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<title>Radiation pattern</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Radiation pattern</span></span>

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<p>In the field of <a href="Antenna_(radio)" title="Antenna (radio)">antenna</a> design the term <b>radiation pattern</b> (or <b>antenna pattern</b> or <b>far-field pattern</b>) refers to the <i>directional</i> (angular) dependence of the strength of the <a href="Radio_waves" class="mw-redirect" title="Radio waves">radio waves</a> from the antenna or other source.<sup id="cite_ref-Balanis_1-0" class="reference"><a href="#cite_note-Balanis-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Cheng_2-0" class="reference"><a href="#cite_note-Cheng-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-JordanBalmain1968_3-0" class="reference"><a href="#cite_note-JordanBalmain1968-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Particularly in the fields of <a href="Fiber_optics" class="mw-redirect" title="Fiber optics">fiber optics</a>, <a href="Laser" title="Laser">lasers</a>, and <a href="Integrated_optics" class="mw-redirect" title="Integrated optics">integrated optics</a>, the term radiation pattern may also be used as a synonym for the <b><a href="Near_and_far_field" title="Near and far field">near-field</a> pattern</b> or <b>Fresnel pattern</b>.<sup id="cite_ref-IEEEdict1997_4-0" class="reference"><a href="#cite_note-IEEEdict1997-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This refers to the <i>positional</i> dependence of the <a href="Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a> in the <a href="Near_and_far_field" title="Near and far field">near field</a>, or Fresnel region of the source. The near-field pattern is most commonly defined over a plane placed in front of the source, or over a cylindrical or spherical surface enclosing it.<sup id="cite_ref-Balanis_1-1" class="reference"><a href="#cite_note-Balanis-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-IEEEdict1997_4-1" class="reference"><a href="#cite_note-IEEEdict1997-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>The far-field pattern of an antenna may be determined experimentally at an <a href="Antenna_measurement" title="Antenna measurement">antenna range</a>, or alternatively, the near-field pattern may be found using a <b><a href="Electromagnetic_near-field_scanner" class="mw-redirect" title="Electromagnetic near-field scanner">near-field scanner</a></b>, and the radiation pattern deduced from it by computation.<sup id="cite_ref-Balanis_1-2" class="reference"><a href="#cite_note-Balanis-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The far-field radiation pattern can also be calculated from the antenna shape by computer programs such as <a href="Numerical_Electromagnetics_Code" title="Numerical Electromagnetics Code">NEC</a>. Other software, like <a href="Ansys_HFSS" title="Ansys HFSS">HFSS</a> can also compute the near field.

</p><p>The far field radiation pattern may be represented graphically as a plot of one of a number of related variables, like the <a href="Field_strength" title="Field strength">field strength</a> at a constant (large) radius (an <b>amplitude pattern</b> or <b>field pattern</b>), the power per unit solid angle (<b>power pattern</b>) and the <a href="Antenna_gain" class="mw-redirect" title="Antenna gain">directive gain</a>. Very often, only the relative amplitude is plotted, normalized either to the amplitude on the antenna <a href="Antenna_boresight" title="Antenna boresight">boresight</a>, or to the total radiated power. The plotted quantity may be shown on a linear scale, or in <a href="Decibel" title="Decibel">dB</a>. The plot is typically represented as a three-dimensional graph (as at right), or as separate graphs in the <a href="Vertical_and_horizontal" title="Vertical and horizontal">vertical plane</a> and <a href="Vertical_and_horizontal" title="Vertical and horizontal">horizontal plane</a>. This is often known as a <b>polar diagram</b>.

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<div class="mw-heading mw-heading2"><h2 id="Reciprocity">Reciprocity</h2></div>

<p>It is a fundamental property of antennas that the <b>receiving pattern</b> (sensitivity as a function of direction) of an antenna when used for <a href="Radio_receiver" title="Radio receiver">receiving</a> is identical to the far-field radiation pattern of the antenna when used for <a href="Transmitter" title="Transmitter">transmitting</a>. This is a consequence of the <a href="Reciprocity_(electromagnetism)" title="Reciprocity (electromagnetism)">reciprocity theorem</a> of electromagnetics and is proved below. Therefore, in discussions of radiation patterns the antenna can be viewed as either transmitting or receiving, whichever is more convenient.

