Methodology for Adjusting HF Yagi Gamma Matches

There are many feed systems used in yagis over the years. Gamma matches are not as common as they once were. More typical are beta matches and T matches to convert the low impedance of a yagi to 50 Ω. Moxon (rectangle) yagis and OWA (coupled resonator) designs are becoming more popular for their direct match to 50 Ω coax.

For my home brew stacked 20 meter and 15 meter yagis I originally planned for beta matches. I bought the fibreglass tubes for the split driven element. Then I changed my mind and switched to gamma matches. Although in disfavour for their unbalanced and asymmetric topology the impact on yagi performance is negligible.

For me the deciding factor was a more economical and robust driven element. That is not to say the beta match is deficient however the construction for the home brewer is perhaps more challenging. I am a little less convinced now that I've built, tuned and installed the lower yagis of the stacks.

Gamma matches are finicky creatures. For commercial yagis the challenge for the ham is modest since all the design work has been done. All you need to do is set it up according to the manual and tweak it as necessary to achieve the desired SWR curve. When you design and build from scratch the challenge is of a higher order.

Which matching system is best?

Before delving deeper into gamma matches let's compare it to alternatives. There are both objective and subjective aspects depending on materials, preferred construction method and other factors unique to every builder. This is not about commercial products where the design and construction are outside of your control other than choosing which antenna to purchase.

Just about every common matching network is equivalent -- gamma, beta, T, transmission line transformer, etc. -- in that they are essentially L-networks. There is a C and an L, one parallel (shunt) and one series, sometimes split in half for a balanced feed. The gamma is a little more complex due to the step-up function of the gamma rod and an L in series with the a couple of C (capacitor and shortened element). You could in fact feed the yagi with an L-network although that is rarely done with rotatable yagis.

With respect to achieving the best SWR curve they are essentially equivalent. Any simple matching network is constrained by the range of R and X across the band of an inherently high Q antenna like a yagi. When I first modelled the 5-element 15 meter yagis I built I utilized a beta match for its simplicity and compatibility with EZNEC's stepped diameter correction (SDC) algorithm. Compare that SWR curve with the one measured on the completed yagi with its gamma match and you will see they are the same.

Construction of any of the matching systems is straight-forward. However all of them require unique mechanisms:

• Beta: Series C comes from a driven element shorter than its resonant length and which is adjustable. The driven element must be split and made mechanically robust. Parallel L is with a shorted stub or a coil, with the former easier to adjust with a custom made slider, and is more robust.
• Gamma: Series C should be a variable capacitor, either a component or (as shown and popular) insulated wire inside the gamma rod. The amount of capacitive reactance in the driven element is about the same as for a beta match. The strap and insulated support must be fabricated. After adjustment the capacitor must be robust enough to withstand weather and maintain a stable value. The driven element is continuous and easy to make robust using "plumber's delight" construction.
• T: Similar to a gamma with symmetric halves to provide a balanced feed. It requires a step-down transformer.
• OWA: The driven element is split for a dipole feed, just like the beta match. Adjustment is by varying the lengths of and spacing between the driven element and coupled resonator. NEC2 models can get you close but inaccuracy is unavoidable so field adjustment is required and can take some time. The coupled resonator adds wind load (and cost!) but no gain or pattern improvement; it is not a director.
• Transmission line transformer: Split driven element. Reactance set to zero by adjusting driven element length. Coax of the required value can be difficult to find and so is usually made from parallel runs of 75 Ω coax to boost 25 Ω (typical for a mono-band yagi) to 50 Ω.
• Broadband transformer: Residual reactance must be zero and achieved by adjust the driven element length. A 2:1 ratio is often a good choice since the impedance of a typical full size mono-band yagi is near 25 Ω. If the yagi has a high Q the transformer ferrite core is at risk of overheating at frequencies where the SWR is high.

For me the factors that ultimately convinced me to go for the gamma match were driven element robustness and (I thought) relatively simplicity of construction and tuning. Gamma matches are peculiar creatures that defy intuition and easy answers. I had more difficulty than expected getting them to work. Let's dig a little deeper and learn why.

