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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Reflection coefficient</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">This article is about reflections of waves. For the use of the term with capillary membrames, see <a href="Starling_equation#Reflection_coefficient" title="Starling equation">Starling equation § Reflection coefficient</a>. For intensity ratios, see <a href="Reflectance" title="Reflectance">Reflectance</a>.</div>
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<p>In <a href="Physics" title="Physics">physics</a> and <a href="Electrical_engineering" title="Electrical engineering">electrical engineering</a> the <b>reflection coefficient</b> is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the <a href="Amplitude" title="Amplitude">amplitude</a> of the reflected wave to the incident wave, with each expressed as <a href="Phasor" title="Phasor">phasors</a>. For example, it is used in <a href="Optics" title="Optics">optics</a> to calculate the amount of light that is reflected from a surface with a different index of refraction, such as a glass surface, or in an electrical <a href="Transmission_line" title="Transmission line">transmission line</a> to calculate how much of the <a href="Reflections_of_signals_on_conducting_lines" title="Reflections of signals on conducting lines">electromagnetic wave</a> is reflected by an impedance discontinuity. The reflection coefficient is closely related to the <i><a href="Transmission_coefficient" title="Transmission coefficient">transmission coefficient</a></i>. The <a href="Reflectance" title="Reflectance">reflectance</a> of a system is also sometimes called a reflection coefficient.
</p>
<p>Different disciplines have different applications for the term.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Transmission_lines">Transmission lines</h2></div>
<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Reflections_of_signals_on_conducting_lines" title="Reflections of signals on conducting lines">Reflections of signals on conducting lines</a> and <a href="Signal_reflection" title="Signal reflection">Signal reflection</a></div>
<p>In <a href="Telecommunications" title="Telecommunications">telecommunications</a> and <a href="Transmission_line" title="Transmission line">transmission line</a> theory, the reflection coefficient is the <a href="Ratio" title="Ratio">ratio</a> of the <a href="Complex_amplitude" class="mw-redirect" title="Complex amplitude">complex amplitude</a> of the reflected wave to that of the incident wave. The voltage and current at any point along a transmission line can always be resolved into forward and reflected traveling waves given a specified reference impedance <i>Z<sub>0</sub></i>. The reference impedance used is typically the <a href="Characteristic_impedance" title="Characteristic impedance">characteristic impedance</a> of a transmission line that's involved, but one can speak of reflection coefficient without any actual transmission line being present. In terms of the forward and reflected waves determined by the voltage and current, the reflection coefficient is defined as the <a href="Complex_number" title="Complex number">complex</a> ratio of the voltage of the reflected wave (<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 V^{-}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V^{-}}</annotation>
</semantics>
</math></span><img src="./9fa9ed53745ca8c9a03fbbb6c28c341c067aaa0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.428ex; height:2.509ex;" alt="{\displaystyle V^{-}}" loading="lazy"></span>) to that of the incident wave (<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 V^{+}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V^{+}}</annotation>
</semantics>
</math></span><img src="./4b2aafc830b7719ab820d87c8a8eec52e8756b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.428ex; height:2.509ex;" alt="{\displaystyle V^{+}}" loading="lazy"></span>). This is typically represented with 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 \Gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>
</semantics>
</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> (capital <a href="Gamma" title="Gamma">gamma</a>) and can be written as:
</p>
<dl><dd><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 \Gamma ={\frac {V^{-}}{V^{+}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma ={\frac {V^{-}}{V^{+}}}}</annotation>
</semantics>
</math></span><img src="./33eb5c95e58a153a5ad7de947135819e4184ee23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.815ex; height:5.676ex;" alt="{\displaystyle \Gamma ={\frac {V^{-}}{V^{+}}}}" loading="lazy"></span></dd></dl>
<p>It can also be defined using the <i>currents</i> associated with the reflected and forward waves, but introducing a minus sign to account for the opposite orientations of the two currents:
</p>
<dl><dd><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 \Gamma =-{\frac {I^{-}}{I^{+}}}={\frac {V^{-}}{V^{+}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<msup>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
</mrow>
</msup>
<msup>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma =-{\frac {I^{-}}{I^{+}}}={\frac {V^{-}}{V^{+}}}}</annotation>
</semantics>
</math></span><img src="./2fb3f9b8bcf7f91283b62e88b63c25bd2cb4022a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.285ex; height:5.676ex;" alt="{\displaystyle \Gamma =-{\frac {I^{-}}{I^{+}}}={\frac {V^{-}}{V^{+}}}}" loading="lazy"></span></dd></dl>
<p>The reflection coefficient may also be established using other field or <a href="Electronic_circuit" title="Electronic circuit">circuit</a> pairs of quantities whose product defines power resolvable into a forward and reverse wave. With electromagnetic plane waves, one uses the ratio of the electric fields of the reflected to that of the incident wave (or magnetic fields, again with a minus sign); the ratio of each wave's electric field <i>E</i> to its magnetic field <i>H</i> is the medium's characteristic impedance, <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 Z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{0}}</annotation>
</semantics>
</math></span><img src="./bfcd49ab63d30163ac54e60a8e24ff9ccd7bcd44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{0}}" loading="lazy"></span>, (equal to the <a href="Impedance_of_free_space" title="Impedance of free space">impedance of free space</a> if the medium is a vacuum).<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>In the accompanying figure, a signal source with internal impedance <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 Z_{S}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{S}}</annotation>
</semantics>
</math></span><img src="./f35299f14001027f7c1494d579f8f8a3120e8666.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle Z_{S}}" loading="lazy"></span> possibly followed by a transmission line of characteristic impedance <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 Z_{S}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{S}}</annotation>
