Aperture antenna directivity calculation

[baby_1]

Hello
Here is an uniform E filed distribution aperture antenna E and H far filed euations

and here is directivity calculation approach

my problem is why don't we calculate D with both E(theta) and E(phi)? My mean is:
instead of using

we should use

because both of them are not zero in far-filed zone and not zero at theta=pi/2

 

[marcusl]

$E_0$ is the magnitude of the electric field vector E. If you were to compute the sum of the magnitude squared of the components as you suggest, it should come out as $E_0^2$ as written in the text.

BTW, you have written something different. I assume that you mean $E_\theta^2 + E_\phi^2$ rather than $E^2(\theta)+E^2(\phi)$? The latter is valid but means something else.

 

[baby_1]

Dear marcusl
you are right I wanted to write

As the writer wrote we have

and we assume phi=pi/2 that maximizes U.
and with my approach:

and I didn't know why we assume only

to calculate power intensity ( U) ( as in the book mentioned)?

 

[marcusl]

The book didn't say to use $E_\theta$, it said that the field is maximum at $\theta=0$. Putting that angle into the equations and finding the sum of the squares of the components, as you did, gives the answer.