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?).

 

[Mene]

To calculate the total directivity of an array of antennas, you need to take into account the individual directivity of each antenna as well as the spacing between them. The effective aperture (A_e) of one antenna is a measure of its ability to receive or transmit signals. When multiple antennas are placed together in an array, their effective apertures combine to form the effective aperture of the array (A_t). This means that the total effective aperture of the array is larger than that of a single antenna, resulting in a higher directivity.

The relationship between A_e and A_t is not a simple one-to-one ratio. It depends on the type and configuration of the array. In your example, the spacing between the antennas is 0.25 wavelengths, which means that the array is a half-wavelength dipole array. In this case, the effective aperture of the array (A_t) is equal to the sum of the effective apertures of each individual antenna (A_e). So, for an array of three 10 dBi antennas with a spacing of 0.25 wavelengths, the total effective aperture would be 3 * 0.05L^2 = 0.15L^2.

Using eqn #2, the total directivity of the array would then be D_t = (4 * pi * 0.15L^2) / (L^2) = 12 dBi. This means that the array has a higher directivity compared to a single antenna, which makes it more directional and able to transmit or receive signals in a specific direction with greater strength.

In summary, the total directivity of an array of antennas is calculated by combining the individual directivities of each antenna with the effective aperture of the array, which is a function of the spacing between the antennas. The specific relationship between A_e and A_t depends on the type of array, and for a half-wavelength dipole array, A_t is equal to the sum of A_e.

 

[piercas]

Dear student,

Thank you for your question. The directivity of an array of antennas can be calculated by using the formula D_t = (4 * pi * A_t) / (lambda^2). In this formula, A_t represents the effective aperture of the array, which is a measure of the array's ability to collect or radiate electromagnetic energy. The effective aperture takes into account the number of antennas in the array and their spacing.

To calculate the effective aperture of an array, you can use the formula A_t = N * A_e, where N is the number of antennas in the array and A_e is the effective aperture of one antenna. In your example, N = 3 and A_e = 0.05L^2, so A_t = 3 * 0.05L^2 = 0.15L^2.

Using this value for A_t in the formula for directivity, we get D_t = (4 * pi * 0.15L^2) / (lambda^2) = 1.9. This means that the directivity of the array is 1.9 times greater than the directivity of one antenna.

I hope this helps clarify the relationship between the effective aperture and directivity of an array of antennas. Please let me know if you have any further questions.