Tedium of Single Band Contests

As I write this, the 2022 CQ WW 160 Meter contest is not over. But it's over for me, and has been since Saturday evening. The endless tedium wore me down. Better DX conditions would have helped, since without that the prospect for contacts is severely limited.

Some people like that and will work with whatever the propagation gods (or devils) provide. I can live with that in most contests. You make do since you know that everyone else is in the same situation. Skill and technology are what make winners.

In a single band contest the application of skill is narrower. When the DX isn't there on top band you have two things you can do: call CQ endlessly and trawl the bands (and cluster/skimmer spots) for fresh meat. I can maintain my enthusiasm for a while. Eventually the tedium gets me down. I have regularly bailed early from contests such as these. This isn't a new problem for me, since it was the same when I first got into contesting as a teenager in the 1970s.

My strategies for countering the tedium have been to operate with a handicap like QRP or as a member of a multi-op team. With an inferior signal every QSO is an accomplishment. There are always more stations to work and challenges to overcome. With a kilowatt and a full size vertical on top band the QSOs are not so difficult. After the first 600 to 700 QSOs you have milked at least 80% of the available stations and the rate falls off a cliff.

I had over 700 QSOs in the log when I shut down in the wee hours of the first night of the contest. I worked every available state/province multiplier and over 50 countries, despite the mediocre conditions. Many other countries were heard but they did not hear as well as the hordes calling them. Productivity suffered and the long tedium had begun.

When I resumed Saturday evening the rate improved as many stations made their first foray into the contest. Very soon that pool dried up and the tedium resumed, worse than before. Many contesters love the challenge of scrapping for every contact. The best operators scan the bands or jump on spots with one receiver while listening to answers to their CQ machines on another receiver. A few have sophisticated equipment to allow them to do both concurrently. The rest of us can only listen on the second receiver when the CQ machine isn't transmitting.

I am not happy with that style of operating. DX conditions were worse the second night and that accentuated the tedium. So I quit the contest. I never had the inclination of winning in any category and, frankly, my station, good as it is, is inferior to many others. There is added frustration from the Beverages which allow me to copy too many stations that cannot hear me. Oftentimes a kilowatt is no more productive than QRP. At least the amp keeps the shack warm during winter weekends.

I can sustain my enthusiasm in a multi-op because off times break the tedium. Every time I sit down to do a shift I am refreshed and able to dig in and get the job done. As a single op the tedium is unremitting and it wears me down. I admire the operators who deal with it better than I can.

This problem is common to most single band contests. It also occurs in contests with limited participation. I place ARRL Sweepstakes in a similar category since you can only work stations once regardless of band. The outcome is much the same, but with more spectrum to cover.

For the contests I most enjoy you respond to conditions and ever-changing activity levels by band and mode changes, alternating CQ and S & P, SO2R, watching signal strengths and spectrum scopes, among other strategies. Building and operating a multi-faceted station that permits these tactics sustains my interest and enthusiasm before, during and after the contest. In many respects I enjoy that more than operating in contests.

Single band contests rarely suit my desires. There are exceptions. Examples in 2021 include my 15 meter single band assisted entry in the CQ WW SSB contest and my (unofficial) 20 meter assisted entry in the ARRL DX SSB contest. I had fun putting to the test the recently completed stacks for 15 and 20 meters. I may do the same on 40 meters in one of the upcoming ARRL DX contest weekends to see what I can do with the recently completed 3-element yagi.

I will continue to operate the ARRL and CQ 160 contests but probably never with the objective of winning. To become competitively oriented I would need to make it a multi-op operation. As a single op, it's mostly an opportunity to work DX that is otherwise sparse on top band and to compare my antenna and station capabilities to those of others. The latter spurs my technical side to improve my station. 

Outside of contests, there is now far more top band DX to be worked on FT8, and I do make regular appearances. That's a story I'll defer to a future article.

Yagi Side Resonance

RF is very sociable: it happily interacts with everything in the universe. A few interactions are significant, some are minor and the vast majority are negligible. As hams we discover or predict interactions, then deal with the problematic ones and ignore the rest. Often, interactions are unknown or dismissed, and only addressed when they are too severe to be ignored. 

As deliberately radiating devices, antennas are especially prone to interactions, as the perpetrator or as the victim, or both. As the victim, an antenna's pattern and efficiency can be degraded by interactions. Those interactions can be with structures, utilities, trees, ground other antennas within a couple of wavelengths.

When I designed my 3-element 40 meter yagi I paid particular attention to its interaction as the perpetrator and the 15 meter stack as the victim. It is big and high enough that it is not particularly prone to be a victim. This is helped by the vertical polarization of the 80 and 160 meter antennas: orthogonal antennas interact little.

Here you see a simple 80 meter dipole excited at 3.6 MHz. It is positioned 200' (60 m) laterally from the centre of the 40 meter yagi and 50' (15 m) lower. The 40 meter yagi has a substantial mutual impedance and is happily interacting. How is this possible? 

