# Measurements in far field for radar cross section

I have to measure the S parameter for a rectangular sample with an antenna that works from 2 to 18 GHz. I need to measure a large band from 5 to 15 GHz and I have a problem with the known F.F. (Far Field) formula:

$$R = 2D^2/λ$$

Usually with $$D$$ is referred the aperture of the antenna, that in my case is 100 mm, while I found that for RCS the formula use as $$D$$ value the dimension of the sample (300 mm).

My questions are:

• Why can the F.F. formula be referred to the antenna or to the sample and
• when I need to use one or the other?

The far field formula is simply derived from a general idea that the phase error at the edge of the antenna aperture should be less than $$22.5^\circ$$ which means that $$r'-r < {1\over16}\lambda$$.
This figure is chosen because it gives gain errors of less than some small percentage, I can't remember the figure, it depends on the aperture field distribution too).

Now draw a new diagram with the large target object, and the source antenna, and work out the difference in path length between a ray from the centre of the object, to the centre of the antenna, and a ray between the edge of the object and the opposite edge of the antenna.

The whole path difference $$r'-r$$ should be kept to below $${1\over16}\lambda$$ for some level of accuracy.
You can see from this diagram (slide $$r'$$ up to make one triangle) that the "safe" far-field distance will be
$${2(D_{antenna}+D_{target})^2}\over\lambda$$
This will guarantee that phase errors from any part of the antenna aperture, to any part of the reflecting target, are kept below $$22.5^\circ$$.

You might be able to relax this criterion by half if you're willing to tolerate a larger error, or know something about the aperture distribution of the antenna, or the reflection properties of the target. A sphere, for example, might not reflect from the whole surface, only the middle. Unfortunately a horn antenna will have a fairly uniform field.

$$\lambda$$ in all equations above needs to be calculated at the highest frequency you are measuring (but not the highest specified frequency of the horn).

• @Allergenfree I've updated the formula... it's even further than I first thought, 16 m. – tomnexus May 14 '19 at 18:20
• Thanks a lot for the very good answer, you completely solved my doubt – Allergenfree May 15 '19 at 8:38