Wednesday, December 16, 2015

40 Meter 3-element Yagi: Matching vs Optimization

Yagi performance can be pretty well summarized by the following metrics:
  • Gain
  • F/B
  • SWR
We can model or measure these, across the band of interest, and then compare to make the choice that suits our individual needs. In all cases we should take care to account for and, if possible, minimize losses due to ground, conductor resistance, load equivalent series resistance and matching network. This is as true of all antennas, including dipoles and verticals, not just yagis.

As a general rule you can optimize the performance of a yagi, though usually only of one or two of those three metrics. As you move lower in frequency the optimization challenge increases since the bands grow quite large when expressed as a percentage. Good performance at one frequency is a poor indicator of performance across the entire band.
  • 20 meters: 2.5%
  • 40 meters: 4.3%
  • 80 meters: 14.3%
  • 160 meters: 11.1%
For the lower HF bands the challenge is usually a bit less than stated since, in some parts of the world, the bands are narrower, or we are primarily interested in a narrower slice of spectrum. For me these would be 7.0 to 7.2 MHz (2.9%), 3.5 to 3.8 MHz (8.6%) and 1.8 to 1.9 MHz (5.6%). Even so it is difficult to optimize antennas for these wide bandwidths.

In this article I'll focus on a 3-element, full-size yagi for 40 meters. Getting good performance over the entire band, even up to 7.3 MHz, is achievable without excess compromise. If you've been a regular reader you'll know that 40 meter yagis are of specific interest to me. If your interest differs the lessons can be applied to yagis for other bands.

The model of the yagi is kept simple in this exercise:
  • Boom: 15 meters. This is slightly longer than the standard 48' boom for a 3-element full-size yagi.
  • Elements: 40 mm diameter tubing. Correction for taper can be included before a design is turned into a construction template.
  • Driven element position: Unless specified otherwise, it is slightly offset from centre towards the reflector by 0.2 meters. This provides clearance from the boom-to-mast bracket and usually nets a small amount of additional gain.
  • Parasite tuning is specified as ±X% for a selected centre frequency. The centre frequency is with respect to gain and F/B frequency positioning, not resonance or matching. This method of tuning parasites must not be used for shortened parasitic elements, whether by coil, capacity hat, trap or other technique.
With that preamble out of the way let's dive into some design.

Optimizing the yagi

Gain in a yagi is determined by two parameters: boom length and parasite tuning. You then only need elements distributed along the boom so that there is sufficient coupling to fully exploit the boom length. The tighter the parasites are tuned (small deviation from resonance), the greater the gain. The driven element only affects gain and F/B a small amount, mostly determined by its spacing from the reflector and first director. We are free to adjust driven element length (within reason) in pursuit of a good match.
I first learned these lessons a long time ago from W2PV in his excellent book Yagi Antenna Design. The book is out of print, but well worth picking up on the used market. In a few pages of graphs he illustrates the performance differences in yagis with various boom lengths, element spacing and parasite tuning. Although we now have better modelling software his presentation of the material is still top notch.
Unfortunately, as we maximize gain, while the F/B may suffer somewhat, the SWR rapidly becomes a problem. The reason is that high gain is associated with low radiation resistance. By Ohm's Law this means increased current. The higher current is partially responsible for the increased gain. It is also responsible for increased I²R loss. In a yagi with shortened elements, whether traps and coils, there can be considerable loss. There typically is negligible loss in HF yagis with full-size elements, due to the large diameter tubing. However, whether the elements are short or long the low radiation resistance makes matching difficult.

With loose parasite tuning the matching problem is easier to solve, at the expense of gain. F/B will generally remain good. An example is the optimized 3-element 40 meter yagi presented in the ARRL Antenna Book. The SWR is kept below 2 across the band, and F/B is good, but gain is sacrificed. This is due to the wide ±7.3% parasite tuning. The reflector is 7.3% longer and the director is 7.3% shorter than resonant length at the centre frequency.

Tightening parasite tuning to ±3% gives us a gain greater than 9 dbi (in free space) across the band. This is almost 1.5 db better than the ARRL design. The downside is a very low radiation resistance of ~12 Ω, which is difficult to match and only allows a 2:1 SWR bandwidth of 125 kHz. Optimizing for gain prevents optimizing for match.

If we loosen parasite tuning to around ±6% the radiation resistance rises to ~24 Ω, which is easier to match to 50 Ω coax and increases the 2:1 SWR bandwidth to 200 kHz. Gain is almost 1 db better than the ARRL design.

