To summarize a long story, after the upper 5-element 20 meter yagi of the stack was raised I discovered that the gamma match had somehow slipped out of adjustment after being tuned. This probably happened while weatherproofing the gamma capacitor. Since the antenna is very large and heavy I decided to correct the mismatch and not to take the antenna down for repair. Taking the antenna down, adjusting it and raising it again would take 3 full days with the help of several friends. Correction is the easier and safer alternative.
I developed models to determine the effect of a matching network. A simple L-network is all that is required, and it can be made very efficient. For convenience my plan was to mount the network at the stack switch. Since the network and the gamma mismatch both shift the phase of the upper yagi it is also necessary to bring the yagis back into phase. The existing phasing lines assume that the yagis are identical, per the design of the antennas and stack.
Although the problem I am solving is specifically for my antenna system, the techniques and tools are applicable to other antenna challenges. Since these may be of use to readers, it is worthwhile to describe the resolution in depth.
Danger of calculation versus measurement
I keep so busy with my many projects that I don't always remember what I've done and my notes may be missing or difficult to understand. I had a set of impedance values across the band for the upper yagi in my notes and I assumed they were measurements. It turned out that they were calculated for the length of the phasing line to the stack switch based on measurements at the coax joint on the boom where it connects to the rotation loop.
The impedance is determined by the electrical length of the transmission line. The phasing harness lengths were calculated from the physical lengths and the VF (velocity factor) of the coax. For the upper yagi that includes 11' of RG213 (rotation loop) and 30' of LMR400 (tower run). This is approximately equivalent to 44' of LMR400. The total length is 66' including the 22' length of LMR400 from the feed point to the tower.
Physical lengths are easy to measure and mine were done with great accuracy. Unfortunately the published VF of coax is not always exact for a variety of reasons. When the phasing harness segments are from the same coax reel the VF is usually equal. In other cases, the VF can differ and the yagis will not be properly phased.
Under the misconception that the recorded values were measurements I used TLW to design the L-network. It turned out to be a degenerate case where the network reduced to a single component: a series inductor.
I dutifully built it (see above) and tested it with an analyzer (at right). The box must be closed since the inductance is reduced by the aluminum enclosure. To compensate, the coil is designed with a higher inductance. The required 31 μH of series inductance is what I got after adjusting the coil turns.
I discovered my mistake at 110'. Thoroughly confused and out of time that day I descended the tower.
Measurements
Several days later I went back up the tower with an antenna analyzer to measure the upper yagi impedance at the stack switch. It was a windy day so I had to visually integrate the readings as the elements wiggled and waggled above me. A few ohms either way is not a serious issue since the L-network can be fine tuned once it is installed.
Taking care not to lose the piece of paper with the numbers in the brisk wind I came down and ran them through TLW. With a bit of experimentation the measurements and calculations matched for a 40.5' physical length of LMR400. That's an error of 8%! This implies a VF of 0.78, far less than the specified 0.85. This is so surprising that I plan to take a tape measure up the tower to recheck the lengths, despite having triple-checked them on the ground.
At right are the measurements, taken at every 50 kHz across the 20 meter band. From a cursory examination of the impedance values it might appear to be impossible for a fixed network to properly transform the SWR across the band. Appearances can be deceiving.
Accounting for frequency, one network can transform a disparate set of impedances to a fixed target impedance. That is approximately true in this case where one network restores the designed SWR curve for the 5-element yagi: about 1.5 at the 20 meter band edges and 1 in the vicinity of 14.1 MHz.
Achieving a perfect 50 Ω impedance is not possible nor is it desirable. Careful design should result in a close match to the lower yagi's SWR, and therefore equal power division and near equal SWR when switching among upper, low and both yagis in the stack. There are software tools to ease the design process. Having one antenna exactly 50 Ω across the band does not achieve either objective.
Designing and testing the L-network
An L-network consists of a capacitor and a coil, one in series and the other a shunt on either the 50 Ω or the antenna port. There are two optional configurations, usually called low pass or high pass since they also behave as filters. Its filter performance is unimportant for this application so I chose a low pass network because the calculated L and C values were easy to work with. By this I mean a low value for L (small coil and low loss) and for C close to a capacitor value and power rating in my junk box.
