Ground Losses
Here are my thoughts (right or wrong) about ground losses with end fed half wave vertical antennas. Let's say you just use a ground rod for a return. To return the displacement currents that enter the ground from the field around the antenna there will be a high earth resistance with such a simple ground. Let's use 35 Ohms for an example. For the 1/4 wave length vertical with 35 Ohms of feed point resistance, the input power will be divided between the ground resistance and the feed point resistance. The two resistances will add up to a total feed point resistance of 70 Ohms but the efficiency of the system will only be 50%. For the 1/2 wave length vertical with an assumed feed point resistance of 2450 Ohms the feed point resistance will be 2485 Ohms (2450 + 35) and the power dissipated in the earth at the feed point will only be 1.4%. This gives us an efficiency of 98.6% for the 1/2 wave length vertical. With an efficiency of 98.6% I don't see any reason to lay down an elaborate radial system for a 1/2 wave length vertical from an efficiency point of view. However, at about a 1/4 wave length from the base of a 1/2 wave length vertical the ground currents once again increase. Of course this pattern will repeat itself out for many wave lengths from the antenna. For this reason a bunch of (>1/2 wave length) radials should make an improvement on ground wave field strengths or on very low elevation sky wave paths by lowering the pseudo-Brewster angle. However, I am skeptical about how much of these ground (or surface) wave currents, if any, return back to the feed point and contribute to the feed point losses. In any case, to reiterate my earlier statement, improving the ground conductivity out for many wavelengths will improve the far field of any antenna, at least at some elevations. This does not mean that an elaborate ground is required for an end fed half wave length vertical antenna to "have something for the antenna to push against" or to "complete the circuit" as is often said. Whether or not you consider ground loss several wave lengths from the antenna as antenna system loss or not is up to you.