It is no surprise that the pattern of the 80 meter dipole is distorted. For an 80 meter yagi with 2 or more elements the pattern degradation would be substantial. This is of particular interest to me since I am considering a wire yagi for 80 meters suspended between my two tall towers, including the one supporting the 40 meter yagi.

A ham in the distant past had the interesting idea of making a yagi into a dipole for the next lower band. Viewed from the side a yagi is a short dipole with large capacitance hats. A typical 3-element yagi resonates at about half that of the design frequency. That is, a 3-element 20 meter yagi is resonant within or near the 40 meter band, and the same is true of a 3-element 40 meter yagi on 80 meters.

The trick is with the feed point. You must excite the boom and not the driven element. Although they are mechanically and electrically connected the 40 meter feed is neutral with respect to the boom. As is common knowledge, the elements of a yagi are electrically independent of an orthogonal boom that crosses their centres.

Yagis with elements insulated from the boom do not have this attribute. At least the outermost parasitic elements -- reflector and last director -- must be electrically connected to the boom. 

Since the boom is a continuous conductor a similar feed system is required as for a yagi with a continuous driven element. A gamma match may work. More often an omega match must be used since the yagi cannot be adjusted to be compatible with a gamma match. In an earlier article I referenced a couple of construction articles by those who have done this.

Were the elements of the yagi insulated from the boom, its 46' (14 m) length would serve pretty well as a dipole on 30 meters.  

For those modelling these effects it will be necessary to add the boom to the model. It is ordinarily omitted from the model because, other than the element-to-boom clamp, the boom is neutral. To model the boom dipole I had to split the centre wire of each element to allow connection of the wires that comprise the boom. SDC (stepped diameter correction) and other NEC2 limitations that confound accurate yagi modelling are a relatively minor source of inaccuracy when the elements are employed as capacitance hats.

Both matching networks are difficult to retrofit to an existing 40 meter yagi due to its size. For example, the calculated gamma rod length for the impedance at 3.6 MHz is 13' (4 m) long. There is no good way to do that without a crane and basket. More capacitive reactance can reduce the length by several feet but that requires rebuilding the 40 meter yagi! Perhaps a "long enough" but short gamma rod and an L-network will get you to 50 Ω.

Notice that there are other resonances, which is not unusual for an antenna with many protuberances. There is a resonance at 14.4 MHz that, due to model inaccuracies, might actually fall within the 20 meter band. I have not done the modelling to determine the magnitude of the 20 meter interaction.

Although the antenna can have a DE feed point for 40 and a boom feed point for 80 meters, it is not advisable to operate both bands concurrently without band pass filters that are up to the challenge. However this is usually only a concern for multi-op and SO2R contesters. Everyone else can utilize two transmission lines (one terminated) or a switch at the antenna.

There are pros and cons in this situation. The major benefit is the possibility of a high dipole on a band where one might otherwise only be able to deploy a wire antenna like an inverted vee at a lower height. The major disadvantage is that a yagi can interact with a horizontal antenna for the lower band when the boom is broadside to that antenna. There is no one solution since every station is different with respect to operating objectives and favoured beam directions.

That lower resonance is there whether you feed the boom or not. Putting that lower band resonance to use as a dipole may involve significant effort. The boom resonance can be defeated in the same manner as for a tower. However I've never heard of someone taking the trouble to do so. Interactions are silent so that their impact is easily mistaken for other things and therefore may never be addressed.

Out of interest I created a boom model for my 5-element 20 meter yagis. The resonance is lower than half of 14 MHz because, with 5 elements, the boom is electrically longer. The boom resonance surprisingly falls right inside the 60 meter band. Of course it almost certainly doesn't but it should be nearby. But I have no interest in that band and the interaction isn't important since it isn't a contest band.

Notice how the elements adjacent to the feed point at the boom centre are very active (directors 1 and 2), but not so for the other 3 elements. There isn't much current flowing at the extremities since the most active elements introduce a current distribution discontinuity. That is in the nature of capacitance hats. There is a similar effect with loading coils. That is why they are effective at their job of shortening an antenna.

More can be said of unusual yagi resonances, how to positively exploit them and the downside of interactions with other antennas. But I won't. Every antenna and station is unique so it's enough to raise awareness of this little understood issue. Look around your antenna farm, whether it is big or small, and consider what might be at play. Most will shrug it off since characterizing the scope of the problem and taking mitigation measures is neither easy nor straight-forward.

If I build the 80 meter wire yagi that I have tentatively inserted into my 2022 plan there are factors that lessen my worry. First, most of the time the boom of the 40 meter yagi will not be parallel to the 80 meter elements. Second, yagis have enough directivity to reduce interaction when they are at different heights. Third, I still have the 80 meter vertical yagi should a problem arise when the 40 meter yagi is pointing in a direction that increases the interaction.

When I get serious about building an 80 meter wire yagi I will develop models to explore the matter in more depth. I have no plan to convert any of my yagis as lower band dipoles since I have enough antennas to meet my needs. My concern is interactions.

