Antenna Array Radiation Pattern and Directivity - Maple Help

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Radiation Pattern and Directivity of an Antenna Array

Introduction

The application calculates the array factor and directivity for a uniform linear antenna array, and then plots the radiation pattern.

 > $\mathrm{restart}:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Standard}\right]\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{plots}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{ScientificConstants}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{ColorTools}\right):$

Parameters

The number of elements in the uniform array:

 > $\mathrm{N}≔10:$

Design frequency:

 > $\mathrm{f__d}≔1⟦\mathrm{GHz}⟧:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$

Permittivity and permeability of free space:

 > $\mathrm{ε__o}≔\mathrm{evalf}\left(\mathrm{Constant}\left(\mathrm{permittivity_of_vacuum},\mathrm{units}\right)\right);$
 ${8.854187815}{}{{10}}^{{-12}}{}⟦\frac{{F}}{{m}}⟧$ (2.1)
 > $\mathrm{μ__o}≔\mathrm{evalf}\left(\mathrm{Constant}\left(\mathrm{permeability_of_vacuum},\mathrm{units}\right)\right)$
 ${1.256637062}{}{{10}}^{{-6}}{}⟦\frac{{H}}{{m}}⟧$ (2.2)

Phase constant:

 > $\mathrm{β__o}≔2\cdot \mathrm{π}\cdot \mathrm{f__d}\cdot \sqrt{\mathrm{μ__o}\cdot \mathrm{ε__o}}$
 $\mathrm{β__o}{≔}{20.95845022}{}⟦\frac{{1}}{{m}}⟧$ (2.3)

Wavelength:

 > $\mathrm{λ__d}≔2\cdot \frac{\mathrm{\pi }}{\mathrm{β__o}}$
 $\mathrm{λ__d}{≔}{0.2997924580}{}⟦{m}⟧$ (2.4)

For a maximum at:

 > $\mathrm{φ__m}≔\frac{\mathrm{\pi }}{3.}:$

Inter-element spacing:

 >
 ${d}{≔}{0.09993081933}{}⟦{m}⟧$ (2.5)

Progressive phase shift between elements:

 > $\mathrm{ψ}≔\mathrm{β__o}\cdot \mathrm{d}\cdot \mathrm{cos}\left(\mathrm{φ__m}\right)$
 ${\mathrm{\psi }}{≔}{1.047197551}$ (2.6)

Calculations

Array factor

 > $\mathrm{AF}≔\mathrm{φ}\to \left|\frac{1}{\mathrm{N}}\cdot \frac{\mathrm{sin}\left(\frac{\mathrm{N}}{2}\cdot \left(\mathrm{β__o}\cdot \mathrm{d}\cdot \mathrm{cos}\left(\mathrm{φ}\right)-\mathrm{ψ}\right)\right)}{\mathrm{sin}\left(\frac{1}{2}\cdot \left(\mathrm{β__o}\cdot \mathrm{d}\cdot \mathrm{cos}\left(\mathrm{φ}\right)-\mathrm{ψ}\right)\right)}\right|:$

 > $\mathrm{polarplot}\left(\mathrm{AF}\left(\mathrm{φ}\right),\mathrm{φ}=0..2\cdot \mathrm{π},\mathrm{filled},\mathrm{size}=\left[700,700\right],\mathrm{color}=\mathrm{Color}\left("RGB",\left[0/255,79/244,121/255\right]\right),\mathrm{thickness}=0,\mathrm{axesfont}=\left[\mathrm{Calibri}\right],\mathrm{adaptive}=\mathrm{false},\mathrm{numpoints}=500,\mathrm{axis}=\left[\mathrm{gridlines}=\left[\mathrm{color}=\mathrm{grey}\right]\right]\right)$

The directivity for this array is calculated from the total power radiated.

 >
 $\mathrm{P__tot}{≔}{1.428780370}$ (3.1)
 > $\mathrm{D__o}≔\frac{4\cdot \mathrm{\pi }}{\mathrm{P__tot}}$
 $\mathrm{D__o}{≔}{8.795173059}$ (3.2)

Which, in dB, is:

 > $10\cdot \mathrm{log10}\left(\mathrm{D__o}\right)$
 ${9.442443893}$ (3.3)
 > 
 Reference Iskander, Magdi F., Electromagnetic Fields and Waves, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1992.