. The antennas at right are both short-boom 3-element tri-band yagis (yes, you can stack multi-band antennas) vertically spaced 20 meters. I modelled this tri-band trap yagi in EZNEC some months ago. If you like you can refer to that article for design details, although that really isn't necessary for what follows.

Before adding the complication of ground it is helpful to first run the model in free space, eliminating the effects of ground on the near field and the additive effect of the two ground reflections (images). The plots (azimuth on the left, elevation on the right) compare feeding just one yagi versus feeding both in phase. The test frequency is 14.1 MHz. At this frequency the 20 meters stacking distance is ~1λ.


The gain of feeding one yagi is 7.14 dbi, which is a bit lower than the 7.28 dbi it has when the other yagi isn't present. That is the effect of mutual impedance between the yagis. In this instance current in the unfed yagi is ~10% that of the fed yagi, which is pretty good so far as stacks go. The azimuth pattern is not distorted because the array is symmetric in that plane. In elevation the single yagi pattern is distorted (upper yagi, in this case) due to vertical asymmetry. Symmetry is restored when both yagis are fed.

The gain with both yagis fed and no phase shift is 10.49 dbi, or an additional 3.35 db. If there were no mutual impedance between yagis the gain ought to be 10.28 dbi, precisely 3 db more than a single yagi. In free space the gain peaks at 0° elevation since there are no ground reflections (images).

Notice that the shape of the azimuth pattern is almost exactly the same as for a single fed yagi, just with higher gain. As should be apparent that it is from the elevation pattern that the energy is taken for the stacking gain: the main lobe is narrower in the elevation plane. Lucky for us vertical stacking is easier since it has both equal compass coverage and improved low-angle radiation. (The latter will become clear when we come to real ground.) That is, good DX performance without the need to make more frequent adjustments to the beam heading than with a single yagi.

Mutual impedance slightly reduces the gain if only one yagi in the stack is fed but results in more than 3 db gain increase when both are fed. So far so good. While I am not going to deal with stacking optimization in this article it should be obvious that 1λ separation is effective for this particular yagi, at least in free space.

Ground effects

Moving from free space to real ground we introduce two additional factors to our stacked array:
  • Near-field interaction with ground
  • Far-field pattern due to ground reflections from all yagis in the stack
In the first case we are concerned with antenna impedance and efficiency. Any significant differences among the yagis due to near-field ground interactions (mutual impedance) can cause problems. These include unequal power division and pattern distortion of the lowest yagi. As an absolute minimum the lowest yagi in the stack should be no less than than λ/2 above ground (10.5 meters at 14 MHz). For best results the lowest yagi should be at least 1λ high.

In the second case the stacking distance and the ratio of stacking distance to height affect the superposition (linear sum) of the space waves and ground reflections. First we take the above array of tri-band yagis and place them over typical medium ground, with the lower yagi at 20 meters height (1λ on 20 meters, and 2λ on 10 meters).

As is typical for a horizontal antenna there are more lobes in the pattern as you go higher. The combined pattern therefore has a couple of notable features:
  • Fewer nulls in the elevation pattern. Unless deliberately designed the nulls between lobes of the two antennas will not coincide. In my judgment this is to be seen as an advantage since switching of stack feed is less often needed to receive stations on high-angle paths.
  • The lowest lobes are the ones with the greatest energy and they typically combine to give the low-angle gain stacks are justifiably famous for achieving. They do not sum exactly since the lobe widths and centres are different due to the different heights. The smaller the stacking distance the better these lobes add.
In this particular example when the two tri-band yagis achieve a maximum gain of nearly 15 dbi at 9° elevation. At very low angles the pattern of the highest yagi dominates the summed pattern.

It should be clear from this discussion that it would be very unusual for stacking gain to be exactly 3 db! This is true even absent near-field ground effects and mutual impedance between closely stacked yagis. Indeed a perfect summation of lobes is not necessarily the best. Although elevation angle peaks can be closely matched when the ratio of stacking distance to height above ground is highest, mutual coupling between yagis can diminish gain.

Counter-example: 4-over-4 on 40 meters

I think it worthwhile to include a stacking example that fails miserably. For this I am using the 4-element 40 meters switchable wire yagi from a recent article. In addition to the original with an apex height of 20 meters (λ/2) there is an identical antenna above it at 40 meters height  (1λ). The effective height of each is several meters lower due to the inverted vee element shape. It is assumed that both yagis are switched in tandem.

The separation between yagis is λ/2, which is very small for an antenna of this gain and boom length. The mutual coupling between antennas is significant though not too bad. SWR is barely affected when only the upper or lower yagi is fed, and is slightly improved when both are fed. Gain is degraded by just under 1 db when only one yagi is fed.

The failure of stacking is readily apparent when we plot the elevation pattern of the stack versus each of the lower and upper yagis.

Here we see the effect of having the lower yagi close to ground, even with a small stacking distance. The main lobe of the lower yagi is of high enough elevation that it adds poorly with that of the upper yagi. The net result is a mere 0.3 db gain at low angles in comparison to the upper alone. Indeed the lower yagi is worsening the pattern of the stack.