Gamma Match Peculiarities & Challenges

After a brief struggle with the gamma match on the newly installed 3-element 40 meter yagi I thought to myself that there has to be a better way of doing this. Unlike many other ways of feeding yagis, the gamma match remains an enigma. Calculating its dimensions is at best an estimate and it can be off by quite a lot. 

We can get by despite this because the gamma match will match a wide range of feed point impedances. A bit of trial and error usually does the trick. It is interesting that it can be so difficult to analyze and accurately predict its behaviour!

The most common style of gamma match is shown below. A reference diagram will help to avoid the possibility of confusion even though most hams have encountered the gamma match.

Unfortunately I do not have the relevant education to dive too deep into the transmission line and antenna theory to properly understand the gamma match. I can get through partway and then I will inevitably hit a wall. I have not been successful, so far, with published material on the gamma match since they tend to touch lightly on important theoretical points, make assumptions that overlook the general case, or crunch through the complex number equations and hyperbolic trigonometry without explaining what is being done and why.

The basic behaviour of a gamma match is as follows. I am paraphrasing an article that like by Healey W3PG in April 1969 QST. His figure 4 is reproduced above. ARRL members can pull the article from the QST archive.

1. As a short or partial folded dipole, the current split on the DE (driven element) and gamma rod multiplies the feed point impedance (both R and X components) by a step-up ratio. The ratio is determined by the tube diameters and spacing. 
2. The impedance is raised before being stepped up because the DE tap point (gamma rod strap) is off centre. This occurs due to the current and voltage gradient along the element, with voltage rising and current declining as you move toward the element tips. The increase isn't large -- a typical value is 5% -- but well worth taking into account.
   
3. The shorted stub formed by the gamma rod and element adds parallel inductive reactance Xp to the stepped up impedance Z₂ at the DE tap point and also transforms the net impedance since it is a transmission line. 
4. In a typical gamma match, the reactance at the feed point is inductive. The series gamma capacitor -Xp cancels that reactance at the design frequency.

When properly configured the feed point impedance is 50 + j0 Ω, at one selected frequency, usually near the centre of frequency range of interest. The SWR curve across the band should be similar to that of other antenna feed systems. As Cebik W4RNL points out in "Some Preliminary Notes on the Gamma Match", and I found with my new 40 meter yagi, getting the gamma match to do that is not as straight-forward as for other common feed systems. The mathematics are more difficult, the physical dimensions are critical and the (dipole) feed point impedance we seek to transform may be impossible to measure. (Search for the Cebik article in a search engine to locate an extant archive.)

I would like to better understand the gamma match. I have most of the mathematical training required but not enough of the electromagnetic physics and engineering. I sometimes joke that I've forgotten more mathematics than most people have ever learned. There is a thin layer of rust coating my neurons. However I never formally studied electromagnetism. 

Bear with me as I plow through what I can of the subject without delving deep into aspects that I am not yet qualified to comment on. After designing a number of yagis and gamma matches, and then modelling, measuring and adjusting them, there are several challenges about gamma matches that I've been contemplating:

• Achieving a perfect 50 + j0 Ω match at one frequency is not difficult. Since a yagi is a high Q antenna its impedance changes rapidly with frequency. There is no gamma match design process that I'm aware of that optimizes the SWR across an amateur band. Is there a best set of driven element length (pre-match R + jX Ω) and gamma dimensions that maximizes the SWR bandwidth? If there is one, how large an improvement is possible?
  
• For a yagi like the one I built for 40 meters that cannot be accurately modelled with NEC2, it would be helpful to reverse the calculations. That is, knowing all the dimensions and the impedance at the gamma match feed point, what is the dipole feed point impedance? Since the element is continuous rather than split there is no good way to measure it directly; I used a split centre for my early experiments. We most often rely on models and heuristics to estimate the impedance to be matched. But I have no accurate model of the antenna impedance and I need to know the actual impedance across the band to fully understand the antenna's behaviour.
  
• Reliability of the gamma match calculations is sensitive to several mechanical design parameters that are difficult to control in practice:

• The "open stub" between the gamma rod beyond the shorting strap and the driven element
• Tube diameter steps along the driven element and gamma rod that are typical of large yagis
• Asymmetric lengths (imbalance) between the coax connector and the connections to the DE and gamma capacitor
• Transmission line effects of a long, thin tubular gamma capacitor versus a fixed capacitor (lumped constant)
• What is the best gamma match topology to allow a minimum number of supplemental switched reactance elements to improve the SWR at the high end of the 40 meter band? The gain and directivity are very good high in the band but the rapidly declining radiation resistance prevents achievement of a low SWR across the full band with a static matching network. Yes, an OWA design with a coupled resonator does far better but that is not the problem I'm trying to solve.
  

The challenge is greater for the 40 meter yagi than I experienced with my yagis for 20, 15 and 10 meters because of the increased mechanical complexity, physical size and the greater bandwidth of the band. For example, there are 4 dimension steps along the gamma match, and the 300 kHz bandwidth of the band is equivalent to 600 kHz at 14 MHz and 900 kHz at 21 MHz.

