So - when or why would a random wire be preferred over a multi-band end fed?
The EFHW has to be a specific length, and only has a
low SWR on harmonics of the base (half-wave) frequency.
Some matching transformers appearing on the web can
have relatively high losses - up to 3 dB.
A random wire can be any length. Too short and the
efficiency drops, but otherwise not particularly critical.
With either antenna, the radiation pattern on each band
depends on the length and arrangement of the wire.
With too long of a wire, the main lobes may be too narrow
for your operating preferences, although they can give
useful gain if they are pointing in the right directions.
If you want to use a wire on frequencies where it isn't
an integer multiple of 1/2 wavelength the high impedance
EFHW transformer won't work very well. That's why your
40m antenna also works 20, 15, and 10m, but not 80m,
60m, 30m, 17m, or 12m.
The use of a transformer to match either sort of wire
is a relatively recent introduction. There has been enough
recent work on high-impedance wideband transformers
that low-loss versions are possible if enough attention
is payed to the ferrite core and the number of turns for
an EFHW.
However, there no "appropriate" transformer that will
match a random wire across a wide frequency range,
especially one shorter than 1/4 wavelength. The feedpoint
impedance can vary from less than 20m ohms (with
considerable reactance) for short wires to thousands
of ohms. The standard 4 : 1 or 9 : 1 transformer will
lower the worst case SWR on bands where the wire is
more than about 3/8 wavelength, but will make it worse
on other bands.
G8JNJ has made a very extensive analysis comparing various transformer types, which may be
worth reading for a better understanding. Typical
losses are as low as 1 dB on some bands, and may
be 3 dB or more on others, even for the better choices.
Additional losses in the coax and tuner also need to
be considered.
There certainly may be times where the convenience
of a remote random end-fed wire overcomes the
potential losses. For example, for a discrete antenna,
where you just set up whatever wire you can manage
and make it work across all bands of interest. That
trade-off is a personal choice, there isn't any "right"
or "wrong" answer.
My preference for either wire length is to feed it directly
from an antenna tuner, instead of using any sort of
matching transformer and coax. Usually I just bring one
end of the wire in through a window and tie it to the tuner
(along with a good ground). An autotuner (or remotely
switched preset tuner) at the feedpoint can be used
when that isn't convenient.
And, over the years, I've generally had best results
with a half wave wire for 80m (~130'), partly because
my interests tend to be on the lower bands more than
the higher ones. I've used wires from 18" to 600' over
the years on HF: while wires as short as 1/8 can be
pressed into service, I'd suggest 1/4 wavelength as a
minimum length with a good ground system, or 3/8
wavelength otherwise, and overall performance seems
to drop off by 4 wavelengths (a ~130' used wire on 10m)
unless the wire is pointing in a desired direction.
A wire over 3/8 wavelength or so should be matchable
with a simple L tuner using one coil and one capacitor,
which I have sometimes implemented as a preset unit
with a band switch: it has to be set up initially, but if
your antenna doesn't change often, then once the
capacitors and coil taps are soldered to the switch
contacts, band changing is simply a matter of
selecting the desired band, which gets around the
problem of autotuners not liking to match high
impedance antennas.