</p><p>There are limits to reciprocity: It applies only to <i><a href="Passive_component" class="mw-redirect" title="Passive component">passive</a></i> antenna elements – <i>active</i> antennas that incorporate amplifiers or other individually powered components are <i>not</i> reciprocal. And even when the antenna is made of exclusively of passive elements, reciprocity only applies to the waves emitted and intercepted by the antenna. Reciprocity does <i>not</i> apply to the distribution of current in the various parts of the antenna generated by the intercepted waves nor currents that create emitted waves: Antenna current profiles typically differ for receiving and transmitting, despite the waves in the <a href="Near_and_far_field" title="Near and far field">far field</a> radiating inward and outward along the same path, with the same overall pattern, just with reversed direction.

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<div class="mw-heading mw-heading2"><h2 id="Typical_patterns">Typical patterns</h2></div>

<p>Since <a href="Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> is <a href="Dipole_radiation" class="mw-redirect" title="Dipole radiation">dipole radiation</a>, it is not possible to build an antenna that radiates coherently equally in all directions, although such a hypothetical <a href="Isotropic_antenna" class="mw-redirect" title="Isotropic antenna">isotropic antenna</a> is used as a reference to calculate <a href="Antenna_gain" class="mw-redirect" title="Antenna gain">antenna gain</a>.

</p><p>The simplest antennas, <a href="Monopole_antenna" title="Monopole antenna">monopole</a> and <a href="Dipole_antenna" title="Dipole antenna">dipole antennas</a>, consist of one or two straight metal rods along a common axis. These <a href="Axial_symmetry" title="Axial symmetry">axially symmetric</a> antennas have radiation patterns with a similar symmetry, called <a href="Omnidirectional_antenna" title="Omnidirectional antenna">omnidirectional</a> patterns; they radiate equal power in all directions perpendicular to the antenna, with the power varying only with the angle to the axis, dropping off to zero on the antenna's axis. This illustrates the general principle that if the shape of an antenna is symmetrical, its radiation pattern will have the same symmetry.

</p><p>In most antennas, the radiation from the different parts of the antenna <a href="Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interferes</a> at some angles; the radiation pattern of the antenna can be considered an <a href="Interference_pattern" class="mw-redirect" title="Interference pattern">interference pattern</a>. This results in minimum or zero radiation at certain angles where the radio waves from the different parts arrive <a href="Out_of_phase" class="mw-redirect" title="Out of phase">out of phase</a>, and <a href="Local_maximum" class="mw-redirect" title="Local maximum">local maxima</a> of radiation at other angles where the radio waves arrive <a href="In_phase" class="mw-redirect" title="In phase">in phase</a>. Therefore, the radiation plot of most antennas shows a pattern of maxima called "<i>lobes</i>" at various angles, separated by "<i><a href="Null_(radio)" title="Null (radio)">nulls</a></i>" at which the radiation goes to zero. The larger the antenna is compared to a wavelength, the more lobes there will be.

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<p>In a <a href="Directional_antenna" title="Directional antenna">directional antenna</a> in which the objective is to emit the radio waves in one particular direction, the antenna is designed to radiate most of its power in the lobe directed in the desired direction. Therefore, in the radiation plot this lobe appears larger than the others; it is called the "<i><a href="Main_lobe" title="Main lobe">main lobe</a></i>". The axis of maximum radiation, passing through the center of the main lobe, is called the "<i>beam axis</i>" or <i><a href="Antenna_boresight" title="Antenna boresight">boresight axis</a></i>". In some antennas, such as split-beam antennas, there may exist more than one major lobe. The other lobes beside the main lobe, representing unwanted radiation in other directions, are called minor lobes. The minor lobes oriented at an angle to the main lobe are called "<i><a href="Side_lobe" class="mw-redirect" title="Side lobe">side lobes</a></i>". The minor lobe in the opposite direction (180°) from the main lobe is called the "<i>back lobe</i>".