Restricting ourselves to the bottom 200 kHz of the band helps to manage the SWR bandwidth but it is still a greater matching challenge compared to yagis for the higher bands, especially since having more than 3 elements makes it easier to achieve a wider SWR bandwidth. 

I ran into issues with the gamma match on the 40 meter yagi that were not as easy to deal with as they were for the higher band yagis I constructed over the past few years. The greater bandwidth requirement is one of the reasons. Since it's cold and snowy this time of winter I am taking the opportunity to better understand the gamma match and plan for improvements. When warmer weather arrives and I can comfortably climb the tower those steps can be taken.

Instrument choice

Let's assume you measure a perfect 50 + j0 Ω with your antenna analyzer connected to the gamma match feed point using a variable capacitor for the gamma capacitor. Now swing the capacitor back and forth until you get, say, -100 Ω and +100 Ω reactance. You will notice that the R value also changes.

There are two potential causes for the varying R. One is that the mismatch causes standing waves through the components of the gamma match and antenna. In practice this is really not an issue for a typical antenna utilizing a gamma match.

The other cause is the measuring instrument. Measuring antennas is usually done with a handheld single port analyzer. These common instruments run the gamut from horrendously inaccurate to very good accuracy. The Rig Experts AA54 I use is middle of the pack in expense and quite accurate for the price. That is, to a point. Every antenna analyzer and VNA will exhibit decreasing accuracy as R and X depart from their nominal port impedance of 50 + j0 Ω.

In the case of the AA54, when I achieved a 50 + j0 Ω match at about 7.085 MHz with 290 pf of series capacitance, I swung the 500 pf variable capacitor from one extreme to the other. R varied by a few ohms, in the range 46 Ω to 54 Ω. 

One investigator who I will not name noticed a far greater range of R when using an inferior instrument and concluded that the various mathematical models of gamma matches were unreliable! I was amused since it was obvious that he was really measuring the poor accuracy of his antenna analyzer and not the antenna. That did not appear to occur to him since he questioned none of the measurements.

Since the antenna analyzer has that inherent inaccuracy it is better in most cases to adjust the capacitor to get X = 0, measure the capacitor's value and calculate the capacitive reactance. The R value, if it's not too far from 50 Ω, will be quite accurate even on poor instruments.

You should invest in an instrument that is equal to the task you require of it or you should modify your use of it to mitigate its limitations. Compared to the problems you'll run into with a poor instrument the modest expense is worthwhile. I know far too many hams, some of whom have reached out to me after reading my blog, that use poor antenna analyzers and then make excuses when they see nonsense readings rather than invest in a better instrument. Please don't be one of those hams.

Gamma section stepped tubing

The DE and gamma rod dimension steps along the gamma match length, and the Z₀ of the 2-wire transmission line they form, are:

• [4"] 7.5" element-to-boom plate & 0.84" rod: Z₀ = ~270 Ω & step-up of ~13
• [26"] 2.375" DE & 0.84" rod: Z₀ = 340 Ω & step-up of 6.0
• [18"] 1.9" DE & 0.84" rod: Z₀ = 350 Ω & step-up of 5.4
• [variable] 1.9" DE & 0.625" rod: Z₀ = 370 Ω & step-up of 5.9

This is mathematically messy. Aside from the boom clamp, most HF gamma matches do not have steps on the DE and gamma rod. Perhaps a weighted average is a good enough estimate for those critical parameters. I hope so since that's what I've been doing.

You can certainly find a match by combining a naive estimate with trial-and-error adjustments. For the majority of cases that is good enough. However, it's a problem if you would like to reverse the gamma match equations to find the dipole impedance of the unmodified antenna. As already said, "plumber's delight" construction of the DE does not allow a direct measurement of the dipole impedance.

Gamma capacitor

There are two common types of series gamma capacitor: lumped constant (fixed or variable capacitor) and cylindrical (insulated wire or tube inside the gamma rod). They are only roughly equivalent, yet few hams consider the inherent peculiarities of cylindrical gamma capacitors. I have run across a few cautionary notes about the cylindrical capacitor, like that by W8JI, but nothing specific. The linked page is long so here's the relevant paragraph:

The above example of decreased power rating is especially important to Amateurs using coaxial cables as capacitors. Voltage is NOT constant along the length of a long coaxial capacitor. Maximum voltage in the component is always HIGHER than the actual voltage across the terminals of the "capacitor", and it is higher than the voltage calculated by the current through the capacitor! Coaxial capacitors or linear stubs used as reactive elements always have significantly lower operating Q, higher power loss, and operate under more electrical stress than a well-designed lumped component. Stubs and linear loading does have the advantage of spreading heat out. You won't notice the heat as much, even though there is a lot more heat energy! Just don't let the smaller temperature rise fool you into thinking the system has less power loss.

