What I am trying to do is figure out what lenght my end fed half wave should be and the lenght of the counterpoise (or the ladder line) should be for a portable operation where I am sitting at a picnic table out in the park working from 5 to 30 watts with battery power. Given that you really can't know how your antenna is working when you just set up outside for some portable work, at least until you have been there a while, I wanted to at least get the theory correct. I understand that the tuner will load up practically anything, so I don't want to rely on just getting a good SWR match.
First, there is no length that your antenna or feedline should
be. There are many
combinations of antenna and feedline lengths that will work, depending on how you plan to
string up your antenna, the distances and directions you want to work, the type of tuner you
are using, etc. The best we can do is to offer suggestions, but you have to make the
final determination of what works best in your situation.
Obviously, if you are using a half-wave end-fed, then it should be half a wavelength long,
though the same matching system will work for multiples of that. In this regard, a half-wave
wire for 40m can also be used on 20m. If you are using a base tuner attached to the wire
then a parallel-tuned circuit can be arranged to match both bands by just adjusting a single
variable capacitor. In that case, adding a quarter wave radial for each band to the tuner
should give a reasonably good system, and you may be able to get by with shorter radials,
possibly just one for 20m, or even a metal object such as a pack frame, but you'll have to
see how that works with your rig.
If you are using a Zepp feed, then a half-wave wire for 40m will work, but the quarter wave
feedline becomes a half wave on 20m. Is this a problem? Well, I've used several Zepp
antennas fed with 50' of 300 ohm twinlead (because that was the only feedline available.)
This is close to a quarter wave on 80m (when corrected for velocity factor) and roughly
a multiple of a half wavelength on other bands. This worked reasonably well running QRP
with a parallel-tuned antenna tuner, but the high impedance might not be a good choice
at higher powers when using a relay-switched auto-tuner. As you suggest, I don't really
have a good comparison to say whether the antenna was working or not, but I was able
to make a number of contacts from Alaska with my Argonaut.
So there are a number of factors to consider:
(1) the impedance to be matched. Most relay-switched auto-tuners don't like high-impedance
loads due to the voltages that can be developed across the relays. An end-fed half-wave
antenna has a high input impedance. I've never had a problem matching one with a manual
tuner, and it isn't difficult to make a purpose-built tuner for one band (and perhaps multiple
bands). Using a quarter wave of feedline will bring this down to a much more reasonable
impedance - you might not even need a tuner, but this only works for specific feedline
lengths. (And limiting yourself to 17' of feedline doesn't let you get the antenna up very
high on 20m.) If your tuner can handle it, you can use other lengths to get your antenna
up higher, though the impedance will vary over a much wider range, and the common
mode currents may not be as well behaved.
(2) the radiation pattern. While a 40m half-wave wire will also have a high feedpoint
impedance on 20m, the radiation patterns are different: maximum radiation on 20m will
be between 15 and 60 degrees to the direction of the wire. If the wire is vertical, then
maximum radiation will be at about 35 degrees above
the horizon. You'll still make
some contacts, but a half wave wire will work better for most likely vertical angles.
When the wires are horizontal there will be a 4-lobed pattern on 20m which should
work well in the directions of the lobes and less well towards the nulls. It makes a good
general purpose antenna, but might not work in a specific direction of interest unless
you align it properly when you string it up.
(3) this brings us back to the important question - what is it about the end-fed half-wave
antenna that lead you to choose it in the first place? If you are thinking of using it
horizontally with a Zepp feed, then center-feed with twinlead to a balanced tuner would
probably be a better choice. Then the maximum radiation is in the same direction for both
40m and 20m, and it can be used on other bands even when the wire is not a multiple of
a half wavelength. You may still have a high impedance load with some combination of
antenna and feedline lengths, but you can simply choose a non-resonant combination that
allows your tuner to work on all bands of interest.
If you are planning to run the wire directly to a tuner at the rig (or a fixed tuner of some
sort connected to the rig via coax) as is often done for end-fed half-wave antennas then
you run into the problem of the radiation pattern of the full wave wire on 20m, besides
having to get the wire over a branch more than 60' off the ground. For 40m it makes
sense to slope the wire or run it over a branch and back down towards the ground, since
even high angle radiation is still useful during the day, but this isn't as effective on 20m
since the average height above ground of the maximum current points is fairly low.
It does make a difference where you are located, however. Over poor ground (very dry
or rocky) then horizontal polarization is likely to work better than vertical. But if you
are in a salt marsh or overlooking the ocean, vertical polarization may be much better.
One of best portable antennas I've ever used was a sloping Zepp-fed long wire over
salt water for one Field Day in Alaska.
Having done a lot of portable operation over the years, I've gravitated to a set of dipoles
on a common feedpoint. It gives me more flexibility in how I install the antenna and I can
optimize the configuration for whatever combination of bands I want. Best of all, once
I tuned up the antennas (>30 years ago now) I haven't used an antenna tuner with them.
Well, except for the cases where I've combined the wires in various other configurations
to suite the circumstances. Generally I string up the feedpoint as high as I reasonably
can and tie the wire ends off to convenient branches, etc. in an inverted vee configuration.
This also has the advantage that it requires only a single high point to install the antenna,
which could be a portable mast in an area that doesn't have trees.
So there is no "one right answer". There are lots of options, all of which will work to some
extent, though they will work differently. Fortunately all of these antennas are cheap and
easy to build, so you can try two or three different ones and see which works best for your
preferences and style of operation.