What could possibly go wrong? Theory vs. practice

I am not an expert on the theory of gamma matches. I will leave that to others. There are ample discussions to be found in various places and software tools to design gamma matches. Some of them might even be correct. But as I said above, gamma matches are peculiar creatures.

Perhaps the clearest description of the gamma match that I've found is by W8II. A survey of theory and design algorithms was done by W4RNL (SK). Going back further there is a good description of the gamma match by W3PG in a 1969 QST article. Predictable gamma match designs are stymied by sensitivities of aspects of their construction:

• Ratio of element to gamma rod diameters: The resulting step up ratio is sensitive to the diameter and spacing, as is the inductance of the shorted stub (L) they comprise.
• Feed impedance: Since it is difficult to measure the impedance of a continuous driven element we usually have to estimate it.
• Gamma capacitor: A long wire inside the gamma rod can display transmission line behaviour since it is a non-negligible fraction of a wavelength. W8JI discusses possible consequences.

As Cebik's theory makes clear there are many wrong design algorithms and right ones are difficult to come by. In his survey the Tolles-Nelson-Leeson algorithm seemed to be best. For my designs I relied on the gammaMW4 software that came with my copy of the ARRL Antenna Book. My experience with it was less than satisfactory, but I had it and it at least got me pointed in the right direction.

With an approximate impedance from the EZNEC model of the 5-element 15 meter yagi I used the software to design the gamma match. Unfortunately it didn't come close when I put it to the test in the field. Trying the software again, after achieving a match in the field, with a range of deviations from the original impedance still did not get close to the experimentally found dimensions. Keep that in mind when you use gamma match design software. The sensitivities mentioned above are significant.

Choosing a design

I bought 1/16" thick mild aluminum alloy 1" wide to make the straps. I formed the strap by manually folding the strap around tubes of the same diameter. The ends were then bent and drilled for stainless fasteners. It took a little experimentation to get the folds and bends in the proper positions to have correct element to rod spacing and good grip to the tubes.

Alternatively you can use a stainless u-bolt and a straight strap. For best long term contact between the strap and tubes a fold at the top and bottom of the strap protects against the weather and increases the contact surface.

A short section of PVC pipe is used for the inner spacer. The ½" gamma rod pierces the pipe. An aluminum strap or a set of UV-resistant cable ties secures the pipe to the element.

There is a coax connector on an aluminum L hanging from the element-to-boom clamp (closeup here). It is sealed on the inside to prevent moisture penetration through it and into the coax. RG213 with the outer jacket and braid removed is the inner "plate" of the gamma capacitor. Measured capacitance is a little over 2 pf per inch (25 to 27 pf per foot) for a ½" OD gamma rod.

There are many component parts to fatigue or fall prey to the weather so you want it to be robust. After matching I took the following steps:

• Sealed the end of the RG213 capacitor to protect against water and high voltage. Arcing probability increases as environmental contaminants accumulate.
• Sealed and taped the inner end of the capacitor to protect against water and capacitance changes due to movement.
• Stranded wire from the coax connector to the RG213 was replaced with a solid wire. Although there is more risk of fatigue the stranded wire developed kinks from repeated stress.

With the lower yagis of the stacks on the tower I want to know how they tolerate several months of cold, ice and wind before putting up the top yagis of the stack. The top yagis are more difficult to lower and fix if faults arise.

Gamma match model

With some difficulty it is possible to model a gamma match using NEC2. I did so using EZNEC and had some success. There are a few rules to follow:

• Close spaced wires (element and gamma rod) must have their segments line up exactly. This must be redone as the lengths and connection positions are adjusted.
• The short wires for the source and straps should have one segment.
• Stepped diameter correction does not work with the gamma match present. I replaced it with the equivalent diameter wire (calculated by SDC without the gamma match present) beyond the shorting strap. The inner section of the driven element must be true diameter since it affects the step-up ratio.
• The gamma capacitor is a load placed at the inner end of the gamma rod. This simulates the effect of sliding the gamma rod outward but without compensation for transmission line effect of the length of RG213. The exposed section of RG213 is modelled to the actual diameter of the wire and polyethylene insulation.