Without worrying about the theory, let's look at a few measurements I took. I used the best instrument I have: the VNWA3 by DG8SAQ. The calibration point is at the end of the short cable terminated with a male BNC (on the right). The calibration error due to the barrel connector, binding posts and short wires to the capacitor is small at 7 MHz. Despite the calibration it will be seen that the 13" from the VNA port to the gamma capacitor is significant.

Before doing the measurement I tested the capacitance read by the VNA with two different high Q capacitors designed for moderate to high RF current. They read almost exactly flat up to 30 MHz, with a gradual rise up to 100 MHz. They are boring charts so I omitted them from this article.

The gamma capacitor is 43" (1.1 m) of PVC jacketed RG213 inside a 6' length of ⅝" × 0.058" tube. I previously measured this capacitor as ~340 pf (8 pf per inch) with a lower quality RLC meter that uses a low frequency signal to measure coils and capacitors. I replaced it with LMR400 in the gamma match of the 40 meter yagi since PE is a superior dielectric material compared to PVC (lower loss).

The gamma capacitor is far from an ideal flat line! I placed markers at points of interest. The RL (return loss) plot hints at the loss due to the PVC dielectric.

At very low frequencies the capacitor value is about what it should be. However, notice the loss shown by the RL. The loss peaks at ~8 MHz before moderating, and then becoming extreme between 80 MHz and 120 MHz. I did not compare a length of LMR400. The test RF capacitors have negligible RL.

The first capacitance peak at 26 MHz is near where I estimated the frequency at which the gamma capacitor is an ¼λ open transmission line stub. The VF within the capacitor is difficult to estimate, and I didn't bother trying to measure it since it requires subtracting out the VF of the VNA coax and connections to the capacitor. The stub begins at the VNA port, not at the gamma capacitor, so it is 56" (43" + 13"), with VF changing at cable junctions. Although I did not verify the source of the high loss 100 MHz resonance, there is one candidate that I strongly suspect.

The ¼λ stub frequency will be different when installed on an antenna. It is far enough from 7 MHz that it is probably not a problem. However, notice how the capacitance rises dramatically at half the stub frequency. That may be noticable.

Open stub beyond the shorting strap

Most analyses of gamma matches ignore the open stub formed by the gamma rod and DE outside the shorting strap. After many trials of building and adjusting gamma matches I feel the same. My model of gamma matches also indicated little effect. I thought it still worth a calculation.

For the 23" of ⅝" gamma rod beyond the strap on the 40 meter yagi DE the calculated parallel reactance is approximately -3700 Ω at 7.1 MHz, for a capacitance of 6 pf. Since the magnitude of the reactance is much larger than the stepped up antenna impedance the effect should be small. It is also swamped by the larger reactances of the gamma match's shorted stub and series capacitor.

That said, it is best to keep the length of the rod only long enough for the gamma match to have a moderate amount of adjustment room. That minimizes any potential effect and reduces wind load.

Lumped reactance vs. transmission line sections

Unlike the cylindrical gamma capacitor, a lumped constant (fixed or variable capacitor) should provide a better all-band match. It's simply more predictable. One factor that may have played a role in my matching woes is that the length of LMR400 is just shy of 60". That would lower the ¼λ stub resonance of the cylindrical capacitor, perhaps as low as 20 MHz, and the capacitance gradient would begin its rise at a proportionately lower frequency. 

There is a slope to the capacitance curve at 7 MHz with the shorter RG213 gamma capacitor. That is in part due to transmission line effects that worsen as the ¼λ stub resonance is approached. The higher capacitance lowers the reactance of the series capacitor. The reactance of any capacitor decreases as the frequency increases since Xc = 1/(2πfC). 

I would like to measure the series capacitance needed to cancel the inductive reactance at several spots across the 40 meter band. Then I can tell whether the sign of the decreasing reactance of a fixed capacitance is the same or opposite to that of the inductive reactance. The first makes the SWR worse and the second makes the SWR better. However, I won't climb the tower in winter to do the measurement.

The deviation of the capacitive reactance across the 40 meter band is likely to be smaller than ±10 Ω from the mean value of ~75 Ω. That's significant but not disastrous. Until I take those measurements I don't know for sure if this effects are responsible for the worse SWR curve with the cylindrical capacitor versus the variable capacitor I used for the initial adjustment.

There may be other effects that a measurement would uncover. There are certainly several quirks shown by the VNA measurement.

There is another curious effect of the cylindrical capacitor we need to consider. When the capacitance is reduced by sliding the gamma rod outward the lead in wire from the feed point is lengthened by the same amount. We are therefore increasing XL while increasing Xc. If the magnitudes are similar the adjustment does not go as expected.

For the capacitor on the 40 meter yagi gamma match we have the following. For a capacitor value of 300 pf, a 1" movement of the gamma rod changes the reactance 1.5 Ω. The inductive reactance of the lead in wire is ~0.7 Ω per inch. The former is pretty exact but the latter is estimated from coil forming equations. Therefore sliding the gamma rod will have about half the expected effect.

