Tuesday, April 29, 2014

Choosing a High-bands Yagi (Part 3) - 3-element Tri-bander

Continuing on from Part 2 of my analysis of small high-bands yagis, it is time to evaluate 3-element tri-band yagis. In this category I include those antenna with 3 elements (reflector, driven and director) that are each active on 20, 15 and 10 meters. Traps for 10 and 15 meters are employed to achieve tri-band resonance. Examples include the TH-3, TA-33, A3S, among others. They typically have a short boom, usually between 3.5 and 4.5 meters.

As with 2-element tri-banders the boom length must be a compromise since it varies by a 2:1 ratio between 20 and 10 meters, as measured in wavelengths. At a typical boom length of 4.2 meters (~14') this is 0.2λ, 0.3λ and 0.4λ on 20, 15 and 10 meters, respectively. Tri-band yagis with booms longer than 4.5 meters typically have additional elements for one or both of 10 and 15 meters since otherwise the boom is too long for effective element coupling, and therefore reduced gain. Examples of these include the TH-6 (the antenna I have stored in my garage) and CL-36, among many others.

Before we delve into the design and evaluation it is worth a moment to review the parameters which effect 3-element (and larger) yagi performance:
  • Boom length: Maximum achievable gain is a function of boom length - the longer the boom the greater the gain. There is likewise an optimum boom length (in wavelengths) for a yagi comprised of a chosen number of elements. For a 3-element yagi the optimum boom length is ~0.35λ, per the reference yagi introduced in Part 1. Maximum F/B is a more complex function of boom length, but shows a periodic behaviour with peaks at odd multiples of λ/4. For a 3-element yagi there is just the one peak at 0.25λ since the next odd multiple is too long a boom for 3 elements.
  • Parasitic element resonance: Maximum gain and F/B are achieved when the resonant frequencies of the parasitic elements are close together -- but not too close -- with lengths of 0.49λ and 0.47λ for the reflector and director, respectively. However this is a poor choice since the gain, F/B and SWR bandwidths are narrowest in this case. Better choices are 0.502λ and 0.463λ, where we sacrifice ~0.5 db of maximum gain and, perhaps, ~5 db of F/B in return for good performance across all 3 bands. The use of traps will further narrow bandwidth performance so we want to start with a reasonably broadband design choice. These choices help to avoid sharp radiation resistance dips at maximum gain which make it difficult to achieve a broadband match.
  • Element separation: It is assumed that for the 3-element yagi the separation between reflector and director elements is equal to the boom length; that is, they are at opposite ends of the boom. The only remaining separation variable is the position of the driven element. To avoid mechanical interference with the mast clamp the driven element is typically offset from the centre position towards the reflector. As the separation between driven and reflector elements is reduced the gain increases and the impedance falls, but only to a certain point. A small offset is optimal.
Notwithstanding the above points it is not my intent to design an "optimal" 3-element trap yagi. What I am doing is creating a benchmark to serve as a proxy for the better commercial products in the category. Since there is strong market demand to optimize SWR (match) performance it is to be expected that products may have an incentive to sacrifice more gain and F/B performance than I'd prefer.

Unfortunately there is no certain way to judge yagi products for their performance. Published specs are incomplete and graphs too often look a little too perfect to be accepted as presented. This is the case even when the specs are mostly true. We can use the laws of physics (electromagnetism) to explore what is theoretically possible (and probable), knowing that no real product can exceed the performance of an ideal antenna. That is, we can use physics as a sanity test. That is one of my purposes for modelling a variety of small yagi antennas. The hope is that the best small tri-band yagis approach the ideal.

Design Process

The design process for a 3-element tri-bander is straight-forward, though more complex than for the 2-element version. Gain and F/B in the 2-element yagi is solely determined by the boom length (for a fixed style of element design). With 3 elements it is necessary to coordinate the relative resonance of the reflector and director elements to achieve a desired gain and F/B profile over the band of interest, as was described above.

Once the desired gain and F/B profiles are selected per the reflector and director tuning we can think about the match. This is done by adjusting the driven element and matching network (e.g. beta or gamma match). This matching process does not affect gain and F/B, provided the driven element does not get excessively short and the matching network is low loss.