As for the gamma match, I designed a network to transform the impedance to 50 Ω at the same frequency, a little above 14.1 MHz. For the gamma match that gave an SWR below 1.5 across the 20 meter band. It is a little more complicated for the mismatched upper yagi. Some experimentation was required.
I used a combination of TLW and SimSmith. Using the actual impedance measurements, I designed an L-network for 14.150 MHz. I plugged the network into SimSmith and checked how it performed against the measured values across the band. The SWR curve was good but not great. For the best SWR the L-network was set to match at 14.100 MHz. The result was an SWR of 1.3 at 14 MHz and 1.7 at 14.350 MHz.
I picked a vintage 100 pf transmitting mica capacitor from my junk box for the network. These capacitors are old and require testing, but I have had good success with them. A modern transmitting ceramic doorknob is a better choice. A variable capacitor can be used but you'll need a larger box and adjustment will require extra work. It really isn't worth the trouble for this application. But if you choose to use a variable capacitor, pay attention to the voltage calculated by TLW. The variable capacitor must meet the requirement for your maximum power, plus a safety margin.
Since the capacitor measured as 102 pf versus the required 96 pf it was necessary to return to SimSmith to check the network performance across the band with the larger capacitance. The capacitance may be even higher due to stray capacitance within the enclosure.
It is no surprise that a perfect SWR of 1 cannot be achieved at the design frequency. The best I got was the depicted 1.1. That is not really a problem since it has negligible effect at the band edges where the SWR is highest. This is typical since the mismatch at the band edges is dominated by deviations of R and X from 50 and 0 Ω, respectively, of the load's complex impedance rather than the small deviation due to the capacitor being off by a few percent. With this network SimSmith calculates the SWR as 1.4 at 14 MHz and 1.8 at 14.350 MHz.
The inductor was designed with Coil by K6STI. It is 7 turns of AWG 12 bare copper wire, with a diameter of around 0.8" (2 cm) and 1.25" long (3 cm). To wind the coil I used a ½" PVC pipe that has an OD of 0.84". The calculated Q is about 350, which is quite good. TLW predicts very little power will be dissipated. I took the old coil out of the enclosure and substituted the new one. A 50 Ω load was connected to the antenna port (the same setup as was shown earlier) and measured by an antenna analyzer at 14.1 MHz.
To adjust the coil I opened the enclosure to squeeze or spread turns. The measurement must be done with the enclosure closed since the aluminum reduces the inductance. For this reason the coil design was for an inductance of a little over 0.5 μH. Once the measured inductance was 0.46 μH the shunt capacitor was installed. It's a tight fit but that's okay; the coil and capacitor can be close, or even touch, and the only effects might be slight increases of C and L. The coil can be adjusted during final turning of the network.
For tuning the network a simulated complex load is helpful. It is not really required if you enjoy doing this work 110' up the tower! My preference is to minimize the tower work by simulating the load in my workshop. For the measured impedance of 82-j9 Ω I put a 75 Ω resistor in series with a 1000 pf capacitor. Although 75-j11 is not exact it is close enough to test and coarse tune the network.
The simulated load only permits adjustment of the network at the 14.1 MHz design frequency. You would need to build loads to simulate the impedance at other frequencies to fully bench test the network. I didn't do that since it isn't strictly necessary. If the measurements are correct the software calculations for its behaviour over the rest of the band should also be correct.
Testing of the completed network is shown at right. The SWR is a little high because the enclosure is open for the purpose of taking a picture. With the enclosure closed the SWR is 1.1, which is quite good for the inexact simulated load.
The network was further examined with SimSmith by substituting the simulated load and seeing what the measured impedance would be. The impedance measured by the analyzer was within 1 Ω for both R and X. That's less than the accuracy of the analyzer so we can't expect to do any better.
I do have a better instrument (VNWA3) but that degree of accuracy is overkill for what this project requires.
On a final note, look above at the image from TLW. Notice the RF voltage across the shunt capacitor. It is not much higher than for a matched load at a kilowatt. The transformation ratio is small so there are no high impedance points in the network and therefore no especially high voltages.
A physically smaller capacitor than the one selected is not advised. The current is high and you want a capacitor rated for high power RF so that the loss is low and within the physical heat dissipation rating of the component.
