# Distance requirements in antenna arrays?

Antenna array is a broad term and it encompasses at least the following concepts:

1. MIMO or massive MIMO: for spatial multiplexing
2. beamforming: increasing the SNR (at transmitter or receiver) by aligning the signals properly
3. phased array: special case of beamforming for narrowband signals in which the phase can be adjusted to align the signals. Mainly used in transmitters to stear a beam into a certain direction without mechanical movements. Used in RADAR.

Please correct this list if it is wrong or if I have forgotton something. I could also think of cases like increasing bandwidth by having many narrowband antennas in parallel - not sure if this is done in practice?

It is well known that the size of an antenna has to be on the order of lambda/2 but conventionally larger than ~lambda/10.

Suppose for a moment I would have a hypothetical isotropic radiator (an infinitely small antenna able to radiate at any desired frequency).

Are there upper (or lower limits) on the distance between the individual antenna elements in antenna arrays (for the cases listed above)?

Stated differently, can antenna arrays be decreased in size if the antenna elements themselves can be decreased in size?

• Why did you build an array? For example, did you want better directionality than you'd get from a single antenna? Sep 20, 2019 at 1:32
• I did not build any array, this is just a question for understanding.
– divB
Sep 20, 2019 at 2:31

The antenna array spacing and thus size is usually based on the wavelength of the signals of interest, not the radiating element length.

It's a geometry problem in the far field, with the ratio between the antenna separation and the wavelength being the interesting parameter.

Constructive and destructive interference contributes to the pattern and gain that can be accomplished with the array. If the elements are much closer together than lambda/10, the geometric patterns of wavefronts for each antenna are too similar to create useful interference patterns, and the array acts more like a single antenna at those separations and frequencies. As the antennas are separated into each others far-field, the number of lobes in the pattern increases, thus managing and aiming the pattern becomes far more complicated.

• Ok, thank you that was my assumption. Do you have a pointer where this is described in more detail (maybe with a radiation pattern or so)? You say when they're closer than ~lambda/10 they are too similar. Why 10; why doesn't happen this already at ~lambda/2? Finally, would you be able to comment on if this applies only to MIMO, massive MIMO, beamforming or phase arrays or if it applied to all of them?
– divB
Sep 20, 2019 at 2:36
• Same geometry applies to all of the above, just more of the same. lambda/2 is a great spacing for a deep null sideways (vs. broadside) given an in-phase feed. Sep 20, 2019 at 2:43

can antenna arrays be decreased in size if the antenna elements themselves can be decreased in size?

No, they cannot. The antenna arrays work by constructive and destructive interference where, in one direction the radiation from antennas of the array is in the same phase, leading to a maximum in radiation, and in one direction the phases are opposite, leading to a null in the radiation. The interference pattern is a function of the phase difference of these different radiated fields and if the antennas would be in the same locations, there would be no interference.

The analytical way of analyzing an antenna array is called array factor. For N identical antennas, the array factor is:

$$AF = \Sigma_{i=1}^N w_i e^{-j k r_i}$$

where $$\w_i\$$ is the weight of the specific antenna, $$\ r_i\$$ is the vector defining the position of the antenna, and $$\k\$$ is the wave vector.

If you just want to get a good gut feeling on how the locations and phase differences affect, have a look at the Python code I used to answer another SE question: https://nbviewer.jupyter.org/gist/Diimu/71010f1ffee95e6fe0082ad1dffda5c3/analytical_dipoles.ipynb . The tool uses ideal Hertz's dipoles and neglects their mutual coupling, but can be used to model surprisingly complicated things.

If you don't have Python installed, you can try it out in e.g., https://jupyter.org/try . The python is a quite easy language to play with if you have any programming experience. Below you can see the directivity pattern of an array of two horizontal electric dipole antennas that are fed with the same phase and amplitude. The distance is varied from 0 to 1 wavelength in a loop. As you can see, the case where the distance is $$\0.1\lambda\$$ the radiation pattern is virtually identical with $$\0\lambda\$$. At $$\0.5\lambda\$$ you see a high directivity as the two dipoles cancel each other out in the direction of the displacement and amplify each other $$\90^\circ\$$ away from that.

For further reading see below, or pretty much any antenna theory book:

The minimum array size is a function of the antenna radiation pattern, the signal wavelength and the performance goals for the array. For very close spacing, the antennas may even interact with each other. In some cases, smaller antennas with close spacing can degrade array performance by introducing coupling between the antennas or unwanted sidelobe and backlobes from the smaller antenna radiation patterns.

To visualize how the spacing affects the radiation from the array, consider the case of a an isotropic antenna, one with the same radiation in all directions. In this case, the array factor equation describes the radiation pattern from the array.

$$AF = \Sigma_{i=1}^N w_i e^{-j k r_i}$$

However, this equation can be hard to visualize. It provides the far field view of the signal. One way to see what is happening in an array is to visualize the waves emitted from each antenna and how they constructively and destructively interact. For example, a single isotropic antenna will emits wave in all directions. In a single plane, the wave will look like ripples on a pond.

When several antennas are combined, and spaced out, the waves will constructively interfere and destructively interfere. This is usually setup to create a beam. The image below shows 5 antennas (as white dots) forming a beam.

TO see how spacing affects the radiation patterns from an array of 5 isotropic antennas, the animation below shows how the radiation pattern changes with antenna spacing.

There are some notes on how to recreate the visualizations here - http://exnumerus.blogspot.com/2021/02/visualizing-3d-sine-wave-using-blender.html