Keep in mind that a lot of the work has already been done for us. Starting with the venerable work of the NBS many years ago, extensive theoretical modelling by W2PV around 1980, and more recent work by many others utilizing state-of-the-art software and high-speed computers. I am leaning on that work when I select design parameters. There is no need for me, or anyone, to design a yagi from scratch. It would be a waste of time. There are no undiscovered secrets left to be ferreted out by aspiring mavericks.

With all the forgoing in mind here is how I went about the design:
  • Choose the resonant frequencies for the reflector and director, in relation to a centre frequency. You can either look up the performance figures or experiment with EZNEC or a similar tool. If you experiment always tune both reflector and director so that they maintain the same ratio with respect to the centre frequency. This will simplify matching. Note that as in the reference yagi it is not possible in a 3-element yagi to get the maximum gain and F/B anywhere near the same frequency, so choose the centre frequency with care.
  • Develop two antenna models: one for the yagi and one for element tuning. Once an element is inserted into the yagi it is very difficult to tune it due to mutual coupling. It is better to tune it in a separate model and copy it to the yagi model. Do the element model in free space.
  • For the trap elements we are using it is necessary to tune the reflector and director on all 3 bands. Calculate the target resonant frequencies on each band and get the element resonance as close to those frequencies as possible. Resonance is the frequency where the reactance is zero, not where the SWR is minimum. Don't fuss over it too since there are other factors not included in the model that affect element tuning: tubing taper schedule; element-to-boom clamps; trap dimension; coupling to other antennas, etc.
  • With all 3 elements in the model the performance curves can be shifted up or down in frequency by changing resonance of both parasitic elements by the same amount.
  • Alter the placement of the driven element and resonance of the parasitic elements in small increments to adjust performance (gain and F/B). Remember to do all resonance tuning in the element model and copy its parameters to the yagi model. You may want separate models for the director and reflector (as I did) to speed the design process.
  • Adjust the driven element in the yagi model to place the SWR curve consistent with your design objective. Add a beta match or similar network if the feed point impedance is too low for a direct match to 50 Ω coax -- this is mandatory in the antenna we are designing. Tuning of the driven element and match has little to no affect on gain and F/B performance, which is why it is done last.
Of course I say you can do all this work but few will and will even want to do so. Since this is my project I will cut to the chase and present the antenna that I designed by this process. As with any yagi antenna it is a compromise, but one that I believe reflects the best computer-designed small tri-band yagis on the market. I will only present the performance figures since the physical dimensions and trap designs are unlikely to be an exact reflection of any commercial product. For example, my model has no boom, and that has an effect on element length. Further, the trap (load) model has no physical dimension, and although the trap performance might be similar the design is almost certainly different from all commercial products.

I chose the following centre frequencies to optimize gain and F/B for CW and lower SSB segment of each band. These frequencies require careful selection since the relationship of optimum frequency for gain and F/B are boom length dependent, and are therefore different on all 3 bands. They are derived from the models developed by W2PV.
  • 20 meters: 14.000 MHz
  • 15 meters: 21.200 MHz
  • 10 meters: 28.600 MHz
Other design parameters:
  • Boom length: 4.2 meters: 0.2λ on 20 meters, 0.3λ on 15 meters and 0.4λ on 10 meters
  • Driven element spacing: 1.9 meters from the reflector, which provides enough space at boom centre for a boom-to-mast clamp but does not result in excessive torque due to wind load asymmetry
  • Reflector element resonance: 13.440, 20.350, 27.450 MHz (0.960 of centre frequency, or 0.502λ length equivalent)
  • Director element resonance:  14.560, 22.050, 29.740 MHz (1.040 of centre frequency, or 0.463λ length equivalent)
TH-3MK4 (from online manual)
The parasitic element lengths can be made similar on all bands despite the different boom lengths on each band as measured in wavelengths. They could be further optimized for the respective boom lengths on each band but since the potential improvements are minor I am opting to keep it simple.

The final design looks a lot like the Hy-Gain TH-3MK4, the schematic of which is shown here. Of course other 3-element tri-band trap yagis look similar and, probably, have similar performance. My choice of the TH-3 as a model is not a recommendation of this antenna. It's just that I know Hy-Gain best and their traps seem to measure well. I have never used a TH-3.