It is clearly better to not stack in this instance. The best result comes from simply raising the wire yagi to the greatest possible height. It should come as no surprise that in the super-stations that do use stacks on 40 meters that the upper yagi is typically at 60 meters height (1.5λ). You need to do that to get worthwhile stacking gain. The same reasoning applies to higher bands, with respect to wavelength.

Yagi tuning, power splitting and switching

Yagis of all types can be stacked, provided that the stacking distance is sufficient that the mutual impedance is small. However there are good reasons to use identical antennas that are identically tuned.
  • Power splitting: The impedance of off yagis in the stack should be made as identical as possible, across the band(s) of interest. Impedance differences, and even equal but highly reactive impedances, will unequally split source power. Inequalities will show up in the far-field pattern with the yagi getting more power dominating the pattern and a reduction in stack gain.
  • Switching: Many stacks feature switching between the full stack and individual antennas within the stack, to allow real-time selection of best matches the path's elevation angle. An impedance difference seen by the transmitter could require tuning after every switch action. This is undesirable.
The simplest setting is where just one yagi is selected. Power does not have to be split, and relays connect the main transmission line to the phasing line connected to the selected yagi. Both lines should be 50 Ω.

When two or more yagis are selected we need a way to equally split the source power and to ensure the yagis are fed in phase. Variations are permitted to create alternative patterns but that is rarely done. We'll stick with the dominant arrangement that calls for equal power and phase.

Equally splitting the power to a two-yagi stack can be as simple as a coaxial T-connector. However that is insufficient in most cases since this places the loads in parallel and, if the impedances have been matched as they ought to be, the summed impedance is half that of an individual yagi. For the ideal 50 Ω case (at resonance) this gives a net impedance of 25 Ω, or an SWR of 2 for a 50 Ω source. The SWR will be worse at all other frequencies.

A transformer solves the problem. Perhaps the simplest is the λ/4 coaxial transformer. If a λ/4 section of 70 Ω coax is connected to each arm of the T-connector the 50 Ω impedance of the yagi is transformed to 100 Ω. I used this type of feed in my design of additive 40 meters slopers.

In parallel (via the T-connector) and with the transformers the summed impedance is the desired 50 Ω. The remainder of the phasing lines (between the yagi and coaxial transformer) are 50 Ω.

For a single band two-yagi stack that is permanently wired (not switchable) the use of λ/4 coaxial transformers is the best and most efficient choice. Even if switching is desired this is still not a bad choice, if we provide a means to cut the transformers out of the circuit when an individual yagi is selected. This can be done by coiling the transformer at the switching box on the tower and using relays on both transformer ports.

In every other stacking scenario it is better to use a high-efficiency broadband toroidal transformer (unun). This includes multi-band yagis and stacks with more than 2 yagis. For two-yagi stacks the transformer is 2:1 (50 Ω to 25 Ω) and 3:1 (50 Ω to 17 Ω) for three-yagi stacks. Two 2:1 transformers or a 4:1 transformer can be used for a four-yagi stack, depending on the desired switching configuration.

There are many commercial products (e.g. Comtek) that include the transformer and relays in one box and a switching unit for the shack. DC power is typically delivered by separate 3 or 4 conductor cable to accommodate the several operating modes. The relays cut the transformers out of the circuit when a single yagi is selected. While these devices aren't difficult to build most hams choose to buy since they are not very expensive and save effort and time better spent building and maintaining these monster antennas.

Phasing

Whatever transformer option is selected it is vital that the phasing lines to the yagis be of equal length. Pattern addition will only give the modelled gain if the yagis are fed in phase. A substantial phase error will decrease gain and otherwise degrade the pattern.

A reasonable maximum error tolerance is 1%, which is equivalent to 3.6° phase shift between yagis for 1λ electrical length of phasing line. The same accuracy goal should be used for λ/4 coaxial transformers. Not only must the lengths be equal the lines should be the same brand and vintage to ensure that the velocity factors are the same. Especially avoid solid polyethylene (0.66) in one and foam (0.8 to 0.9) in the other. For example, a 1% error in a 5 meters long phasing line made from RG-213 is 5 cm (2"), which is easily done.

If  λ/4 coaxial transformers are used these must be directly connected to the switch box, not to the yagis. For permanently wired stacks the transformers can form part of the runs from the T-connector to the yagis. They will not reach all the way since, even with foam coax, the transformers only span 0.4λ, and the yagis are sure to be well over 0.5λ apart.

It should be no surprise that the switching unit and transformer are typically mounted on the tower between and equidistant from the yagis in a two-yagi stack.

Final

Well, wasn't that fun? Stacking may seem like it's only for the big guns yet I know many smaller stations that utilize stacks for the high bands, especially on 10 meters. Even with a 25 meters high tower plus mast it is very possible to get most of the benefit of stacking as low as the 20 meters band. Maybe it isn't so much of a dream after all.

My next article will be my year end retrospective and a look forward to 2015. Unlike my plans for 2014 this coming year is less-well planned since I am running up against the constraints of what I can reasonably build on my suburban property.

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