The rate of change of capacitive reactance depends on gamma capacitor construction, frequency and length relative to wavelength. I have run into adjustment trouble with a couple of yagis due to the magnitudes of XL and Xc being too close. One solution is to alter the DE length so that a different capacitance is needed. The rate of change of a capacitor's reactance is inversely proportional to its value, and in that way the rates of change can be made to diverge.

Methods to reverse the gamma match calculations

The best way to determine the dipole feed point impedance of yagi is to temporarily substitute a split centre to the DE. That is how I conducted my experiments in 2020 with the 40 meter dipole. That isn't always possible or effective use of time and materials, especially for the size of a 40 meter yagi element.

With exact measurements of the feed point impedance and gamma match dimensions there are several methods to calculate the dipole impedance of the antenna:

• Reverse the gamma match design equations and solve for the feed point: Z₀ = R + jX
• Build equations from the fundamentals to calculate the dipole impedance Z₀
• Using an existing gamma match calculator, manually enter R and X values until the design parameters match the dimensions of the built gamma match; alternatively, automate the process with a software algorithm

I gave up on the gamma match calculator distributed with the ARRL Antenna Book because it is difficult to use. Each iteration requires re-entry of all the parameters, which is absurd. It is also not possible to inspect the algorithm. For my most recent calculations I used the TNL (Tolles, Nelson, Leeson) algorithm programmed by Cebik in an Excel spreadsheet. 

Cebik is gone but his articles and files have been archived in several places. I won't give any URLs because they likely have a short lifetime. Use a search engine to locate copies extant at the time of your search. After all, you could be reading this article years after it was written.

I converted the Excel spreadsheet to Open Office and plugged in the dimensions of the gamma match. I hoped to see the best fit with the NEC2 model of the yagi's dipole impedance. That didn't happen.

I manually adjusted the dipole Ra and Xa values in the spreadsheet until it churned out the measurements of the matched antenna to within a few percent. The final values of Ra and Xa that resulted in the actual gamma match dimensions do not look realistic (see the screenshot below). Even so, the TNL algorithm's estimate based on the modelled impedance was a pretty good starting point to build and adjust the yagi's gamma match.

As previously discussed, there are anomalies introduced by the cylindrical gamma capacitor and other components of the built antenna and match. For example, the precise location of the feed point (coax connector and wire leads) and using 2" as the average DE diameter. I found that small changes to Ra and Xa cause relatively large changes to the gamma rod length Lgr and gamma capacitor Cs.

The values of Ra and Xa are not correct. Ra ought to be closer to 20 Ω and Xa nearly twice as large. But I don't really know, and that's why I want the ability, with reasonable accuracy, to reverse the gamma match equations. I copied out the equations from the spreadsheet cells to see how difficult it would be to reverse them. I successfully reversed the less complex of the equations but without better fundamental knowledge of RF network theory I am unlikely to be successful. Besides, the manually discovered Ra and Xa diverge from reality enough that I doubt the work would be worthwhile.

I heard that there are improvements to the version of the TNL equations encapsulated in Cebik's now old spreadsheet. Unfortunately I don't have them and a cursory search didn't turn them up. The sensitivity of the equations to diverse physical parameters are also a concern. I don't plan to pursue the reversing project further at this time.

Next steps

Gamma matches are fun, if you like that sort of thing. They are useful, simple and terribly enigmatic. The opacity of their behaviour may delight the many hams of the trial-and-error brigade while horrifying those of us who prefer exact specifications and predictable behaviour. I keep making them even though they frustrate me to no end. I wish there was a more accurate design procedure.

When warm weather returns I will climb the tower and take a host of measurements at the feed point of the 40 meter yagi. I have two purposes. One is to improve the SWR bandwidth. The second is to determine the dipole impedance across the band so that I can verify my model of the antenna. The latter is not particularly necessary since the antenna performs very well. But I'm curious.

With the measurements in hand I will redesign the gamma match to improve the SWR curve and design a switching system to extend the low SWR range to cover more of the SSB band segment. There is no rush since it works well on CW and CW is my most common mode. I rarely venture onto 40 meter phone except during a handful of contests.

If I get very ambitious I'll resume my attempt to reverse the gamma match equations.

2022: Chasing Loose Ends

It is time for my annual look back on the year that was and year that has begun. You'll find that this is a tradition when you look back in this blog for January of each year.

Undertaking the construction, maintenance and use of an amateur radio station of the size that mine has become does not happen by accident. Well, you might accidentally arrive at a worthy destination by directionless wandering, but I'm getting too old to waste time. So I plan.

A plan is worthless unless it has clear and measurable milestones to track your progress. That makes many people so uncomfortable that they conveniently forget their stated intentions or they "move the goalposts" and hope that no one notices. People do notice, though in most cases they will remain diplomatically silent. I am too honest with myself to engage in self-deception. Putting my plans in the blog at the start of each year helps to keep me honest.

