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.