Soliloquy . ..

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^ My onmi-directional antenna looks like the above ...

^ If I were to attempt to place a reflector on this antenna,
^ it would, I trust, be electrically connected to the center
^ radials but insulated from the brackets? How would I
^ determine the size and spacing?

My comments were in response to your query on directionality and they are
probably not useful in your system. I am not an antenna expert, only an
amateur experimenter, and I have not seen any documentation on reflectors and
directors on anything except half-wave dipoles. Although, as I move around
town and examine TV antennas it appears to me that they are making some sort
of use of reflectors, directors, or both.

But the function of reflectors and directors, on the receive side, is to
increase the amount of the desired frequency reaching the antenna. They may
not be very useful if the intent is to scan a wide range of frequencies.

N2AH appears to have your answer.

Frank