So, how did I do in 2021? For those with time to waste and a penchant for fact checking have all the raw data at hand: my stated plan and where I ended up on December 31. I am not surprised that my success rate for the line items is not high. I like to aim high but I won't sacrifice my life to get there.

What is important is that the major projects were completed. The items in my plan are not intended to be equally weighted. The following statement from my 2021 plan remains valid:

"Beyond 2021 no major antenna projects are planned. There will continue to be refinements and improvements, and I will likely dabble with experimental antennas and related projects."

By far the largest project was designing, building and raising the 3-element 40 meter yagi. The 5-element 10 meter yagis were a lesser challenge that consumed more time than I anticipated. Difficulties getting the rotatable, side mounted TH6 working properly took time I'd have preferred to spend elsewhere. Nevertheless the year was reasonably successful.

One important change to the plan was that the rotatable side mount was done for the TH6 and not the XM240. I left the small 40 meter yagi where it was because I determined it was more productive to have it fully rotatable (360°) and the TH6 was really only needed for the shorted North American paths and as a rapid choice to work Caribbean and Central America multipliers and to work South America. By doing it this way I was able to avoid building the more complex swing arm (300° rotation) for side mounting the XM240.

Lesser projects completed in 2021 included:

• New amplifier for high power SO2R (with BPF) and 6 meter DXing
• Correcting the mismatch of the upper 5-element 20 meter yagi
• Reversible east-west 156 meter Beverage
  

Of these, the only one I have yet to discuss is the BPF (band pass filters). There is a reason for the deferral, and I hope to correct the omission in the not too distant future. Numerous repairs are omitted from the list since those are unavoidable maintenance tasks for those with large stations. There's always work to be done. 

To give examples of the maintenance that must be done, this week I repaired a Beverage that a tree fell onto. A dead limb of that tree is hung up in another tree, out of my reach, that is certain to fall in the near future. On the 15/20 meter tower the prop pitch motor has become unreliable in cold weather. It will need service (again), but I will live with the problem until warm weather returns.

2021 projects that were deleted or deferred included:

• Stack switch for the 10 meter yagis
• More radials for the 160 meter antenna
• Efficiency improvements for the 80 meter vertical yagi
• VHF antennas: longer boom 6 meter antenna, and an antenna for 2 meter DXing
• Antennas for the WARC bands: 30, 17 and 12 meters

The 10 meter yagis were raised late in the year while the 40 meter yagi was under construction. The latter was the priority so the 10 meter stack switch was put aside to the new year. All the parts are on hand and construction is straight forward since it is no different than for the 15 meter and 20 meter stack switches.

Laying more radials for the shunt-fed tower on 160 meters is easy enough, if I had the wire on hand. I didn't so it didn't happen. Eking a fraction of a decibel on 160 meter was not a priority. Of even less urgency was improvement to the 80 meter array. I have alternative ideas to mull.

The same lack of urgency applies to VHF and WARC antennas. Maybe in 2022.

I previously published an annotated version of the following photo on the blog and on my QRZ.com page. It will be interesting to compare how different the station looks a year from now. I doubt that it'll looks very different from this visual perspective. Most of the changes will be in the shack, electronics on the tower and antennas, and relatively minor antenna changes or additions.

Many hams have told me that it is impressive. I suppose it is, however I get exhausted looking at it. It was a lot of work over the past 5 years since moving to this rural QTH. What I can celebrate is that the bulk of the big projects are now done. All of it must be maintained, and that is a never ending job. 

I didn't get here with a firm plan from the outset, as long time readers will know. There were too many uncertainties in my mind about what I really wanted and what I could realistically accomplish. I wanted "big" with no clear definition of what that meant. There were too many alternatives to explore. 

Each next step was well considered but not always the step after that. This is not a commercial enterprise so I allowed myself to be influenced by whims and opportunities, and by the adventure of exploration. Every ham with a large antenna farm has their own story.

Future projects will be less arduous. I am contemplating the addition of a small tower in future to avoid interaction and mechanical challenges of antennas on the existing towers. I also want a small tower with few encumbrances for antenna experiments. The station will not reach stasis for a long time since there will be new antennas to explore and improvements made to existing ones.

With a sigh of relief that I've made it this far, let's look at what I have planned for 2022. Although the projects will be less visually impressive they will have a impact on contest competitiveness, among other advantages.

• Prop pitch rotators: I am having more difficulties with the motor that turns the 15 and 20 meter yagis. The gearbox is the problem this time. Service will wait until warm weather returns. I must also finish building the direction pot for this rotator and improve the one for the chain drive prop pitch motor on the 40/10 tower. Once those are done a new controller in the shack is needed to make them ergonomic and more reliable. The controller will be home brew, including software.
  
• 160 meters: There are two options to improve my top band signal. One is to shunt feed the other tower and feed them for switchable end-fire and broadside configurations. The second is a parasitic array using a wire driven element between the towers or adding wire elements to the existing antennas. There are pros and cons to be weighed. A large consideration is for ease of deployment each fall and removal in the spring for haying.
  