</p><p>Minor lobes usually represent radiation in undesired directions, so in directional antennas a design goal is usually to reduce the minor lobes. Side lobes are normally the largest of the minor lobes. The level of minor lobes is usually expressed as a ratio of the power density in the lobe in question to that of the major lobe. This ratio is often termed the side lobe ratio or side lobe level. Side lobe levels of −20 dB or greater are usually not desirable in many applications. Attainment of a side lobe level smaller than −30 dB usually requires very careful design and construction. In most radar systems, for example, low side lobe ratios are very important to minimize false target indications through the side lobes.

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<div class="mw-heading mw-heading2"><h2 id="Proof_of_reciprocity">Proof of reciprocity</h2></div>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Antenna_(radio)" title="Antenna (radio)">Antennas</a></th></tr><tr><td class="sidebar-image" style="margin-bottom:0.5em;"><span typeof="mw:File"></span></td></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="Antenna_types" title="Antenna types">Common types</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid#aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Dipole_antenna" title="Dipole antenna">Dipole</a></li>

<li><a href="Fractal_antenna" title="Fractal antenna">Fractal</a></li>

<li><a href="Monopole_antenna" title="Monopole antenna">Monopole</a></li>

<li><a href="Satellite_dish" title="Satellite dish">Satellite dish</a></li>

<li><a href="Television_antenna" title="Television antenna">Television</a></li>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Components</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid#aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Balun" title="Balun">Balun</a></li>

<li><a href="Block_upconverter" title="Block upconverter">Block upconverter</a></li>

<li><a href="Coaxial_cable" title="Coaxial cable">Coaxial cable</a></li>

<li><a href="Counterpoise_(ground_system)" title="Counterpoise (ground system)">Counterpoise (ground system)</a></li>

<li><a href="Low-noise_block_downconverter" title="Low-noise block downconverter">Low-noise block downconverter</a></li>

<li><a href="Passive_radiator" class="mw-redirect" title="Passive radiator">Passive radiator</a></li>

<li><a href="Receiver_(radio)" class="mw-redirect" title="Receiver (radio)">Receiver</a></li>

<li><a href="Antenna_rotator" title="Antenna rotator">Rotator</a></li>

<li><a href="Transmitter" title="Transmitter">Transmitter</a></li>

<li><a href="Antenna_tuner" title="Antenna tuner">Tuner</a></li>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Systems</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid#aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Antenna_farm" title="Antenna farm">Antenna farm</a></li>

<li><a href="Amateur_radio" title="Amateur radio">Amateur radio</a></li>

<li><a href="Cellular_network" title="Cellular network">Cellular network</a></li>

<li><a href="Hotspot_(Wi-Fi)" class="mw-redirect" title="Hotspot (Wi-Fi)">Hotspot</a></li>

<li><a href="Municipal_wireless_network" title="Municipal wireless network">Municipal wireless network</a></li>

<li><a href="Radio" title="Radio">Radio</a></li>

<li><a href="Radio_masts_and_towers" title="Radio masts and towers">Radio masts and towers</a></li>

<li><a href="Wireless" title="Wireless">Wireless</a></li></ul>

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</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Safety and regulation</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid#aaa;border-bottom:1px solid #aaa;">

<ul><li><a href="Wireless_device_radiation_and_health" title="Wireless device radiation and health">Wireless device radiation and health</a></li>

<li><a href="Wireless_electronic_devices_and_health" class="mw-redirect" title="Wireless electronic devices and health">Wireless electronic devices and health</a></li>

<li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="International_Telecommunication_Union" title="International Telecommunication Union">International Telecommunication Union</a><br>(<a href="ITU_Radio_Regulations" title="ITU Radio Regulations">Radio Regulations</a>)</div></li>

<li><a href="World_Radiocommunication_Conference" title="World Radiocommunication Conference">World Radiocommunication Conference</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span class="nowrap">Radiation sources / regions</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid#aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Antenna_boresight" title="Antenna boresight">Boresight</a></li>

