This is the second of a two-part series. Part 1 can be found (Q&As Say the Most Basic How-to-Dos Remain Elusive, Part 1) here.
What's a decent equation that tells me how much I can gradually tweak the antenna's resonant frequency by simply adjusting its physical length? You can't do that with the traditional W3DZZ trap design.
It's an illuminating yet strange question to me, in that it involves a degree of roundabout logic. Let's approach the question as a novice. Assume you know a "standard" half-wave dipole, as popularly quoted in the literature over the years, is approximately 135 feet long at 3.5 MHz, 66 feet at 7 MHz, 34 feet at 14 MHz, 22 feet at 21 MHz, and 17 feet at 28 MHz, and that those lengths are accurate to within a few percent (derived from l = 468/f, where l is the length of a resonant half-wave dipole at frequency f). Now we need to operate our trap dipole antenna as a half-wave dipole on 3.5 MHz, three half-waves at 7 MHz and 14 MHz, five half-waves at 21 MHz, and seven half-waves at 28 MHz. Now say you make a reasonable (but not quite right) assumption that the equivalent electrical length of your antenna for a given frequency will be the product of the antenna's half-wave length times the half-wave multiplier. So the antenna's equivalent electrical length needs to be 135 feet at 3.5 MHz, 198 feet at 7 MHz, 102 feet at 14 MHz, 110 feet at 21 MHz, and 119 feet at 28 MHz.
If you plot the required reactance of the parallel LC network that you need to get these lengths for a given band, versus the difference between the antenna's assumed multiwave resonant length and the antenna's actual length (about 122 feet), there's a fairly straight-line relationship. We minimize curve fitting slightly at 28 MHz (10 meters), where in practice typical changes in L and C have little effect. Or you can just see the plot in Figure 2 for L = 8.9 microhenries and C = 40 pF. The values in this particular case are suited for a "tape measure" (i.e., very-thick-wire) antenna. We use a different solution set of L and C for this example, in part to demonstrate that we'll reach the same conclusion.
This plot of trap reactance versus the difference between required and actual antenna length suggests that the basic dipole equation is adequate for deriving the change in f versus l for an antenna with traps.
However, the corresponding plot (not shown) for the old W3DZZ trap dipole would be a somewhat uneven line not considering 7 MHz (40 meters). And it would basically show a discontinuity if you included 40 meters, because it uses its traps as an isolating element for that band. So it would be difficult to render judgment on its "straight-line" profile directly. That aside, we know that antenna was too short to begin with, and making very small changes in parameters with the W3DZZ generally creates a brand-new round of long cut-and-try experimentation.
So does a fairly straight-line relationship suggest our new trap antenna will behave similarly to a regular (no traps) dipole antenna? Can we simply adjust the length of our trap dipole, giving us another degree of freedom to tweak frequency smoothly?
Let's differentiate the dipole's general equation, which is actually l = 492 (n – 0.05)/f, where n is the number of half-waves at the frequency of operation. We assign certain values of n to evaluate what will happen at a corresponding operating frequency, f, but we do not consider n as a function of f. Thus we treat 492 (n – 0.05) as a constant. After differentiation and rearrangement of the variables, we get the finite-difference equation:
How good an approximation is this (no traps) dipole equation? To find out, let's adjust the resonant frequency of the antenna simply by changing its length slightly. If Δl = -1 foot, then Δf = 26 kHz at the low end of the 3.5 MHz band, 34 kHz at 7 MHz, 135 kHz at 14 MHz, 181 kHz at 21 MHz, and 229 kHz at 28 MHz. Note again that we set n = 1 at 3.5 MHz, n = 3 at 7 MHz and at 14 MHz, n = 5 at 21 MHz, and n = 7 at 28 MHz.
The resulting Δf values are in fairly good agreement with the EZNEC modeling program, which predicts that Δf for our 122-foot trap dipole (14-gauge wire with L = 11 microhenries, C = 31.2 picofarads) will rise about 25, 30, 100, 160, and 230 kHz, respectively. Thus, qualitatively, our equation is good enough, and we see that, within limits, we can secure a monotonic but not a directly proportional change in resonant frequency from band to band (as many initially tend to believe) as we adjust the trap antenna's physical length. In practice, this iterative adjustment of frequency with length allows us to fine-tune the antenna quickly on all five bands during initial installation in the field.
I gave your equation set to the numerical software (Systems of Nonlinear Equations, v1), and it didn't find any solutions. What's happening?
How did you program them in? Trust the software, but be aware of its limitations. Software using such techniques as the Method of Undetermined Coefficients and Newton-Raphson approximations can be very touchy when it tries to converge on answers that are numerically very small (much, much less than 1). Rarely, the software may correctly determine the value for some or most of the variables; others will be incorrect because, in setting up your equations, you've unwittingly broken a few program rules for effective numerical iteration. In iterative approximation, the software occasionally encounters a fraction having a small numerator divided by a small denominator. Then the software may not be able to converge on a solution, especially if the unknown variable is in the denominator of some system term. I can't be sure, but this may be one problem you encountered; the program couldn't solve for L, which is measured in microhenries.
One tip an old mathematician gave me years ago, and which I generally apply today to help get around problems with tiny numbers, is to place as many unknown variables as possible in the numerator of your equations. The "harmonic equation" set as originally shown in an April post is of the form:
So if you had L in the denominator of the term as shown, place it in the numerator. To virtually eliminate the computational "blow-up" possibility when working with tiny denominators, also express such terms as 1/2πf for each f as a small decimal number, and place it in the User Constants section of the program for multiplying by L. The numerical software should successfully calculate the value of all unknowns, and the actual value of L will be the reciprocal of what the software displays.