Editing Phased array (section)

==Formulation==
[[File:Array frame.svg|thumb|398x398px|Coordinate frame of phased array used in calculation of array factor, directivity, and gain.]]

=== Array factor ===
The total [[directivity]] of a phased array will be a result of the gain of the individual array elements, and the directivity due their positioning in an array. This latter component is closely tied (but not equal to<ref name=":3">{{Cite web |title=Antenna Arrays: A Computational Approach |url=https://ieeexplore.ieee.org/book/5599319 |access-date=2023-05-20 |website=[[IEEE]]}}</ref>) to the [[array factor]].<ref name=":4">{{cite book
 | last  = Balanis
 | first = Constantine A. 
 | title  = Antenna Theory: Analysis and Design, 4th Ed.
 | publisher = John Wiley & Sons
 | date   = 2015
 | url    = https://books.google.com/books?id=PTFcCwAAQBAJ&q=%22phased%20array%22&pg=PA303
 | isbn   = 978-1119178989
 }}</ref>{{page needed|date=July 2023}}<ref name=":3" /> In a (rectangular) planar phased array, of dimensions <math>M\times N</math>, with inter-element spacing <math>d_{x}</math> and <math>d_{y}</math>, respectively, the array factor can be calculated accordingly<ref name="Balanis" /><ref name=":4" />{{page needed|date=July 2023}}:[[File:Phase array sweep.webm|thumb|Radiation pattern of phased array containing 7 emitters spaced a quarter wavelength apart, showing the beam switching direction.  The phase shift between adjacent emitters is switched from 45 degrees to −45 degrees|398x398px]]<math display="block">AF=\sum_{n=1}^{N}I_{n1}\left[\sum_{m=1}^{M}I_{m1}\mathrm{e}^{j\left(m-1\right)\left(kd_{x}\sin\theta\cos\phi+\beta_{x}\right)}\right]\mathrm{e}^{j\left(n-1\right)\left(kd_{y}\sin\theta\sin\phi+\beta_{y}\right)}</math>

Here, <math>\theta</math> and <math>\phi</math> are the directions which we are taking the array factor in, in the coordinate frame depicted to the right. The factors <math>\beta_{x}</math> and <math>\beta_{y}</math> are the ''progressive phase shift'' that is used to steer the beam electronically. The factors <math>I_{n1}</math> and <math>I_{m1}</math> are the excitation coefficients of the individual elements.

Beam steering is indicated in the same coordinate frame, however the direction of steering is indicated with <math>\theta_{0}</math> and <math>\phi_{0}</math>, which is used in calculation of progressive phase:

:<math>\beta_{x}=-kd_{x}\sin\theta_{0}\cos\phi_{0}</math>

:<math>\beta_{y}=-kd_{y}\sin\theta_{0}\sin\phi_{0}</math>

In all above equations, the value <math>k</math> describes the [[wavenumber]] of the frequency used in transmission.

These equations can be solved to predict the nulls, main lobe, and grating lobes of the array. Referring to the exponents in the array factor equation, we can say that major and grating lobes will occur at integer <math>m,n=0,1,2,\dots</math>  solutions to the following equations:<ref name="Balanis" /><ref name=":4" />{{page needed|date=July 2023}}

:<math>kd_{x}\sin\theta\cos\phi+\beta_{x}=\pm2m\pi</math>

:<math>kd_{y}\sin\theta\sin\phi+\beta_{y}=\pm2n\pi</math>

=== Worked example ===
It is common in engineering to provide phased array <math>AF</math> values in [[Decibel|decibels]] through <math>AF_{dB}=10\log_{10}AF</math>. Recalling the complex exponential in the array factor equation above, often, what is ''really'' meant by array factor is the magnitude of the summed [[phasor]] produced at the end of array factor calculation. With this, we can produce the following equation:<math display="block">AF_{dB}=10\log_{10}\left\|\sum_{n=1}^{N}I_{1n}\left[\sum_{m=1}^{M}I_{m1}\mathrm{e}^{j\left(m-1\right)\left(kd_{x}\sin\theta\cos\phi+\beta_{x}\right)}\right]\mathrm{e}^{j\left(n-1\right)\left(kd_{y}\sin\theta\sin\phi+\beta_{y}\right)}\right\|</math>For the ease of visualization, we will analyze array factor given an input ''azimuth and elevation'', which we will map to the array frame <math>\theta</math> and <math>\phi</math> through the following conversion:

:<math>\theta=\arccos\left(\cos\left(\theta_{az}\right)\sin\left(\theta_{el}\right)\right)</math>

:<math>\phi=\arctan2\left(\sin\left(\theta_{el}\right),\sin\left(\theta_{az}\cos\left(\theta_{el}\right)\right)\right)</math>

This represents a coordinate frame whose <math>\mathbf{x}</math> axis is aligned with the array <math>\mathbf{z}</math> axis, and whose <math>\mathbf{y}</math> axis is aligned with the array <math>\mathbf{x}</math> axis.

If we consider a <math>16\times16</math> phased array, this process provides the following values for <math>AF_{dB}</math>, when steering to bore-sight (<math>\theta_{0}=0^{\circ}</math>,<math>\phi_{0}=0^{\circ}</math>):
{|
|[[File:16x16 0.250 lambda spacing planar array factor.png|frameless|400x400px]]
|[[File:16x16 0p500 lambda spacing planar array factor.png|frameless|400x400px]]
|-
|[[File:16x16 1p0 lambda spacing planar array factor.png|frameless|400x400px]]
|[[File:16x16 2p0 lambda spacing planar array factor.png|frameless|400x400px]]
|}
These values have been clipped to have a minimum <math>AF</math> of -50 dB, however, in reality, null points in the array factor pattern will have values significantly smaller than this.