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I am trying to build an antenna for receiving NOAA satellites and I need to build two 137,5MHz dipoles. When I enter the frequency and width of the wires in a calculator, I obtain that $λ/2=1.09\,\mathrm{m}$ and the length of the dipole should be 1.05 m. (It would be great if by the way someone could explain why this difference occurs.)

However, does that mean that the total length of the antenna must be that or that each leg must be 1.05/2 m? And does the separation between the legs affect the antenna measurements in a noticeable way? If so, is it the smaller the better?

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The most general answer to this question is that you neither include nor exclude the gap between the elements. Rather, whatever is going on there is part of the antenna design. Physics doesn't care which things we call “antenna elements” and which things we call “wiring” — the shape of all conducting elements matter. To match theory and reality thoroughly, we would calculate using an antenna model which includes not just the gap but the shape of the wires that come from “across” that gap to the feed-line. Simulation tools can do this quite well, insofar as you can enter an accurate model and spend the computation time.

In practice, for amateur-built antennas, you should use construction techniques which allow the feed point (the gap and wiring I referred to above, and everything that is neither antenna elements nor feed line) to be made much smaller than the wavelength. At HF this can be bolts and wires; at low UHF you might solder carefully shaped rigid wires and matching components directly to a connector; at higher frequencies the antenna and feed might be a single unit using printed circuit manufacturing.

As long as the feed point is small in this way, the effect of choosing different answers to the question of where to measure from will be just one small part of the many effects which cause real antenna performance to deviate from simple models. At which point the practical answer is to make your antenna a bit long, measure it with an analyzer, and trim it down to hit your target.

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