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flat-stock-, as opposed to round-coil mobile antenna....

You are right about the surface area contributing to bandwidth. But it is very frequency dependent, and for typical HF frequencies, that surface area gets to be quite large before any significant increase is noticeable. It's a characteristic of radiation, and since a coil does very little radiating, it's almost a useless quality.
The other side of that surface area and bandwidth thingy is that the surface area is not a sure way of increasing bandwidth without considering a number of other characteristics of the antenna, such as length, or 'resonant length' to be more accurate, which does not stay the same (length decreases as diameter increases). Think in terms of fractions of a wavelength at a particular frequency. "Short'n'fat" or "tall'n'skinny" basically the same results, but just how practical is it? Guess it just depends on what you got to work with?
- 'Doc
 

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