It isn't clear (to me, anyway) why you are specifically seeking an air-cored solution. What is the problem with a ferrite core? You can get a very good match with a 3:1 turns ratio, or if you really want to get precise then increase the turns so you get M:N ratio closer to SQRT(10).
Ah, that's easy. You want a tuned impedance matching network, or perhaps a tuned transformer, or a tapped resonator, etc...
And I did find an amazing report of a modestly wideband (15%) 50-ohm to 1-ohm transformer, working from 1200 to 1400 MHz.
One thing, keep in mind that 0.2nH of inductance is more than 0.5 ohms at 500MHz. Watch out. Do you know what you're doing?
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I switch to ferrite core now. Could you please tell me what is the upper frequency limit of ferrite core? I heard from someone that the limit is 200MHz.
Note also the links to course notes, transmission line and Z-matching SW by Trueman at Concordia and Pozar at Amherst;
RL
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Awww ferk - misread that bit didn't I ....
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