Above: Matching a 20m EFHW by using a quarter-wave transmission line transformer so as to provide for high power handling (500 watts no problem) and to handle a wide range of EFHW feedpoint impedances, all with minimal loss.
If we assume that the feedpoint impedance of an EFHW is 4000 ohms resistive, and we want to match that to the 50-ohm output impedance of our transmitter, we can calculate the characteristic impedance of the needed quarter-wave line as follows: Zo = Sqrt [Zl * Zin] = Sqrt(4000*50) = Sqrt(200000) = 447 ohms. Commercially available 450-ohm window line actually has a characteristic impedance of around 360-400 ohms, but that is close enough for our purposes.
In the screen shot above, TLDetails shows that 16-ish feet of a window line such as Wireman 552 (nominal Zo of 380 ohms) can be used to give us an input VSWR of 1.29 with a loss of only about .3 dB in the quarter-wave transformer. We would probably want to use a current choke between the transformer and any coax feedline leading to our radio's output, but since we have already matched the system to 50 ohms, a 1:1 choke made from a coil of coax (with or without a toroid core) would give us low loss. The insertion loss would be roughly equal to the matched loss of the short length of coax used, plus connector loss, or roughly .2 dB total if we wound the device using RG-58. This means that our total system loss (matching circuit + current choke) is only about .5 dB.
It might even be possible to feed the end of the matching transformer directly into the transmitter without a current choke (e.g., through a BNC-banana adapter) if we are using QRP power levels. At higher power levels I'd go with the current choke to keep RF off the rig's chasis, keyer and mic lines, and of course... us!
Although this is a single-band matching solution, it tolerates wide excursions of the EFHW feedpoint impedance, which is nice since it will vary quite a bit depending on how it is deployed, ground characteristics, etc. If it ended up being anywhere within the range of 1500-5000 ohms we would still end up with a VSWR of 2:1 or better.
Note that we can also make our own 400-ish ohm open wire line and use that for the transformer instead. N5ESE shows a nifty technique for doing that here: http://www.n5ese.com/openfeed.htm
In practice we would cut the parallel line somewhat longer than a quarter-wave (maybe 20% longer), leave the far end open, then snip pieces off of it until we get an acceptable match. If using an antenna analyzer that can display complex impedance, we should initially see inductive reactance (X with a positive sign). This will decrease as we snip pieces off until the reactance reaches zero at the optimum length. If we snip too much, then the sign of X would flip negative indicating capacitive reactance. Or, we could start by adjusting the frequency of the antenna analyzer to find the frequency at which X=0. Snip pieces of the end of the line until that frequency moves to our desired frequency.
The disadvantages of this matching technique vs. the more widely used toroid-based matching transformer are:
1) The parallel line used in the matching transformer would need to be kept up off the ground.
2) It is a single band solution. A well-designed/well-made toroid-based transformer would be broadband enough to provide an acceptable match on multiple bands (assuming that the antenna wire length is set to a half-wave or even multiple thereof on each band).
The advantages are:
1) Much higher power handling capability while remaining lightweight. A toroid-based transformer designed to handle high power levels would need to be beefy and heavy.
2) Low insertion loss, significantly lower than some of the not-so-well-made toroid-based EFHW matching networks that people describe out on the Internet (e.g., the ones that use low-Q components, see Tom W8JI's write-up for more info: http://www.w8ji.com/2end-fed_1_2_wave_matching_system_end%20feed.htm )