A side effect of the close wire spacing is that the model fails the average gain test. However, as explained in the EZNEC manual that if it is known that there are no true losses in the model it is safe to add the lost power to the far field calculations of the pattern.

When the -2.0 db of the average gain (shown when calculating a 3D plot) is added back the gain and other pattern measurements were essentially identical to the original yagi model. Pattern symmetry is maintained (as shown by Cebik) despite the asymmetric construction of the driven element. I'll come back to this shortly.

Below is a closeup of the model and compared to the real gamma match. The letter labels were chosen to be self-explanatory.

Notice the green dots (hard to see) that show the segment edges. The blue squares are wire connections. The red circle is the source and the red square is the capacitor. Let's list the active components of the gamma match that affect the tuning:

• Length of the shorted stub L is determined by E, R and S.
• There is a second and inner section the same transmission line between the source and the capacitor C.
• There is an open transmission line stub outboard of the strap between the end of the gamma rod and the element. It is designated as a small capacitance 'c'.
• Moving the strap affects L and c. Moving the rod affects C, c and, by a small amount, L.

With careful manipulation of the model the match was achieved. The resulting SWR curve is virtually identical to the original beta match and measurement of the real antenna.

With the model thus constructed I was able to explore how various adjustments affect the match and compare it to physical measurements. There was close agreement, telling me that the model is effective. The objective is to develop a deterministic and rapid process to reach a match. Without this insight my first attempts at adjusting real antennas became a confusing and lengthy chore.

One aspect is the myth that the gamma match causes an asymmetry in the yagi's pattern. You can see in the plot above that this is not so, or more accurately that the amount of asymmetry is negligible. You'll end up with far more asymmetry due to interactions with guys and other antennas. The model provides a clue.

Driven element centre is at the source (wire #14). The full half element to the right is #12 and the portion of the half element to the left, beyond the gamma match, is #11. The current in #10 is ~2.2 A with the source I chose. The current in the gamma rod #16 is ~1.0 A and 170° out of phase. This would be almost complete cancellation if the currents were equal (as we'd expect in a transmission line section) so we are left with ~ 1.2 A.

The current of #12 at element centre is ~1.25 A, almost identical to the net current and in phase with #10 (10° difference). That argues for excellent symmetry especially when viewed from the perspective of mutual impedance with the parasitic elements. Indeed, the EZNEC calculated currents in all the parasitic elements are symmetric to at least 3 decimals which is symmetric considering the numerical precision we can expect from the calculating engine.

Current in #17, the dangling end of the gamma rod, is very close to 0 so it really doesn't do much of anything. That would appear to confirm the conclusion of Cebik and others, therefore we can ignore its length during adjustment of the match. If it were an effective open stub its current should be a sizable fraction of that in #11.

The small mismatches of current and phase mentioned may be artifacts of the model since NEC2 is stressed by tightly coupled wires. That is, the actual situation may be even better but without NEC4 and deeper study I don't really know. For now I'll accept these figures since they appear to correspond well to the real world

Of course you must use a common mode choke to ensure the best pattern integrity. It is possible to model the outer coax sheath as long wires connected to the element centre, and I have done this in the past. But with the boom also connected and the unpredictable influence of the coax running along the tower and adjacent to other cables a model is impractical. Install a choke and forget about it, whether the ultimate or one that's good enough.

With the benefit of what I learned the first time I attempted gamma match tuning and with the this model the adjustment of the permanent gamma matches proceeded quickly. That's the subject of the next section. A few more insights from the model will be sprinkled in the following discussion as appropriate. I plan to play with model a little longer to see what more it can teach me.

