Editing Phased array (section)

=== 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.