How to make a Tune-A-Stick resonate

DB> With all that inductance, theoretically it should take a
DB> gnat's whisker's touch to adjust a piddling small capacitance to
DB> tune it, particularly at higher freqs.

It would probably be useful to start off by estimating just how much inductance you have. The formula for that is pretty simple:

L = F x n(^2) x d microhenries

Where
L = inductance,
F = form factor, the ratio of diameter to length (d/l),
n = number of turns,
d = diamter of coil (inches)

You'll wind up with a lot of zeroes in the arithmetic, but this is the simplest way to state the formula in a fast text file.

DB> I do know, however, that each pair of turns
DB> creates a parasitic capacitor (so that'll be
DB> about 766 of 'em), and that this makes the Tune-A-Stick (TAS)
DB> self-resonant. I don't know how to determine what the
DB> capacitance might be.

Every inductor has some self-resonant frequency, but at the sort of dimensions I recall you mentioned, the added capacitance per turn is probably not particularly significant in the HF range.
More significant to practical use would be the series-resonant frequency range you can tune the stick to (meaning) connecting the variable capacitor in series with the Tune-A-Stick. (One position of the MFJ tuner effectively removes its inductance from the circuit, for an easy way to have a capacitor only in series.)

You can estimate that frequency range by computing the series resonant frequency at the minimum (10 uuFd) and maximum (324 uuFd) values of the capacitor in the MFJ box. The formula for estimating those frequencies is:

                  1
        f = ----------------
                   ________
            6.28 \/ L x C

Where:
f = frequency in kilohertz,
L = inductance (from preceding) in henries,
C = capacitance (min and max) in farads.

Again, using the arithmetic stated this simply will give you a lot of zeroes to work with.

But, once you know that, you'll have some notion of where in the band the Tune-A-Stick is resonating, and approximately what capacitor position should be optimum.
But what happens if you discover the Tune-A-Stick has too much inductance? You have the inductor in the MFJ box and its switch to use to reduce the inductance. Just like resistors with DC, inductors in parallel will reduce the total inductance. So, you can hook up the tuner's switchable inductor in parallel with the whole length of the Tune-A-Stick to try obtaining an inductance that can be resonated in the 3-30 mHz region. Now, if that turns out to be a very small amount, it can start to be a short circuit at HF, so you may need to put a ridiculously large capacitance in series with the tapped inductor, something that looks like zero ohms to the RF current, but blocks the DC path through the switchable inductor, to prevent that. Several hundred uuFd should do the trick - and that's nothing more than another variable capacitor with its plates closed, if you have one lying around.

DB> I have read somewhere that above and below the resonant freq
DB> (fo) the circuit takes on different properties, becoming more
DB> like a capacitor one way, an inductor the other. I can't
DB> remember which.

That will become important only after you are sure you have gotten the Tune-A-Stick to be resonant.

DB> The upshot is that it does not consistently display any
DB> "peaking".

Probably needing some understanding of it as indicated above. That sort of thing isn't a problem. If something is inductive, you then tune it with series capacitance, while if it is capacitive, you tune it with series inductance. (Of course. most inductors aren't readily adjustable, so that gets changed in lumps, as in the tapped inductor of the MFJ box.)

DB> The big difference is in the amount of rejected signals, such as
DB> the computer or local AM bcs (like the one on the hill above
DB> me). No filter needed.

That's the big advantage you've already obtained, and being able to resonate the Tune-A-Stick will help that advantage by maximizing the RF current in it.

DB> I am going to work on the tuner, or "waffle box" (since it
DB> waffles about the signals ), this weekend. I had removed the
DB> tapped inductor from the MFJ, but I think I erred in that it
DB> should be connected in parallel with the TAS, thus bringing the
DB> total L down to something the 240pF varicap can tune more
DB> surely.

Sounds like you're on that path already. Do let me know what sort of results you get. The preceding is in the hope it will help you get the Tune-A-Stick into resonance, or keep it in a meaningful frequency range to resonate it.

On a longer view, if you get it resonant, then I suspect it will exhibit some directionality off its end, and perhaps be useful to "point at" the desired signal, to offer some (perhaps rather small) amount of rejection of unwanted RF as well as local noise, like a helix does at VHF or UHF.
Do keep reporting what results you get. Sounds like you're on a good path there.