Avoiding Maximum Gain

Before getting into the modelling results for the "optimum" design it is worthwhile to explore what it means to design for maximum gain, and why this is a bad idea. Just for the fun of it I ran the model with identical elements, all being copies of the driven element. This is a symmetrical yagi that will exhibit 0 db of F/B but has bidirectional gain.

The antenna resonated at 14.140 MHz. At right is the free-space elevation pattern of the antenna. Notice that the gain is quite poor, worse than a dipole, but is reasonably directive. Where did the power go?

At resonance the impedance is 1.9 + j0 Ω. This is exceptionally low. The radiation resistance rises steeply on either side of resonance, so this is a sharp dip. Not only does this make it difficult to design an effective, low-loss matching network the low R value is an invitation to other losses. In particular the traps (modelled at 0.3 Ω ESR, the same as in Part 2) contribute -3.8 db of loss. Even with large aluminum tubing for the elements we get -0.3 db of loss. If there were no losses the bidirectional (broadside) gain would be 5.1 dbi.

There are similar losses on 15 meters, where the traps are most active. On 10 meters the trap loss is small, however the aluminum I²R losses are still present.

This demonstrates why a yagi with 3 or more elements should never be designed for maximum gain. The gain bandwidth is narrow and the target gain cannot be achieved in any case due to losses caused by the low radiation resistance. This is true even in an antenna without traps or other loads.

Trouble with traps

In the interests of sharing what I learned I will confess that the process I came up with above to tune the parasitic trap element did not work well. While I did suspect the possibility of problems, especially with director tuning, I was unprepared for the magnitude of the resulting errors. In the case of 10 meters the performance was mediocre and the performance curves were shifted 1 MHz higher. The same happened on 15 meters, although the curves shifted less than 200 kHz higher. On 20 meters the frequencies came out right but the performance was poorer than expected.

What happened? I know the design process works on yagis comprised of unloaded elements, and in fact got a textbook result with the reference yagi I developed in Part 1. To understand the source of the problem it is useful to make a brief detour to look at how yagis work.

The far-field pattern of a yagi is the superposition of radiation from all of the antenna elements (plus ground reflections). When these are in phase in a specific direction they add, resulting in gain. The amount of gain depends on how close their phases are, and their magnitudes as well. When they are exactly out of phase (180° difference) and the magnitudes are equal there is no net radiation in that direction.

The phase and magnitude of the radiation from a parasitic element depends on:
  • Separation: It takes time for the field from the driven element to propagate to the parasite, a time that is proportional to distance. The phase shift is proportional to the separation measured in wavelengths of the radiation. For example, for 0.2λ separation the phase differential is 72°. That is, by the time the field reaches the parasite the phase angle at the driven element has advanced 72°. In all other directions the phase relationship differs, and therefore the field strength is dependent on the elevation and azimuth angles.
  • Reactance: In general, the closer the parasite resonant frequency (where reactance is 0 Ω) is to the driven element the greater the magnitude of the induced current. When the parasite is tuned to another frequency the reactance causes the magnitude of the induced current to fall and, importantly, the phase of the induced current will be shifted. This determines the phase and magnitude of the radiation from the parasite.

The dynamics are actually more complex since it is a recursive process. Radiation from the parasites induces current in all the other elements, including the driven element, which then combine with the currents on those elements and are again radiated. This is a continuous process while the source is active and energy from the induced fields is available (not yet launched into space or dissipated by I²R losses). EZNEC shows the data for the steady-state current magnitude and phase for every wire segment in the model.

Another important facet of the dynamics is that the mutual coupling lowers system impedance and thus raises the total current in the system for a given source power. Gain is therefore not only a function of superposition but also the system impedance. (EZNEC normalizes the current to 1 A at the source by adjusting the source power unless you tell it to keep the power constant instead.)

When we specify a parasitic element by length relative to a resonant λ/2 element (or the equivalent resonant frequency) it is merely a calculation shortcut. What matters is the reactance. In a loaded element, whether done with coils, capacitance hats or traps, that shortcut is invalid. That's where I went wrong on the first try. Once again it's those traps making my task more difficult.