The low risk of flash over to the aluminum enclosure permits a tight fit, and that will come in handy during installation on the stack switch. The high power test would have to wait for installation since there is no easy way to simulate a high power complex impedance load for bench testing.
The time had come to go for a climb.
Installation of the network
Through the miracle of software and bench simulation the work on the tower was brief and successful. I plugged the coax to the upper yagi into the antenna port of the network and the analyzer to the other port. Take care because the network is not symmetrical. Label the ports if that help you to remember.
I fiddled with the coil to see if I could do better. For reasons described earlier, it was not to be. The match was near perfect at 14.1 MHz and behaved as calculated by SimSmith elsewhere across the band. It is a little high at the top end of the SSB segment, a place I only venture during popular SSB contests with heavy activity. I can live with it.
The enclosure I chose for the L-network was no accident. It fits very nicely onto the stack switch antenna port with a male UHF barrel connector without striking the connector for the other (lower) antenna port. Weatherproofing is a challenge since there is little space between the upper and lower yagi ports to wind tape. I will have to improve the temporary job before autumn's cool, wet weather.
The enclosure is oriented to allow water to leak out the bottom edges. The bottom two screws were not installed to help with that. Tape placed across the top edges limits water incursion. There are better enclosures available but I used what I had on hand, and it really is good enough. It's 110' in the air and no one will see it but me.
When in lower + upper stack mode (BIP) the parallel SWR via the stack switch L-network is even better (see picture to the right).
I tested the completed system with a kilowatt to be sure there are no weak components or poor arc tolerance. All is well.
Phase compensation
The misadjusted gamma match and the upper yagi's matching network both exhibit a phase shift. In this case they are in the same direction so they add. A delay line for the lower yagi is required. Should I ever fix the gamma match on the upper yagi the network and delay line must be removed.
Measuring the phase of the antennas is impractical, so I rely on calculation. Unfortunately, it is easier to calculate the delay line length in software than in reality. On the other hand, forward gain is not overly sensitive to modest phase error. Where we can lose is with the size of minor forward lobes, as shown in the previous article on this topic.
Phase shift in the L-network can be accurately calculated. You can see this (above) for both TLW and SimSmith. The phase shift in the mis-tuned gamma match is more difficult to ascertain. Modelling of the net reactance at the feed point is the best bet, and what I previously did with a model. That is what I will use: varying between 10° to 20° across the 20 meter band.
The L-network phase shift (before substituting a 100 pf capacitor) is 39° at 14.1 MHz, and it, too, varies with frequency. Since this network is almost identical to that in the modelling exercise, and I have had difficulty developing a model that corresponds well with the real antenna measurements, I estimate the phase shift ranges between 49° and 58° across the band, including that of the gamma match.
They add because both phase shifts have the same sign, which is to advance the signal to the upper yagi. Correction therefore requires a delay line of around 54° to the lower yagi. This is a almost exactly what was found for the earlier modelling exercise, and that is not surprising. Software modelling works and it can save a lot of time.
For the average phase shift of 54° the length of the delay line is 0.15λ, or 3.2 meters in free space at 14.1 MHz. That must be multiplied by the VF of the coax used for the delay line. For example, 2.1 meters (6.9') for RG213 and 2.7 meters (8.9') for LMR400. All that said, the length is not critical since the phase shift is frequency dependent. No fixed length can be a perfect solution, and an average value is good enough in this application.
My plan is to use RG213 and insert it at the tail from the lower yagi's feed point at the tower. Should I not repair the gamma match on the upper yagi and the delay line continues to be used, the RG213 will eventually do double duty as a rotation loop. I hope to make the lower yagi rotatable in the next year or two. It is currently fixed on Europe.
This is easy to implement, and yet I haven't done so yet. The degradation of the stack pattern without the delay line is small, but with careful A-B testing on the air it is noticable. The modelled deficit is about 1 db of gain and the appearance of a higher angle minor forward lobe.
Good performance without the delay line may seem odd but when you add in the larger uncertainty of terrain on the yagi patterns the phase error is not a major concern. SWR is the more important concern since amplifier tuning can be a problem when switching among the stack's 3 modes. Correcting the mismatch also restores the equal power division required for optimum stack performance.
Time is a concern now that autumn is fast approaching and my many antenna projects require my attention. Those are the priority now that the SWR of the 20 meter stack has been corrected. The delay line can wait a few more weeks.
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