2 + 3 = ?? : Stacking Dissimilar Yagis

You don't often see stacks made from dissimilar yagis. Using identical yagis simplifies their design and optimizes overall performance. That's how I built my stacks for 10, 15 and 20 meters, right down to the matching networks. The only difference is the booms, which have a negligible impact on antenna performance, instead being chosen for their mechanical properties.

On 40 meters it is rare to see stacks of other than 2-element yagis since 3+ element yagis are so large and expensive. Consider just the tower to support two of those behemoths. I have a 3-element 40 meter yagi and while I love its performance I will not build another. It is more usual to make 40 meter stacks from 2-element yagis. Indeed, my understanding is that W6NL designed the Moxon (XM240 conversion most often) as the foundation for an effective and efficient 40 meter yagi without incurring a large mechanical challenge.

I never considered stacking the XM240 and 3-element yagi even though they are on the same tower and are at heights (λ/2 and λ) and separation (λ/2) that make it feasible. They almost always point in different directions so the only application would have been to "spray" in two directions, such as Europe and the US. The XM240 can't be pointed to Europe since the side mount only allows rotation between southeast and west.

As I've mentioned on the blog, I am in the process of building a 2-element reversible Moxon for 40 meters. It is intended to replace the XM240 on the same rotatable side mount.  Its performance will be superior to the coil-loaded XM240 and since it is reversible it can point over 260° of the compass, including Europe and Asia.

To do so is not without its challenges, and its potential performance is less than with identical yagis. Let's review those challenges, great and small. These apply to stacking of any dissimilar yagis.

• Gain: Long yagis have greater gain than short yagis. The 2-element Moxon has less gain than its big brother and the deficit increases as you move higher in frequency. When you add a little to a lot the sum is less than one might expect. For example, if the antenna gains are 8 and 6 dbi and the main lobes are ideally combined, the net gain is 10 dbi. That's below the nominal 3 db stacking gain of 11 dbi when both have 8 dbi gain. Keep in mind that we're working with a logarithmic scale so you can't simply add and subtract decibels!
• Phase: Different yagis has different feed systems and the driven elements are not equidistant from the tower. Both contribute to phase differences. The relative phase must be determined and compensation built into the stack system. It may not be achievable across the band with a fixed compensation network (e.g. delay line).
• Impedance: Here we are concerned with power division and phase; that is, what SWR affects but not SWR itself. Dissimilar yagis are certain to have different impedances (R and X) across the band of interest. Networks to precisely compensate for those differences can be quite difficult to achieve in practice, and more difficult to have their phases track together.

Both of my yagis, the 3-element and the reversible Moxon, require NEC5 for accurate modelling so I'll be using it throughout. There are so many segments when the models are combined that run times can be long. 

To begin, let's review the gain and SWR of both antennas. I will do this using  EZNEC medium ground, the lower Moxon at 75' (22.8 m) and the upper 3-element yagi at 150' (45.5 m). Those are the approximate heights where they are (or will be) placed on my tower. Interactions with other antennas and guys are not included in the model, but are expected to have only minor effects in my station configuration.

Gain and impedance were first modelled in free space to confirm that ground has only a small effect on the individual yagis. As expected, ground has a negligible effect on the higher and longer 3-element yagi and only a small impact on the shorter Moxon.

Since the yagis are individually fed for this exercise, as they would be with a stack switch in a real deployment, gain, F/B and SWR are not equal to what they are modelled in isolation from other antennas. That is, there is mutual coupling. The λ/2 separated is enough to ensure the effects are modest, but should not be ignored. A single EZNEC model containing both antennas ensures that their interaction is included. 

λ/2 is usually considered to be the minimum separation recommended for best stack performance, however reality is more complex than can be represented by a simple heuristic. Other heights and separations can be enlightening for those who have yet to put up the towers and antennas. My station, already built, has constraints that I am staying within for this analysis.

This chart should not be a surprise. A Moxon isn't a magical antenna. Its gain is slightly lower than that of a conventional 2-element yagi but with better SWR and F/B -- please note that F/B is excluded from this analysis since our focus is stacking performance. 3-element yagis typically have gain that increases with frequency, until the radiation resistance dives and the SWR soars. The 3-element does poorly above 7.2 MHz, which matters in the Americas but not elsewhere. I rarely use the 3-element yagi above 7.2 MHz since it is primarily a DX antenna at its great height. That is by design.

The disparities between the yagis increase with frequency. That carries over when the antennas are stacked, as we'll discover in the present analysis. For now, note that the gains and are not so far apart and the matches excellent in the CW segment. The divergence becomes wide in the SSB segment, especially above the Americas segment between 7.2 and 7.3 MHz.

