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I disagree with the idea that he was talking about broadcast antennas only, although he is definitely referring directly to ground mounted quarter wavelength antennas.  The .pdf I referred to earlier and the book you looked it up in (I have it as well, I just used the .pdf because it is free to everyone) were both written for ham radio operators, and while there is a ham band near the broadcast AM band, what was talked about applies to every ground mounted quarter wavelength antenna, and to an extent every quarter wavelength antenna that has the potential for ground losses (or losses in general).  This includes mobile quarter wavelength antennas, which are notorious for ground losses.

 



The example I'm sure you have seen me use is for a full length quarter wavelength antenna, where X=0 and R=50 we have 28% in losses (happy to show you the math behind it).  If you instead have X=0 and R=36 then you have 0% in losses but suffer from an SWR close to 1.4:1, which is in the range of 2.8% power reflected (but, importantly, not necessarily lost).


The principles that govern full length quarter wavelength antennas are the same weather you are talking about 1 MHz or 1 Ghz antennas.  The only things that change as the frequency changes is the size of the overall antenna.


All that being said, I still don't have a good answer to tell you how to figure out what R should be for a given shortened antenna aside from as the antenna gets shorter R at resonance goes down (unless there is some other corrective matching involved), this is why I stick with the full length quarter wavelength antenna for my examples, their R is a well known value.  As measured at the antenna lowering the R value while staying at resonance (either via bonding the vehicle or adding more radials to a ground mounted antenna or whatever) has the same effect on efficiency.


I suppose you could create an antenna system that shows the actual R for a quarter wavelength antenna (at least very close), then simply put in a shortened version of the antenna for the same frequency for comparison.  This should give you an accurate R for said antenna.



The DB