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it's well known that in a linear and equispaced antenna array, the array factor is equal to:enter image description here

enter image description here

The array is then called uniform if $|I_n| = I$ is the same for all elements. The phase terms $\angle I_n$ is chosen to decide the peak direction of the beam.

My question is: which current is $I_n$? I'm supposing that each cell is connected to a (physical or equivalent) current source. So, is $I_n$ the incident current (let's call it $a$) on the elements ports or the sum of incident and reflected current ($a-b$, where $b$ is the reflected current)?

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$I_n$ is the current on the element, the current which gives rise the to the fields. It has a magnitude and phase. From that you can derive the array patterns.

This is a big simplification from a real-world array, but is useful to allow you to derive some array details in a clean environment. For example:

  • it's a two dimensional design
  • it doesn't consider the element patterns (they produce the same fields in all directions)
  • element length is 1 (they're just unit length elements of current out of the page)
  • there must be some feed that produces the current, but that's not what's being investigated
  • if all $I_n$ are the same, then there's no coupling (or it's taken care of by the feed somehow)
  • The elements are short enough that they can be approximated by a simple short element of current. No current distribution or self-impedance etc.
  • The only things you need to know to describe an element and calculate its contribution are:
    • Current magnitude, $|I_n|$
    • Current phase, $\angle I_n$
    • X-position in terms of wavelength, $knd$
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