It is also notable that the elevation angle of the forward lobes are far apart. This is of course normal for a vertical stack though perhaps not to this degree. The difference is usually a benefit of stacking since it allows the filling of nulls in the elevation patterns of the individual yagis. Yet stacking gain -- often what we want most of all -- is impaired by a wide divergence of forward lobe elevation angles. Heights of λ/2 and λ with a λ/2 separation limits stacking benefits. 

Having noted all of the forgoing, the stacking prognosis is not great. Nevertheless let's proceed and see what we can do with it and where the deficiencies lie in the calculated performance. 

Phase matching

In this array there are several factors for achieving phase matching:

• Feed system
• Feed point impedance 
• Power splitting
• Coax phasing lines
• Elevation angle (antenna height)

Let's briefly look at each of these. The diagram will be an aid to the discussion -- I didn't bother with detailed annotations since they should be evident. While this article focusses on my unusual stacking scenario, I've done this type of analysis before for my existing stacks. The planning and design process is strongly recommended before ordering towers and materials and beginning construction. A major investment should not be made without solid evidence of desired performance.

The superposition of direct and reflected waves from the yagis determines the lobes and nulls in the stack pattern. Yagi coupling (mutual impedance) with each other and the ground have their effects as well, in accord with their separation and height (relative to wavelength). That is elementary. There may be surprises from what we expect since the 40 meter wavelength is quite long, which brings the yagis closer to ground and to each other.

The feed system for the 3-element yagi was originally modelled as a beta match since that is simple in EZNEC and delivers reliable results. It was changed to a gamma match since that can affect the feed phase angle. The EZNEC (NEC5) model I developed closely matches the dimensions and behaviour of the real gamma match on the yagi. 

The Moxon driven element is of course directly connected to the driven element. The reversing electronics are not included since that does not affect the model. That said, while the reversible Moxon is insensitive to the lengths of the coax from the central switch to each element, they should be the same length when the antenna is part of a stack.

When this measures are taken and the yagis are fed via a power splitter that provides a common current source, the feed points will be in phase, or close enough to ensure stack performance. It must be so since one side of the coax directly connects to the centre of the driven element of both. This was confirmed by inspection of the Currents table in EZNEC. Other feed point matching networks may not be so straight forward.

This desired outcome requires that the impedances are equal and that the phasing lines preserve phase (equal electrical length). That is more difficult than it might appear. Consider how power is split. We want the power splitter to split the power equally since only then is optimum stacking gain achieved in the cases where the yagi are identical. The case I'm evaluating is more challenging. 

In the model I used an ideal 2:1 transformer as a power splitter and impedance matching network. The 50 Ω source is on one side of the transformer and parallel 50 Ω feeds to the phasing lines are on the other side, which sum to 25 Ω only when the SWR is 1 to both yagis. An L-network can be used in place of the transformer, as I've successfully employed for my 20, 15 and 10 meter stacks. 

While a transformer can be broadband, an L-network is rarely suitable for more than one band. For those who choose to stack multi-band yagis a transformer is the right choice. For single band use, such as in the present case, an L-network is compact, easy to design and build, and can be more efficient.

In both cases, equal power splitting and impedance transformation are only achieved when the SWR at the matching device is close to 1. That is, an excellent 50 Ω match. For the dissimilar stack that criterion is only satisfied below 7.150 MHz. For identical yagis with identical SWR curves the power will split equally but efficiency will fall; that is, more power is converted to heat in the transformer or L-network.

The phasing lines from the splitter do more than just preserve phase for the dissimilar array. Even for that it is necessary to adjust for the different horizontal locations of the driven elements. For example, in the dissimilar yagi stack that additional distance is 8' (2.5 m) -- the DE is near the centre of the 3-element yagi and at the end of Moxon boom. 

Since the VF of coax is less than 1, the phasing line extension must be shorter than that. For RG213 it would be 5.3' (0.66 × 8). The extra length is a delay line that goes on the yagi (the Moxon in this case) with the driven element further away from the forward direction.

CMC (common mode chokes) should be identical or their differences accounted for in the model. For example, the lengths of coax wound on ferrite toroids become part of the phasing lines.

That simple delay line calculation assumes an elevation angle close to 0°, in the plane of the booms. That is close enough for my high band stacks since they are high enough (again, relative to wavelength) that the elevation angle of both yagis are quite low. This is not true for the 40 meter stack since they are lower, again relative to wavelength. This applies to both similar and dissimilar stacks.

