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Someone can explain to me the difference in terms of performance between a flared rectangular waveguide antenna (precisely a pyramidal horn) and a not flared rectangular waveguide antenna?

In horn antennas, is there is a better impedance matching with the free space?

In some books I found only the explanation of horns, but I didn't find anything about the reason to make a flare on a rectangular waveguide to improve its performance. The only thing that I found is that flaring is mostly used at high frequencies, like microwaves.

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Short Answer (tons of books, web stuff):

Yes it provides a better match to free space TEM mode. It allows you to adjust the gain/directivity of the radiated field as well.

There are other variants on the wall geometry (exponential vs straight) flare that give improvements as well, but are generally more expensive to manufacture.

The primary reason horn antennas are not used below ~1GHz are primarily because of cost/size constraints. There are waveguides down to about 320MHz that are about 2' x 1', so a waveguide can become rather large at that point, a horn antenna even more so.

Since the mechanical dimensions and deformation resistance are critical, manufacturing costs for these sizes are high and applications are mostly reserved for high powered UHF RF transmitter outputs.

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