Getting a match: to be methodical you need a method

On the first tuning of the gamma matches I mounted a 150 pf variable capacitor to the gamma rod. There were only two adjustments: strap position and capacitor value. Potential confounding factors mentioned above were thus eliminated. Even so I quickly ran into trouble. It was my own fault since I trusted the gamma match software and believed I could adjust the gamma match to perfection with a little bit of tweaking. I was very wrong.

The first problem was the coax. Some is needed since my system for lifting the antenna on the tram line to get it high enough for measurement requires coax down to the ground and the antenna analyzer. The chart below is from one of my first tuning attempts. All that I'm doing is moving the shorting strap in one inch increments with the capacitor value fixed. Zo is the impedance at the feed point and Zm is the impedance measured by the analyzer at the end of the coax.

It is commonly recommended that you use a length of coax that is an exact electrical multiple of λ/2. For example, LMR400 with a VF (velocity factor) of 0.85 should be 19.8' (6.04 m) long or a multiple thereof. If you do this the impedance at the far end is the same as at the antenna. Otherwise you get something else, which will confuse you if you aren't aware of the issue.

There are infinite combinations of R and X value for every SWR greater than 1. Indeed, if you try to tune the gamma match by only measuring SWR you could be there for a very long time! I tried trial-and-error at first because I mistakenly thought I could adjust the match with a few tweaks from the initial software calculated values.

With the 29' length of LMR400 the Rm and Xm values seemed utterly random. Tweaking only resulted in more nonsense measurements. Rm and Xm bear no resemblance to Ro and Xo even though the SWRs are identical. I used TLW to convert Zm to Zo. Only then could I begin to methodically adjust the gamma match.

This is an example of an impedance "rotation" from one of my test values. Since the length of coax is close to an odd multiple of /4 this is a worst case situation. The R value is approximately the reciprocal of the measured value with respect to 50 Ω while X "flips" sign with a smaller or larger magnitude. The Zm = 27 - j31 Ω value is actually quite good since we have only to cancel the series reactance (Xo).

Look again at the chart. Notice that Ro increases as the shorting strap moves outward. As discussed earlier, the gamma match and its cousins are essentially odd looking L-networks. Change C or L and both R and X change. However, the rate of change is not the same. I talked about the utility of this behaviour for the L-network in my 80 meter vertical yagi. We can put the same principle to use here.

The basic gamma match tuning method I use is as follows:

1. Move the strap until Ro is approximately 50 Ω.
2. Adjust the capacitor to bring Xo near 0 Ω.
3. Since each adjustment for Ro and Xo affects the other repeat steps 1 and 2. A few iterations will bring you to 50 + j0 Ω.

Of course it isn't quite that simple. For example, you must have the antenna high enough or, alternatively, pointing upward in the proper fashion for the impedance measurement to hold when the yagi is placed on the tower. Small yagis with the driven element accessible from the tower and for VHF/UHF yagis that can be tuned near the ground you can do this job more easily that for my 20 meter and 15 meter long boom yagis.

When I used software to determine the gamma match dimensions the position of the shorting strap was quite different on the real yagi. It had to be moved outward by about 1' (30 cm). For my 15 meter yagi this meant the strap had to reformed to fit the ¾" tube instead of the 1" tube.

There is another way to change the R value without moving the strap: change the resonant frequency of the driven element. Sliding the element tips in and out changes the series capacitance of the element and therefore Zo. I used the very same technique on the 80 meter yagi to adjust the L-network with a relay to accommodate the different impedances of its yagi and omni-directional modes. The only difference here is that the capacitor is in series (like in a beta match) and not a shunt.

To avoid having to make the gamma rod longer I changed the length of the driven element to increase Ro. You can deduce the general procedure from the following 3 measurements I made while leaving the strap position and gamma capacitor unchanged:

• -1": 36 + j13.5 Ω
• +0": 47.3 + j17.5 Ω
• +½": 51 + j24.5 Ω

Adjusting the driven element in this fashion has no impact on the yagi's pattern. It would take a far larger change to do that, enough to significantly affect the mutual impedance with the parasitic elements. Choose whatever combination of strap position and element length that works best for the mechanical design of the element and gamma match of your yagi.

