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Grasping illumination pattern

02 May 2016  | John Dunn

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A colleague once asked me about the nominal illumination pattern of a radar antenna. His premise was that the antenna's emitted RF emerged as a cone which would impinge on a plane whose area was not perpendicular to the centre axis of that cone but was instead slanted with respect to that centre axis. As a result, the illuminated area of that plane would be an ellipse having a major radius, call that "A", and a minor radius, call that "B". He asked me how would one find the area of that hypothetical ellipse? Maybe just look it up?

Well, yes, we could just look it up but to actually do this calculation, we can set up an integration process that will find the area of one fourth of that hypothetical ellipse having radii "A" and "B".

Having gotten this far, we next go to a table of integrals such as this where we find the following:

We then proceed with the calculation as follows.

Since the calculation is the area for one fourth of the ellipse, the area of the total ellipse is four times the calculation's result which comes to Area = pi * A * B.

As we noted of course, we could just look up the formula for the area of the ellipse and take it on faith. However, I'm usually a bit more comfortable with a formula if I know where it came from, trusting of course that the integral table I used is irrefutably correct.

In any case though, if we look at the special case of A = B = R where "R" is the radius of the resultant circle, we have Area = pi *R² and at that point, we pay our due homage to Euclid and Isaac Newton.

About the author
John Dunn contributed this article.

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