Look again at the diagram above. The wavefront from the lower yagi has to travel farther for correct phasing. Thus the delay in its phasing line must be longer. By coincidence that works out to 8' or 9' of RG213, as determined experimentally with EZNEC. I put that value into the transmission line table.

Impedance mismatch is another confounding factor that affects more than equal power division. There is an infinite set of R and X pairs for any SWR other than 1. They form a circle on a Smith chart. The feed point impedance will not be the same as the impedance presented at the power splitter (transformer or L-network). That has to be calculated. 

The parallel complex sum of the impedances for significant disparities will cause unequal power splitting. The impedance transformation due to the phasing lines and deviation in the behaviour of the matching network (power splitter) are acutely sensitive to the magnitude of the SWR. The power split and feed point phase difference can be inspected with EZNEC. 

Open the Currents table and carefully compare the current magnitude and phase in the wires at the feed points. I say 'carefully' since the current directions (signs of the phase) may be opposite to each other. This is explained in the Currents section of the EZNEC manual. The phases in the two feed points above are close but not equal since some difference at the feed points is necessary to achieve phase alignment at the common wavefront at the stack main lobe's elevation angle. That was determined experimentally in the model, which is often the easiest way of doing it!

Any residual imbalance can be corrected with delay lines and compensation networks. If the values are fixed, as is typical, stack performance will be frequency sensitive, assuming (as is almost certainly the case with dissimilar yagis) the SWR/impedance difference of the yagis varies with frequency. This is evident in the earlier plot of each yagi's gain and SWR for the 40 meter dissimilar yagi stack I am evaluating.

The significant of the power and phase imbalance can be calculated in the model. Therefore let's do that to evaluate the performance. Then we can discuss the difficulty and value of mitigation measures. 

Performance

There is perhaps no better way to demonstrate the expected performance of the dissimilar stack than to compare the modelled patterns for the individual antennas (Moxon and 3-element yagi) and when combined.

The phasing lines in these patterns was optimized for 7.05 MHz, mainly for CW use. The result is modest stacking gain from 7.0 to 7.1 MHz. This is the range where the SWR is low for both the Moxon and the high 3-element yagi. The line to the Moxon is 9' (2.7 m) longer. It is certainly debatable whether a stacking gain of little over 1 db is beneficial. Well, it can be in competitive environments like contests and DXpedition pile ups, but perhaps not for other situations.

By the time the frequency rises to 7.150 MHz the performance of the stack is not good, by any standard. For SSB use there is little stacking benefit, or none at all. Recall from the first chart that at higher frequencies the gain of the two yagis diverges quite a lot and the SWR of the 3-element becomes extreme above 7.2 MHz.

That is the case for a fixed phasing system. In the adjacent 7.2 MHz plot one change has been made: a longer phasing line to the Moxon to optimize performance at that frequency. It has been increased from 9' to 21' (6.4 m). 

The pattern is certainly improved though perhaps not enough be of value. The gain is now equal for the stack and for the 3-element yagi alone. The only useful change is null filling, yet that can be accomplished by the lower Moxon on its own.

What was accomplished by restoring phase alignment disrupted by the high SWR of the 3-element yagi was to prevent the Moxon subtracting from the main lobe gain. That could not improve stack gain beyond that of the 3-element yagi alone because the Moxon gain that high in the band is 3 to 4 db below that of the 3-element yagi.

It is likely that the situation can be further improved with a network to lower the SWR of the 3-element higher in the band, say centred on 7.2 MHz. I've modelled networks of that nature and the low SWR bandwidth is small, often no better than 50 kHz for an SWR below 2. That makes the 3-element yagi more useful for SSB contests but, again, stacking gain really isn't there.

A switched delay line is needed to compensate for the phase shift introduced by the matching network. I don't see the worth of bothering since, as demonstrated, there is little to no benefit of stacking. The yagis are too dissimilar in the top half of the 40 meter band.

Conclusions

After all of this discussion and calculation, what will I do? The short answer is that I don't know. It isn't difficult to stack the yagis so I might, if only out of curiosity. All I need to do is measure the phasing lines and build a stack switch, and make a few changes to my antenna switching software. If it's only useful for CW, that's acceptable since that's my primary interest.

I was already considering an auxiliary switch on the tower for these yagis. That would free up one run of Heliax and open a port on the 40 meter auxiliary switch for the XM240. I am pondering whether to place the XM240 on the other tower fixed south as a multiplier antenna, similar to the TH6 for the high bands. Or I may sell it. The reversing feature of the Moxon might make the XM240 superfluous.

My immediate priority is to swap the Moxon for the XM240, make sure it is working, and only then consider how to get the most out of it. Stacking can be deferred for several months or more.