<li><a href="Focal_cloud" title="Focal cloud">Focal cloud</a></li>

<li><a href="Ground_plane" title="Ground plane">Ground plane</a></li>

<li><a href="Near_and_far_field" title="Near and far field">Near and far field</a></li>

<li><a href="Vertical_plane" class="mw-redirect" title="Vertical plane">Vertical plane</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Characteristics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid#aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Array_gain" title="Array gain">Array gain</a></li>

<li><a href="Directivity" title="Directivity">Directivity</a></li>

<li><a href="Antenna_efficiency" class="mw-redirect" title="Antenna efficiency">Efficiency</a></li>

<li><a href="Electrical_length" title="Electrical length">Electrical length</a></li>

<li><a href="Antenna_equivalent_radius" title="Antenna equivalent radius">Equivalent radius</a></li>

<li><a href="Antenna_factor" title="Antenna factor">Factor</a></li>

<li><a href="Friis_transmission_equation" title="Friis transmission equation">Friis transmission equation</a></li>

<li><a href="Antenna_height_considerations" class="mw-redirect" title="Antenna height considerations">Height</a></li>

<li><a href="Radiation_resistance" title="Radiation resistance">Radiation resistance</a></li>

<li><a href="Radio_propagation" title="Radio propagation">Radio propagation</a></li>

<li><a href="Radio_spectrum" title="Radio spectrum">Radio spectrum</a></li>

<li><a href="Signal-to-noise_ratio" title="Signal-to-noise ratio">Signal-to-noise ratio</a></li>

<li><a href="Spurious_emission" title="Spurious emission">Spurious emission</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Techniques</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid#aaa;border-bottom:1px solid #aaa;"><div class="hlist">

<ul><li><a href="Beam_steering" title="Beam steering">Beam steering</a></li>

<li><a href="Beamforming" title="Beamforming">Beamforming</a></li>

<li><a href="Small_cell" title="Small cell">Small cell</a></li></ul>

</div>

<ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="Bell_Laboratories_Layered_Space-Time" title="Bell Laboratories Layered Space-Time">Bell Laboratories Layered<br>Space-Time (BLAST)</a></div></li>

<li>Massive <a href="MIMO" title="MIMO">Multiple-input multiple-output (MIMO)</a></li></ul>

<ul><li><a href="Reconfigurable_antenna" title="Reconfigurable antenna">Reconfiguration</a></li>

<li><a href="Spread_spectrum" title="Spread spectrum">Spread spectrum</a></li></ul>

</div>

<ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="WSDMA" title="WSDMA">Wideband Space Division<br>Multiple Access (WSDMA)</a></div></li></ul></div></div></td>

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<p>For a complete proof, see the <a href="Reciprocity_(electromagnetism)" title="Reciprocity (electromagnetism)">reciprocity (electromagnetism)</a> article. Here, we present a common simple proof limited to the approximation of two antennas separated by a large distance compared to the size of the antenna, in a homogeneous medium. The first antenna is the test antenna whose patterns are to be investigated; this antenna is free to point in any direction. The second antenna is a reference antenna, which points rigidly at the first antenna.

</p><p>Each antenna is alternately connected to a transmitter having a particular source impedance, and a receiver having the same input impedance (the impedance may differ between the two antennas).

</p><p>It is assumed that the two antennas are sufficiently far apart that the properties of the transmitting antenna are not affected by the load placed upon it by the receiving antenna. Consequently, the amount of power transferred from the transmitter to the receiver can be expressed as the product of two independent factors; one depending on the directional properties of the transmitting antenna, and the other depending on the directional properties of the receiving antenna.

