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I want to build a pyramidal horn telescope as a science project but I’m having some trouble understanding the waveguide dimensions.

This documentation lists the dimensions of a paint thinner can be used (a = 16.8 cm and b = 10.48 cm). Although I understand the purpose for these dimensions it’s quite unclear to me why these values are used and how one could reach to their own custom values that would still work for the desired frequency of 1.42 GHz. Could any value be used that will yield the desired gain and cutoff frequency? Have I misunderstood something? Any help would be appreciated.

Thanks in advance!

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Yes, you can use your own custom values as long as they give the required cutoff frequency and gain. Some important fundamental concepts for the horn antenna are given below to help

For directivity what primarily matters is the area of the open end of the horn antenna. If I assume the area of the open end of the horn is \$ A \$ then the directivity \$ D \$ is given as \$ D = \dfrac{4\pi A}{\lambda^2} \$. The directivity of the antenna is similar to gain but it does not account for reflections from the antenna or for losses in the antena.

Generally the gain of the antenna is given as \$ G = e_r e_c D \$. Here \$ e_r \$ is the reflection efficiency of the antenna. This accounts for the fact that part of the wave being fed into the antenna will be reflected back due to impedance mismatch. Meanwhile \$ e_c \$ accounts for the conductive losses in the antenna. \$ e_c \$ is more of a material constraint and something you probably cannot impact too significantly.

The last consideration I would suggest is to think about the length of the horn. Generally, a longer horn will allow for a slower transition from the impedance of the waveguide to the impedance of freespace leading to fewer reflections. So I longer horn will have fewer reflections in theory. Since you have a relatively small bandwidth of interest, I think the suggested quarter wavelength from the document attached should be fine. Still, if you do try to make a larger aperture, it would probably be wise also to lengthen the horn a bit

If you have access to a library, Constantine Balanis has a great book called "antenna theory analysis and design," which could address many of your questions in more detail.

Edit: For the waveguide, I would not vary too far from the values suggested by the document; otherwise, you risk reflections at the coaxial to waveguide transition.

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