The relatively simple algebraic solution we applied in Use MacGyver’s Methods, Part 3 to what appeared to be a complex antenna problem opens the door to further visualization techniques. Now we'll step up to an example where you can simply gauge how a given design proposal might fare even before you apply the meat of the design equations. Let's see how this largely forgotten technique can save you a lot of time in certain cases, such as a tuned-circuit application.
Take the multiband resonant antenna as an example. It should be easy to determine whether a given five-band design is likely to work before you ever set pencil and paper (or computer program) to it. All you need to do is look at it. What's the length of the antenna? What are the values of the L and C components in the traps/loading elements? If you know these things and some fundamental engineering, you're ready to apply the method.
Here's how to proceed. Examine the characteristics of the antenna's parallel LC tuned circuits, i.e.,
Now just above a trap's resonant frequency (i.e., resonance = where the capacitive reactance equals the inductive reactance), the parallel tuned circuit has extremely high capacitive reactance. Thereafter the reactance decreases exponentially with increasing frequency. Conversely, immediately below the trap's resonant frequency, the tuned circuit exhibits extremely high inductive reactance, which decreases exponentially with decreasing frequency.
All you need to know now is the antenna's required length at its various frequencies of operation, versus its actual length. Let's analyze, for example, one of the earlier popular “trap” antennas for five-band operation, which appeared in 1955 — Buchanan's W3DZZ dipole. The length of that antenna, which touted resonant operation in the 3.5, 7, 14, 21, and 28 MHz frequency segments, was 108 feet. Will it work as advertised?
This antenna was designed to operate as a half-wave dipole on 3.5 and 7 MHz, three-half waves on 14 MHz, five-half waves on 21 MHz, and seven-half waves on 28 MHz. Now let's use a well-known relation for long wires to determine the approximate required length for a given wire working at a specific number of half-waves (n) at a given frequency:
From this equation, we see that the approximate length of a resonant half-wave (n = 1) dipole is 133.5 feet at 3.5 MHz, 66.8 feet at 7 MHz ((n = 1), 103.7 feet at 14 MHz ((n = 3), 116 feet at 21 MHz ((n = 5), and 122.1 feet ((n = 7) at 28 MHz.
Now the basic configuration of the 108-foot long W3DZZ antenna places the traps, which are resonant at about 7 MHz, at each end of a 64-foot section of wire at the antenna's center (in practice, a 66.8 foot wire as determined by the equation above looks to be a better length). Thus each trap provides a very high impedance (i.e., insulator) for a 7 MHz half-wave dipole fed at its center. We therefore expect this section of electrically isolated wire to resonate somewhere in the 7 MHz band.
But what about the other frequency bands? Well, the traps provide inductive reactance below 7 MHz, which is equivalent to adding length to the antenna. So at 3.5 MHz, the traps, which act as inductive loads, essentially extend the physical length of the antenna from 108 feet to at least 133.5 feet. We therefore conclude that we should be able to resonate the antenna somewhere in the 3.5 MHz band.
What happens at 14 MHz? At this frequency, the traps provide capacitive reactance and ideally will provide enough reactance to reduce the effective length of the antenna from 108 feet to near 103.7 feet. So we presumably can resonate the antenna in the 14 MHz region as well.
However, we encounter serious issues as we approach 21 and 28 MHz. The antenna needs to be about 116 feet long at 21 MHz, as mentioned. But the tuned circuits provide capacitive reactance at this frequency, and so they effectively reduce the physical length of the antenna to below 108 feet. Thus we wouldn't expect this antenna to be resonant at 21 MHz.
The same is true at 28 MHz. The antenna needs to be about 122 feet long. But its physical length is just 108 feet, and any capacitive reactance provided by the traps would tend to subtract from the 108 feet. Again, we wouldn't expect it to work. Indeed, the resonant frequencies for the antenna on these two top bands are in the region of 22.5 MHz and 32 MHz, respectively, as indicated by the EZNEC program's antenna analysis (also see ARRL Antenna Book, 19th edition, p. 7-10).
The antenna would need a little tweaking on 14 MHz, too; the resonant frequency appears to be about 14.7 MHz. But even without EZNEC, we've discovered that, beyond the specific values of L and C , we need to calculate for the traps and where the traps need to be placed, the 108-foot long antenna was a bit too short to begin with.
So we see by simple inspection and a bit of basic engineering that this popular antenna would likely be functional at the lower frequencies and probably be OK on the middle frequency band. It would not work as intended on the two higher frequency bands.
Over the years, some users have claimed success with the W3DZZ antenna with low SWR on all bands, but in retrospect the antenna's touted “resonance” at the higher frequencies may have come about because the feedline somehow became part of the antenna and added sufficient length to make the antenna appear resonant at 14, 21, and 28 MHz. Or perhaps there were some anomalies in the construction of the traps that worked in the user's favor.
That doesn't take away from the brilliance of Buchanan's original design. It did bring attention, however, to securing better techniques for feedline-to-antenna coupling to ensure that the transmission line carried out its sole purpose of delivering power to the antenna without becoming part of it. Alternatively, we might consider that errors in measuring technique provided overly optimistic claims for antenna performance. But that's a subject for another time.