</p><p>For the transmitting antenna, by the definition of gain, <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 G}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>, the radiation power density at a distance <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 r}">

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<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> from the antenna (i.e. the power passing through unit area) is

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {W} (\theta ,\Phi )={\frac {\mathrm {G} (\theta ,\Phi )}{4\pi r^{2}}}P_{t}}</annotation>

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</math></span><img src="./e8c97fdbcc15a66bd2ff6d437ba3dd213427bed3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.69ex; height:6.009ex;" alt="{\displaystyle \mathrm {W} (\theta ,\Phi )={\frac {\mathrm {G} (\theta ,\Phi )}{4\pi r^{2}}}P_{t}}" loading="lazy"></span>.</dd></dl>

<p>Here, the angles <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 \theta }">

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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> 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 \Phi }">

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<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>

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</math></span><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" loading="lazy"></span> indicate a dependence on direction from the antenna, 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 P_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{t}}</annotation>

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</math></span><img src="./7bdf246d27d8dd80dc45c1a1eaac69d42ce532d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.318ex; height:2.509ex;" alt="{\displaystyle P_{t}}" loading="lazy"></span> stands for the power the transmitter would deliver into a matched load. The gain <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 G}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> may be broken down into three factors; the <a href="Antenna_gain" class="mw-redirect" title="Antenna gain">antenna gain</a> (the directional redistribution of the power), the <a href="Radiation_efficiency" title="Radiation efficiency">radiation efficiency</a> (accounting for ohmic losses in the antenna), and lastly the loss due to mismatch between the antenna and transmitter. Strictly, to include the mismatch, it should be called the <b>realized gain</b>,<sup id="cite_ref-IEEEdict1997_4-2" class="reference"><a href="#cite_note-IEEEdict1997-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> but this is not common usage.

</p><p>For the receiving antenna, the power delivered to the receiver is

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<annotation encoding="application/x-tex">{\displaystyle P_{r}=\mathrm {A} (\theta ,\Phi )W\,}</annotation>

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</math></span><img src="./7fcc8cb8def0648456b593d2239ff4900ffcf229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.742ex; height:2.843ex;" alt="{\displaystyle P_{r}=\mathrm {A} (\theta ,\Phi )W\,}" loading="lazy"></span>.</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math></span><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> is the power density of the incident radiation, 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 A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> is the <a href="Antenna_aperture" class="mw-redirect" title="Antenna aperture">antenna aperture</a> or effective area of the antenna (the area the antenna would need to occupy in order to intercept the observed captured power). The directional arguments are now relative to the receiving antenna, and again <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}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> is taken to include ohmic and mismatch losses.

</p><p>Putting these expressions together, the power transferred from transmitter to receiver is

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<annotation encoding="application/x-tex">{\displaystyle P_{r}=A{\frac {G}{4\pi r^{2}}}P_{t}}</annotation>

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</math></span><img src="./8d43b30e739dd9fb92b1ade53ace778b5fbcbdfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.059ex; height:5.676ex;" alt="{\displaystyle P_{r}=A{\frac {G}{4\pi r^{2}}}P_{t}}" loading="lazy"></span>,</dd></dl>

<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 G}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> 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 A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> are directionally dependent properties of the transmitting and receiving antennas respectively. For transmission from the reference

antenna (2), to the test antenna (1), that is

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<annotation encoding="application/x-tex">{\displaystyle P_{1r}=\mathrm {A_{1}} (\theta ,\Phi ){\frac {G_{2}}{4\pi r^{2}}}P_{2t}}</annotation>

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</math></span><img src="./3de54637997415a84c0ad3eeca41f10adea690dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.369ex; height:5.676ex;" alt="{\displaystyle P_{1r}=\mathrm {A_{1}} (\theta ,\Phi ){\frac {G_{2}}{4\pi r^{2}}}P_{2t}}" loading="lazy"></span>,</dd></dl>

<p>and for transmission in the opposite direction

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<annotation encoding="application/x-tex">{\displaystyle P_{2r}=A_{2}{\frac {\mathrm {G_{1}} (\theta ,\Phi )}{4\pi r^{2}}}P_{1t}}</annotation>

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</math></span><img src="./ebcbdb0904de5203a830f29b22680a53043cb7ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.651ex; height:6.009ex;" alt="{\displaystyle P_{2r}=A_{2}{\frac {\mathrm {G_{1}} (\theta ,\Phi )}{4\pi r^{2}}}P_{1t}}" loading="lazy"></span>.</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math></span><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"></span> and effective area <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_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation>

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</math></span><img src="./3ec73b8bc9abc3efb934f5a6ec2803713771f4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}" loading="lazy"></span> of antenna 2 are fixed, because the orientation of this antenna is fixed with respect to the first.

