Directivity of array of antennas

[radiodude]

Homework Statement

I am trying to understand how the total directivity of an array of antennas is calculated.

Homework Equations

1. A_e = (D * lambda^2)/(4 * pi)
where:
A_e = effective aperature (or area of antenna)

D = Directivity of one antenna
lambda = wavelength

2. D_t = (4 * pi * A_t) / (lambda^2)
where:
D_t = directivity of array
A_t = effective aperature (of array)
lambda = wavelength

The Attempt at a Solution

What I am confused on is how aperature comes into play because if I have X antennas spaced Y wavelengths apart for the array, total aperature (A_t in eqn #2) is what will change. I just don't know how to get A_t from A_e if I know what the antenna spacing is.


As an example, let's say I know the directivity of one antenna is 10 (which 10 dBi). But if I have 3 of them and let's say they're spaced 0.25 wavelengths apart, what is the directivity of the array?

From eqn #1, A_e would be (10 * (.25L)^2) / (4 * 3.14) = 0.0497L^2 = 0.05L^2
(I'm using L for lambda)

So what is the aperature of the array if I know the aperature of one antenna is 0.05L^2 ? Once I have that I would be able to use eqn #2 to figure out the array's total directivity. But the relationship between A_e and A_t is unknown to me.

Thanks.

 

[radiodude]

If my post was too long, I guess what I'm asking is this:

If I know the aperture of a single antenna, how do you figure out the total aperture if you had multiple antennas?

I'm tempted to just multiply the aperture of one antenna by the number of antennas but something tells me there's more to it than that (e.g., don't you have to account for the arrangement of the antennas somehow?).