The charts at right plot the impedance components (R & X) for the tightly-tuned yagi optimized for gain versus the more loosely tuned yagi. The gain difference is between 0.5 and 0.75 db, and falls to nil at 7.3 MHz. F/B is slightly better for the gain optimized yagi at the high end of the band.

As is typical for 3-element yagis the peak F/B is below the design frequency and the peak gain is above the design frequency. The parasites can be tuned in tandem to shift these curves to where you want them. I chose 7.1 MHz as the design frequency since the resulting F/B and gain curves best meet my needs.

As should be obvious from the impedance chart the radiation resistance is the reason for the reduced bandwidth of the gain optimized yagi; the reactance curves are similar. To best transform the impedance to 50 Ω it helps to have the ratio of X to R as small as possible. This is why the SWR bandwidth of the gain optimized yagi is so poor. Even for the loosely tuned yagi the SWR bandwidth is only 2/3 of the band. That is, in North America; in Europe it will cover the entire 40 meter band.

Some improvement is possible with larger diameter tubing since the reactance changes less with frequency. That is impractical for a 40 meter yagi, for reasons of expense, weight and wind load. In any case the improvement too small to be worth the effort.

The model uses constant diameter aluminum tubing, which is clearly unrealistic. Tapered tubing will alter the element tuning but, after adjustment for the taper, the performance results will not be appreciably different. Since this is a preliminary study, not a construction article, I kept the model simple.

Passive matching networks

There are several common matching networks used to transform the impedance to 50 Ω. They are all similar in that they use transmission line sections in combination with an off-resonance driven element to construct what is, in essence, an L network with a shunt inductance and a series capacitance.
Gain and F/B performance are only negligibly affected by these matching networks. The gamma and T networks are often preferred in large yagis since these do not require insulating and sectioning the driven element for a dipole feed.

Using EZNEC I modeled a few of the aforementioned matching networks. The matches they deliver are indistinguishable. The only real difference is due to an offset in the curves of ~25 kHz that I was too lazy to tune out in the model. Since the gamma and T are more difficult to model with EZNEC, and will have a similar results, I chose to omit them.

An L network made from coils and capacitors would be little different. Actually it would be better but the difference is so small as to be inconsequential. The lesson is that if you use a passive matching network you should choose the one that is easiest to build and adjust, or that has some other feature you value. For example, a gamma match doesn't require the driven element to be split or insulated from the boom. In a 40 meter yagi this can be advantageous.

I modelled the λ/4 transformer with two parallel 6.93 meters lengths of RG-11 (35 Ω). The λ/12 sections are 2.31 meter lengths of RG-213 connected to the driven element, and two parallel 2.31 meter lengths of RG-213 (25 Ω) from there to the 50 Ω transmission line. Parallel matching sections of these types can be constructed with two T-connectors. The beta match hairpin is 1.1 meters of 300 Ω transmission line (1.0 μH) made from aluminum tubing. The driven element shortened to centre the SWR curve.

All these feed systems require a common mode choke or suitable 1:1 balun to prevent common mode current on the transmission line and, likely, degraded F/B.

Active networks

To fully tame the SWR at the feed point an active network can be used. This can be something quite simple if "good enough" results are acceptable. To demonstrate, I took the beta match design from above, which gives a low SWR in the lower part of the band, and added relay activated components at the feed point.

It was quick work in EZNEC to experimentally find the component values I wanted. A pair of 1,500 pf capacitors in series with each half of the driven element lower the SWR below 2 in the high end of the band. You should not use only one capacitor since that would unbalance the driven element. A DPST relay (or a pair of SPST relays) unshorts the capacitors. Use the opposite arrangement (normally open relay contacts) if you primarily use SSB. The capacitors should be transmitting "door knob" type for this high power application.

Alternatively, coils can be used instead of capacitors, if you find they are easier to work with. In this case we would leave the tuning the driven element as before and short the coils when operating above 7.2 MHz. I didn't bother to calculate or model the required coil values.

The SWR does not dip as low as in the passive network since the capacitors (or coils) alter the matching network, and do not simply cancel the capacitive reactance of the shortened driven element. A fully switched network, with coils and capacitors, is needed to optimize the SWR for both switch positions. I chose to keep it simple since that is sufficient to my needs.