</p><p>Now for a given disposition of the antennas, the <a href="Reciprocity_(electromagnetism)" title="Reciprocity (electromagnetism)">reciprocity theorem</a> requires that the power transfer is equally effective in each direction, i.e.

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<annotation encoding="application/x-tex">{\displaystyle {\frac {P_{1r}}{P_{2t}}}={\frac {P_{2r}}{P_{1t}}}}</annotation>

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</math></span><img src="./e5358887652ea7ce74b86143f113c97fa0fe5a43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.347ex; height:5.676ex;" alt="{\displaystyle {\frac {P_{1r}}{P_{2t}}}={\frac {P_{2r}}{P_{1t}}}}" loading="lazy"></span>,</dd></dl>

<p>whence

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {A_{1}} (\theta ,\Phi )}{\mathrm {G_{1}} (\theta ,\Phi )}}={\frac {A_{2}}{G_{2}}}}</annotation>

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</math></span><img src="./e59759798726fbc742139b6832eb49f5092225cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.142ex; height:6.509ex;" alt="{\displaystyle {\frac {\mathrm {A_{1}} (\theta ,\Phi )}{\mathrm {G_{1}} (\theta ,\Phi )}}={\frac {A_{2}}{G_{2}}}}" loading="lazy"></span>.</dd></dl>

<p>But the right hand side of this equation is fixed (because the orientation of antenna 2 is fixed), and so

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {A_{1}} (\theta ,\Phi )}{\mathrm {G_{1}} (\theta ,\Phi )}}=\mathrm {constant} }</annotation>

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</math></span><img src="./aecf8bc0ef73f125506f3bd233478e06ee440e78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.093ex; height:6.509ex;" alt="{\displaystyle {\frac {\mathrm {A_{1}} (\theta ,\Phi )}{\mathrm {G_{1}} (\theta ,\Phi )}}=\mathrm {constant} }" loading="lazy"></span>,</dd></dl>

<p>i.e. the directional dependence of the (receiving) effective aperture and the (transmitting) gain are identical (QED). Furthermore, the constant of proportionality is the same irrespective of the nature of the antenna, and so must be the same for all antennas. Analysis of a particular antenna (such as a <a href="Hertzian_dipole" class="mw-redirect" title="Hertzian dipole">Hertzian dipole</a>), shows that this constant is <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 {\frac {\lambda ^{2}}{4\pi }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\lambda ^{2}}{4\pi }}}</annotation>

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</math></span><img src="./c8ccd2b087e4d6eb5256cfde0e36ab8009170edb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.331ex; height:5.676ex;" alt="{\displaystyle {\frac {\lambda ^{2}}{4\pi }}}" loading="lazy"></span>, 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 \lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> is the free-space wavelength. Hence, for any antenna the gain and the effective aperture are related by

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {A} (\theta ,\Phi )={\frac {\lambda ^{2}\mathrm {G} (\theta ,\Phi )}{4\pi }}}</annotation>

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</math></span><img src="./4c931187744fa5d6a13eb415620c74c4897a40d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.135ex; height:5.843ex;" alt="{\displaystyle \mathrm {A} (\theta ,\Phi )={\frac {\lambda ^{2}\mathrm {G} (\theta ,\Phi )}{4\pi }}}" loading="lazy"></span>.</dd></dl>

<p>Even for a receiving antenna, it is more usual to state the gain than to specify the effective aperture. The power delivered to the receiver is therefore more usually written as

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<annotation encoding="application/x-tex">{\displaystyle P_{r}={\frac {\lambda ^{2}G_{r}G_{t}}{(4\pi r)^{2}}}P_{t}}</annotation>

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</math></span><img src="./28bc742dedfff4740f5b3fbbfa86bd14b4525630.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.581ex; height:6.509ex;" alt="{\displaystyle P_{r}={\frac {\lambda ^{2}G_{r}G_{t}}{(4\pi r)^{2}}}P_{t}}" loading="lazy"></span></dd></dl>

<p>(see <a href="Link_budget" title="Link budget">link budget</a>). The effective aperture is however of interest for comparison with the actual physical size of the antenna.