Because of the complex contribution to phase by the traps it is not possible to tune the parasitic elements by frequency or length. I worked around this by modelling a tube, without traps, to the target director and reflector resonant frequencies on each band (as shown above). I then read off the reactance at the centre frequency -- the X in Z = R + jX.
  • 20 meters: Reflector [+41 Ω]; Director [-37 Ω]
  • 15 meters: Reflector [+38 Ω]; Director [-35 Ω]
  • 10 meters: Reflector [+36 Ω]; Director [-33 Ω]
I retuned the models of the reflector and director, adjusting them so that their impedances showed these reactances at the centre frequencies of all 3 bands. With some trepidation I copied these elements over to the yagi model and observed the gain and F/B across the bands. Lucky for me this worked beautifully.

Small frequency shifts were still required on 2 bands but otherwise the gain and F/B performance were per the textbook. I then proceeded to set the trap ESR to 0.3 Ω (just as in the 2-element yagi) and match the system to 50 Ω.


To avoid boring you with the processdetails, I will directly proceed to the results. These are visible in the collection of charts on the right.

I used the same chart structure as for the 2-element antenna to make comparison easier. I will do a more direct comparison at the end of this series (Part 5?) if you don't want to bother flipping back and forth as you read.

All measurements are made in free space. All of these antennas, including the 2-element version will respond to height above real ground in much the same way so this approach avoids confusing height for raw performance. We will come back to antenna height in a later article in this series.

My observations on the performance models:
  • Gain and F/B bandwidth are superior to the 2-element tri-bander model. Although the maximum gain on 15 and 20 meters is only modestly better, it is available over more of each band.
  • Gain on 10 meters approximates that of a full-size 3-element yagi, reaching a maximum of more than 9 dbi.
  • 15 meters is, again, the worst performer. Bandwidth is narrow, both for performance and SWR (not shown). At the top end of the band the yagi reverses direction, as indicated by negative F/B. Peak F/B is excellent since the boom is close to an odd multiple of λ/4 (0.3λ).
  • Loss due to trap ESR (equivalent series resistance) is, as always, highest around the frequency of maximum gain. In the case of 3-element yagis this frequency is high in the band, whereas it is low in the band for 2-element yagis. This is a general rule for 2 and 3-element yagis, and is unrelated to the use of traps.
You may have noticed that I did not plot SWR. Coming up with a beta match that would give a low SWR on all 3 bands became tedious and I gave up after working on it for a time. Surely that ideal beta match does exist -- most commercial products have done so -- but there was no point in discovering it in this model since I am not planning on building this antenna and it has no impact on gain and F/B performance.

I got close enough to the ideal by playing with beta match parameters (stub impedance and length, and driven element length) to discover the implication for match bandwidth. It is good on 10 and 20 but not good on 15 meters. Once again it is that middle band that causes problems, just as it did on the 2-element tri-bander and even the trap dipole. I do know that Hy-Gain uses different 15 meter traps on the driven and parasitic elements so they presumably have learned something that I have not.

Trap ESR Loss

That last point brings us to the impact of traps on performance. I did the same sensitivity analysis as for the 2-element tri-band yagi. You can go back to Part 2 to see what I had to say on the topic. Here I will simply present the charts that show the impact of trap ESR.

As a reminder, I used 0.3 Ω as the traps ESR for the above performance analysis since that appears to be in the range of the best commercial traps in tri-band yagis. An ESR of 0 Ω represents a lossless trap, which is useful as a theoretical benchmark but is impossible to achieve in the real world.

Loss is highest where gain is maximum since that is typically where the radiation resistance of the antenna is lowest. On 20 meters this is the high end of the band and on 10 and 15 meters it occurs mid-band. On 15 meters the loss is especially high since the radiation resistance is below 10 Ω over a large section of the band.

Apart from 15 meters the impact of trap ESR is modest. Traps are a great convenience in designing a compact tri-band yagi with good performance, but this is a cost that must be paid.

Note: The charts say the reflector tuning is 1.05 Fc, but it is 1.04 Fc as stated earlier in the design process. Or (more correctly) tuning is per the exact reactance values stated above.

Next up...

In Part 4 I will evaluate the Spiderbeam category of multi-band yagi. This is a promising antenna since it has no traps or aluminum tubing. However, as we will see, there are still compromises and trade-offs.

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