A remote or shack-based tuner can also be used. If you choose this active tuning option you ought to calculate the transmission line loss due to the high SWR. Low loss transmission line (e.g. Heliax) may be appropriate.

Coupled resonator

There is another method of achieving a broadband match: the coupled resonator. This requires another full-size element that is tuned to a higher frequency than the driven element, and is positioned close to the driven element. Parasite position and tuning remain unchanged.

A coupled resonator should not be confused with a Moxon. Here the coupled resonator is equivalent to a second driven element. There are designs that use dual driven elements to extend the SWR bandwidth, but a coupled resonator is simpler.

By virtue of the tight coupling with the driven element the current is, in a fashion, split between them. With the subsequently lower current the feed point resistance is raised. By careful tuning and positioning of the driven element and coupled resonator the antenna can present a low SWR across the 40 meter band.
Note: Some label the coupled resonator the first director. That is misleading. True, it is a parasitic element. However it has little impact on gain and F/B when inserted into a conventionally designed yagi. The difference it does make in this respect has more to do with the current distribution, which effectively alters the spacing between the (dual) driven element and the adjacent parasitic elements.
The preliminary design I came up with places the two elements 1 meter apart, each 0.5 meters from the boom centre. The driven element is lengthened to resonate lower in the band and the coupled resonator high in the band. Current in the coupled resonator is 45% that of the driven element at 7.0 MHz and rises steadily to 105% at 7.3 MHz where it is close to resonance.

SWR is below 2 across the band. It was necessary spread the tuning of the parasites a small amount to tame the SWR above 7.25 MHz. However I only gave up 0.1 db of gain and actually improved the F/B compared to the yagi designs discussed earlier. Better results should be possible.

I did briefly attempt to use a coupled resonator to match the gain optimized yagi, with only partial success. I could not get the SWR below 2 across the entire band, but did increase it to over 200 kHz, as compared to the 125 kHz with a conventional matching network.

Although this design "works" it is in no way optimum. This is merely an early attempt to test the use of a coupled resonator in a large antenna of this type, and form a base from which to test other designs. There are commercial antennas that use the same concept, and there are the WA3FET OWA designs that have gone through a substantial amount of optimization. If the concept intrigues you, check those out. Perusing the web I noticed that K9CT has an antenna that is similar to the above design. Yes, it's very big!

The driven element requires a dipole feed: split element and isolated from the boom. The coupled resonator can be mounted directly to the boom. A common mode choke or suitable 1:1 balun is required to prevent common mode current on the transmission line.

The main disadvantage of the coupled resonator is the mechanical load and cost of the additional element. For most it may be better to use an active matching network as discussed above.


Modern contest stations value flexibility. There is an attraction to a high-performance 40 meter yagi that requires no tuning of station equipment, transceiver or amplifier, when changing frequency or when switching antennas. An assured low SWR under all situations delivers that flexibility.

As we've seen there is a price to be paid for this flexibility. It comes as reduced gain, higher cost and physical size. What would be acceptable on higher bands is less so on 40 meters. Yet it is a band that requires high performance if one is to do well in contesting, and in competitive DXing. This is increasingly true as sunspots continue to decline over the next few years.

Well, for the interim I do have an XM240 in storage, ready to be raised once I have a QTH and tower in place. That antenna is ultimately intended as a second antenna for 40 meters. So I plan for something larger as the primary antenna. The work I've done for this article is part of that planning.

To summarize, here are my choices as I see them:
  • Wind and ice load: The coupled resonator design adds substantial load. Even so it is arguable that one more element added near the centre of a full-size yagi is not a large step to take.
  • Cost: Full-size elements and more than 3 elements substantially adds to the cost of the antenna. That is, if the antenna is to survive the elements and be reliably available when contest weekends arrive.
  • Matching: Active and passive impedance matching options each have their pros and cons. Passive, low-SWR broadband matches are preferable. This favours the coupled resonator. A switched set of loads to extend the range of a conventional matching network may be my best choice. Initially I could use low loss coax and a tuner in the shack. The tuner could be bypassed for operation between 7.0 and 7.2 MHz. A coupled resonator can be added later if I decide to take that step.
  • Gain: Sacrificing less than 1 db of gain for a better match is acceptable. I will avoid sacrificing more than that.
When I next revisit this topic I will look at the possibilities of long boom yagis with shortened elements. These are popular antennas since the gain sacrificed is compensated by substantially less cost and wind and ice load. But it can get complicated.

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