</p>

<div class="mw-heading mw-heading3"><h3 id="Practical_consequences">Practical consequences</h3></div>

<ul><li>When determining the pattern of a receiving antenna by computer simulation, it is not necessary to perform a calculation for every possible angle of incidence. Instead, the radiation pattern of the antenna is determined by a single simulation, and the receiving pattern inferred by reciprocity.</li>

<li>When determining the pattern of an <a href="Antenna_measurement" title="Antenna measurement">antenna by measurement</a>, the antenna may be either receiving or transmitting, whichever is more convenient.</li>

<li>For a practical antenna, the side lobe level should be minimum, it is necessary to have the maximum directivity.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul>

<ul><li><a href="Antenna_modeling" class="mw-redirect" title="Antenna modeling">Antenna modeling</a></li>

<li><a href="E-plane_and_H-plane" title="E-plane and H-plane">E-plane and H-plane</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-Balanis-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Balanis_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Balanis_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Balanis_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Constantine A. Balanis: "Antenna Theory, Analysis and Design", John Wiley & Sons, Inc., 2nd ed. 1982 <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-59268-4</bdi></span>

</li>

<li id="cite_note-Cheng-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cheng_2-0">^</a></b></span> <span class="reference-text">David K Cheng: "Field and Wave Electromagnetics", Addison-Wesley Publishing Company Inc., Edition 2, 1998. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-52820-7</bdi></span>

</li>

<li id="cite_note-JordanBalmain1968-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-JordanBalmain1968_3-0">^</a></b></span> <span class="reference-text">Edward C. Jordan & Keith G. Balmain; "Electromagnetic Waves and Radiating Systems" (2nd ed. 1968) Prentice-Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>81-203-0054-8</bdi></span>

</li>

<li id="cite_note-IEEEdict1997-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-IEEEdict1997_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IEEEdict1997_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-IEEEdict1997_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Institute of Electrical and Electronics Engineers, "The IEEE standard dictionary of electrical and electronics terms"; 6th ed. New York, N.Y., Institute of Electrical and Electronics Engineers, c1997. IEEE Std 100-1996. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-55937-833-6</bdi> [ed. Standards Coordinating Committee 10, Terms and Definitions; Jane Radatz, (chair)]</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFSinghSalgotra2016" class="citation journal cs1">Singh, Urvinder; Salgotra, Rohit (20 July 2016). "Synthesis of linear antenna array using flower pollination algorithm". <i>Neural Computing and Applications</i>. <b>29</b> (2): <span class="nowrap">435–</span>445. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00521-016-2457-7">10.1007/s00521-016-2457-7</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:22745168">22745168</a>.</cite></span>

</li>

</ol></div>

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</style><span class="citation FS1037C MS188"><span class="noviewer" typeof="mw:File"><span></span></span> This article incorporates <a href="Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <cite class="citation cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220122224547/https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm"><i>Federal Standard 1037C</i></a>. <a href="General_Services_Administration" title="General Services Administration">General Services Administration</a>. Archived from <a rel="nofollow" class="external text" href="https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm">the original</a> on 2022-01-22.</cite> (in support of <a href="MIL-STD-188" title="MIL-STD-188">MIL-STD-188</a>).</span>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.astronwireless.com/topic-archives-antenna-radiation-patterns.asp">Understanding and Using Antenna Radiation Patterns By Joseph H. Reisert</a></li>

<li>Explanation of the term “<a rel="nofollow" class="external text" href="https://www.radartutorial.eu/06.antennas/an12.en.html">Two-Way beamwidth</a>”</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-01-05" href="https://en.wikipedia.org/wiki/?title=Radiation_pattern&oldid=1267481628">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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