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<title>Array factor</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Array factor</span></span>
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<p>An array is simply a group of objects, and the array factor is a measure of how much a specific characteristic changes because of the grouping. This phenomenon is observed when antennas are grouped together. The radiation (or reception) pattern of the antenna group is considerably different from that of a single antenna. This is due to the constructive and destructive interference properties of radio waves. A well designed antenna array, allows the broadcast power to be directed to where it is needed most.
</p><p>These antenna arrays are typically one dimensional, as seen on collinear dipole arrays, or two dimensional as on military phased arrays.
</p><p>In order to simplify the mathematics, a number of assumptions are typically made:
</p>
<pre> 1. all radiators are equal in every respect
2. all radiators are uniformly spaced
3. the signal phase shift between radiators is constant.
</pre>
<p>The array factor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AF}">
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<annotation encoding="application/x-tex">{\displaystyle AF}</annotation>
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</math></span><img src="./ec38c50a848adbe1b80128b1be057a81cc712ce8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.484ex; height:2.176ex;" alt="{\displaystyle AF}" loading="lazy"></span> is the complex-valued <a href="Far-field" class="mw-redirect" title="Far-field">far-field</a> <a href="Radiation_pattern" title="Radiation pattern">radiation pattern</a> obtained for an array of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> isotropic radiators located at <a href="Coordinates" class="mw-redirect" title="Coordinates">coordinates</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{n}}">
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<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{n}}</annotation>
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</math></span><img src="./5b8b1f42b7c38dec5c4349f9eed65acf57b9c46a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.442ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{n}}" loading="lazy"></span>, as determined by:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AF({\hat {r}})=\sum _{n=1}^{N}a_{n}e^{jk{\hat {r}}\cdot {\vec {r}}_{n}},}">
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<annotation encoding="application/x-tex">{\displaystyle AF({\hat {r}})=\sum _{n=1}^{N}a_{n}e^{jk{\hat {r}}\cdot {\vec {r}}_{n}},}</annotation>
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</math></span><img src="./52cb89ca14d4ea827d0b66cf463f42e4e8cda0de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.646ex; height:7.343ex;" alt="{\displaystyle AF({\hat {r}})=\sum _{n=1}^{N}a_{n}e^{jk{\hat {r}}\cdot {\vec {r}}_{n}},}" loading="lazy"></span>
</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}">
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<annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation>
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</math></span><img src="./790f9209748c2dca7ed7b81932c37c02af1dbc31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" loading="lazy"></span> are the complex-valued excitation coefficients, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {r}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\hat {r}}}</annotation>
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</math></span><img src="./1009619964ce33a4a02aaa7cf82adc0fb0a50f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {r}}}" loading="lazy"></span> is the direction <a href="Unit_vector" title="Unit vector">unit vector</a>. The array factor is defined in the transmitting mode,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> with the time convention <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{j\omega t}}">
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<annotation encoding="application/x-tex">{\displaystyle e^{j\omega t}}</annotation>
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</math></span><img src="./84f8673593e85aacbb2741a859a596219082cb46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.609ex; height:2.676ex;" alt="{\displaystyle e^{j\omega t}}" loading="lazy"></span>. A corresponding expression can be derived for the receiving mode, where a negative sign appears in the exponential factors, as derived in reference.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFBalanis" class="citation book cs1">Balanis, C. A. <i>Antenna Theory, Analysis and Design</i> (3&nbsp;ed.). p.&nbsp;291.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation journal cs1">"IEEE Standard for definitions of terms for antennas". <i>IEEE STD</i>. 2014.</cite></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFFrid2020" class="citation book cs1">Frid, Henrik (2020). <a rel="nofollow" class="external text" href="http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A1392934&amp;dswid=5174"><i>Analysis and Optimization of Installed Antenna Performance</i></a>. Stockholm, Sweden: KTH (PhD thesis). pp.&nbsp;<span class="nowrap">36–</span>39. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-91-7873-447-4</bdi>.</cite></span>
</li>
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Array_antenna" class="mw-redirect" title="Array antenna">Array antenna</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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