Now we need to talk about the gamma capacitor. Its adjustment can be straight-forward or confusing depending on your approach. Sliding the gamma rod to adjust the capacitor affects the "stray" lengths at both ends of the transmission line formed by the element and gamma rod. Refer to the annotated model shown above.

I would like to tell you that I now understand how sliding the gamma rod is affected by those end effects but I cannot. At first I thought I had found something when, very close to a match, sliding the rod had the opposite effect on the capacitive reactance. That was interesting but wrong. It turns out that the length of LMR400 was really 30', not the 29' I first measured. However, this is almost exactly λ/2 on 20 meters so tuning the 20 meter yagi was quicker: no need to convert Zm to Zo.

When I fed the measurements into TLW the effect disappeared. My conclusion, in agreement with the experts and my model, is not to worry about it since the effect either doesn't exist or is too small to matter. I must also caution that the accuracy of the antenna analyzer is paramount otherwise you will to your dismay discover you are dealing with a fictional reactances. My weapon of choice is the RigExpert AA54 which I find to be very good for my antenna projects. There are other equally good products and there are many that are far worse.

We now need to discuss what the capacitor does and how to adjust it. Unlike inductance an increase in capacitor value reduces the reactance (for a constant frequency). Recall that the formula for capacitive reactance is:

X = 1 / (f × π × C)

For f in MHz and C in pF change the numerator to 1,000,000.

Adjusting the capacitor to exactly cancel the residual inductive reactance can be difficult since a small change has a large effect. Let's say we have Z = 48 + j25 Ω when the capacitor value is 40 pf -- this is approximately the case for my 15 meter yagi. We also know (as discussed earlier) that the capacitor I am using changes at a rate of ~2.1 pf / inch (~0.85 pf / cm).

By how much should we change the 40 pf capacitance to cancel the 25 Ω of inductive reactance? This is easy to explore with spreadsheet software, and I wrote one years ago for antenna projects. For C = 40 pf at 21.1 MHz we have X = 189 Ω. To increase X to 214 Ω [25 - (214 - 189)] we need to decrease C. If we shorten the gamma capacitor by 1" so that C becomes 38 pf the revised X is 199 Ω. That's a 10 Ω increase for a 2 pf (1") decrease.

You can estimate the correct answer with a linear extrapolation to 35 pf and that is indeed just about right. I did the same in the EZNEC model I developed and it agreed exactly. The antenna itself behaved just as the calculation and model predict. With that I was done.

One final note about the capacitor. It is a good idea to make the stripped length of RG213 longer than you expect. If you discover you need more capacitance you will have to make a longer one; save the short one in case you later build a higher frequency yagi.

On the other hand if it is too long the gamma rod will have to be overextended and that can cause mechanical instability. What I do is cut short lengths from the RG213 and keep the gamma rod where it is. If you do this don't get overconfident since you may have to increase the capacitance again in the tuning process. Cut off less than you need to until you are very close to a final match.

Parting words

That is my full gamma match tuning procedure. Trust me, speaking from experience, that the task goes far quicker by being methodical. You may get lucky with trial-and-error but that is a low probability outcome.  If you prefer a different method go for it, but do use one.

I now have two properly working large HF yagis and gamma matches, and a methodology to attack the many more yagis I plan to build. Considering how many yagis with gamma matches I've used over the decades it's a little surprising that I learned so little about them. Of course those were almost all commercial yagis with recommended presets from their field tests and only required small tweaks to get them tuned up.

Had I run into serious difficulty I would have reverted to the original plan to use beta matches. Those are far easier to model and are more predictable. However they have their own challenges as I summarized at the start of this article. I am pleased with my choice of the gamma match. For me it was a bonus that I got to learn something new.

Update Feb.21: I have made several minor corrections to units and quantities that slipped in. I don't usually edit to correct typos and grammar but I do want to keep the technical matter accurate.