• VHF: At minimum I need a 2 meter yagi suitable for DXing and occasional contest use. It will go on the Trylon, just below the 6 meter yagi. A superior 6 meter antenna is possible this year but may be deferred to 2023. Extending my DXCC count on 6 meters will not be easy.
  
• WARC band antennas: My primary interest is chasing band-countries for general DXing on 30, 17 and 12 meters. I have no specific plan at present and there are too few major DXpeditions expected this year due to the ongoing pandemic. I use a tuner on the XM240 for 17 meters and the 80 meter vertical for 30 meters, and on 12 meters I load whatever high band antenna is broadside to the station I'm calling. While far from ideal at least it works well enough, for now.
• 40 meters: The matching system for the 3-element yagi needs work to improve the SWR and it needs a switchable network to make it more usable on SSB. It's just a matching problem since performance is excellent.
• 80 meters: I am contemplating a reversible 3-element wire yagi with a rope catenary strung between the towers. A horizontal yagi could pay dividends by being more efficient and to exploit high elevation angle DX paths. It would point northeast to Europe and southwest to cover most of the US. The main lobe of a 3-element yagi is broad enough to work well for most stations I can expect to work. A second 80 meter antenna provides a backup in case one fails before a contest.
• Station automation: I am making slow progress on the hardware and software. Some of what I need should be complete by early spring. The rest can wait until I can begin inviting others to do multi-op contests. COVID's resurgence will delay multi-ops for a while longer.
• Station modernization: A new SDR transceiver will replace the FT950 which has long overstayed its welcome. More extensive software control of equipment and antennas is planned. Other equipment changes are being contemplated.
• Receive antenna: The Beverage system is performing well, although it demands regular maintenance. Completion of the desired 8 directions with a future NW-SE reversible Beverage is doubtful because it's benefits are at best incremental. These antennas are not so directional that more than 6 directions are strictly necessary. Instead I would like a second receive system, probably a multi-direction vertical array. The benefits are diversity reception and for low band multi-op contests. Planning and design can proceed this year, and construction to follow in 2023.
  

This list is vague in comparison to those I've made in previous years. That's deliberate. From here on I will be more relaxed about my station plans. Major projects were never intended to continue forever! Speculative projects for fun and for experimentation will be undertaken when I feel the motivation and interest. 

I will spend more time this year and in subsequent years for non-ham retirement activities.

With that I will end the annual retrospective and look forward. I hope to meet more of you on the air. It's always enjoyable when a contact, during a contest or other times, tells me they read and enjoy the blog. Thank you for following along in my journey. Major contests are on the horizon, the 6 meter summer sporadic E season is only 4 months away and the sunspot count is climbing. Pandemic or not, 2022 will be a great year.

Where Is My Straight Key?

A bunch of the local hams operated SKN (Straight Key Night) on new year's eve. Although I didn't plan to participate I enjoyed reading about their plans and the keys they'd be using. Afterwards came the stories of who they worked. It brought a smile to my face.

It also made me wonder: where is my straight key? I do have one...somewhere. It's very old and not in good condition. Some work would be needed to make it usable. I had an urge to see it again so I dug through boxes of disused gear and parts and I eventually found it.

Doesn't look like much, does it? I have little sentimentality for old things and I am happy to see them go when the time comes. I kept the key because it is the only one that I ever owned. Besides, no one would buy it and I doubt I could give it away. I must have been 14 when I bought it or it was given to me (I don't remember).

My father gave me a scrap of wood of the size I wanted. I sanded and stained the wood, and then screwed down the key. Three plastic feet underneath with pins for nailing into the wood complete the project.

I used it for sending practice until I got my license at the age of 15. One old tube transmitter put a bit of voltage on the paddle and delivered a mild shock when I would accidentally touch the metal lever. It was important to carefully grip the non-ergonomic knob by the rim! I used that key, as you see it, for the first 2 or 3 years that I was licensed.

Then came the contesting bug. The straight key was pushed to side and replaced by an early generation digital keyer with memories. The electronic iambic keyer was a blessing for allaying the fatigue of pounding out hundreds of contacts over a weekend. The key also improved my speed and QSO rate. I never looked back.

I know that I used it from time to time over the years, but those occasions soon became rare. I no longer recall when I last used it. It follows me around as I go about my life, packed in a box most often. It has moved with me numerous times. 

It does not work well. There are a couple of ball bearings missing from a cleaning and oiling incident in the distant past. They popped out, rolled away and apparently not all of them were recovered. The lever wobbles. This is a less than ideal condition to achieve a good "fist".

I always meant to replace the lost bearings but I never did. There was no urgency for an item that I never use and that I am unlikely to ever use. However, I do think about it from time to time. I could buy a new or used key -- they aren't expensive -- yet I haven't done that either. Perhaps I never will.

The memories it elicits have value and that's what really matters. Repairing and using it are less important. Maybe for the next SKN.

Happy New Year!