</semantics>
</math></span><img src="./f35299f14001027f7c1494d579f8f8a3120e8666.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle Z_{S}}" loading="lazy"></span> is represented by its <a href="Th%C3%A9venin_equivalent" class="mw-redirect" title="Thévenin equivalent">Thévenin equivalent</a>, driving the load <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 Z_{L}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{L}}</annotation>
</semantics>
</math></span><img src="./8fb811eeda6b543aafa2f13aa8bbeb80fd77a0c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.939ex; height:2.509ex;" alt="{\displaystyle Z_{L}}" loading="lazy"></span>. For a real (resistive) source impedance <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 Z_{S}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{S}}</annotation>
</semantics>
</math></span><img src="./f35299f14001027f7c1494d579f8f8a3120e8666.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle Z_{S}}" loading="lazy"></span>, if we define <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 \Gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>
</semantics>
</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> using the reference impedance <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 Z_{0}=Z_{S}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{0}=Z_{S}}</annotation>
</semantics>
</math></span><img src="./a0a87f625599e6938f5fccd86c6d25d7a941c147.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.62ex; height:2.509ex;" alt="{\displaystyle Z_{0}=Z_{S}}" loading="lazy"></span> then the source's <a href="Impedance_matching#Reflection-less_matching" title="Impedance matching">maximum power is delivered</a> to a load <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 Z_{L}=Z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{L}=Z_{0}}</annotation>
</semantics>
</math></span><img src="./0e4e1bcfa469e843e07359c395467b7be795a3c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.679ex; height:2.509ex;" alt="{\displaystyle Z_{L}=Z_{0}}" loading="lazy"></span>, in which case <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 \Gamma =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma =0}</annotation>
</semantics>
</math></span><img src="./f9896b8ebf4dd4079563aefd15653a9880687480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.714ex; height:2.176ex;" alt="{\displaystyle \Gamma =0}" loading="lazy"></span> implying no reflected power. More generally, the squared-magnitude of the reflection coefficient <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 |\Gamma |^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Gamma |^{2}}</annotation>
</semantics>
</math></span><img src="./1878b988d2e4dafbb71e4ab635051a612dba36fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.801ex; height:3.343ex;" alt="{\displaystyle |\Gamma |^{2}}" loading="lazy"></span> denotes the proportion of that power that is reflected back to the source, with the power actually delivered toward the load being <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 1-|\Gamma |^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1-|\Gamma |^{2}}</annotation>
</semantics>
</math></span><img src="./dd7e19643c5065373b0eb7fd891c9077559f3f05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.804ex; height:3.343ex;" alt="{\displaystyle 1-|\Gamma |^{2}}" loading="lazy"></span>.
</p><p>Anywhere along an intervening (lossless) transmission line of characteristic impedance <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 Z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{0}}</annotation>
</semantics>
</math></span><img src="./bfcd49ab63d30163ac54e60a8e24ff9ccd7bcd44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{0}}" loading="lazy"></span>, the magnitude of the reflection coefficient <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 |\Gamma |}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Gamma |}</annotation>
</semantics>
</math></span><img src="./0c417bf8f8d99da4c7f0f82ce92f6ba665fffb22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.746ex; height:2.843ex;" alt="{\displaystyle |\Gamma |}" loading="lazy"></span> will remain the same (the powers of the forward and reflected waves stay the same) but with a different phase. In the case of a short circuited load (<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 Z_{L}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{L}=0}</annotation>
</semantics>
</math></span><img src="./5bc00cba1a332143d271849472562697f0fc980e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.2ex; height:2.509ex;" alt="{\displaystyle Z_{L}=0}" loading="lazy"></span>), one finds <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 \Gamma =-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma =-1}</annotation>
</semantics>
</math></span><img src="./5a5b71df0d3d176c3b99ddbd40ecd749fc247aa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.522ex; height:2.343ex;" alt="{\displaystyle \Gamma =-1}" loading="lazy"></span> at the load. This implies the reflected wave having a 180° phase shift (phase reversal) with the voltages of the two waves being opposite at that point and adding to zero (as a short circuit demands).
</p>
<div class="mw-heading mw-heading3"><h3 id="Relation_to_load_impedance">Relation to load impedance</h3></div>
<p>The reflection coefficient is determined by the load impedance at the end of the transmission line, as well as the <a href="Characteristic_impedance" title="Characteristic impedance">characteristic impedance</a> of the line. A load impedance 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 Z_{L}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{L}}</annotation>
</semantics>
</math></span><img src="./8fb811eeda6b543aafa2f13aa8bbeb80fd77a0c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.939ex; height:2.509ex;" alt="{\displaystyle Z_{L}}" loading="lazy"></span> terminating a line with a characteristic impedance 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 Z_{0}\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{0}\,}</annotation>
</semantics>
</math></span><img src="./d21b2afe643f3f5f93e6bcfb280a92d34339d9d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.029ex; height:2.509ex;" alt="{\displaystyle Z_{0}\,}" loading="lazy"></span> will have a reflection coefficient of
</p>
<dl><dd><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 \Gamma ={Z_{L}-Z_{0} \over Z_{L}+Z_{0}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
<mrow>
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma ={Z_{L}-Z_{0} \over Z_{L}+Z_{0}}.}</annotation>
</semantics>
</math></span><img src="./0b2c1eb05aab3011af7ca6923fd719357b1b2db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.455ex; height:5.843ex;" alt="{\displaystyle \Gamma ={Z_{L}-Z_{0} \over Z_{L}+Z_{0}}.}" loading="lazy"></span></dd></dl>
<p>This is the coefficient at the load. The reflection coefficient can also be measured at other points on the line. The <i>magnitude</i> of the reflection coefficient in a lossless transmission line is constant along the line (as are the powers in the forward and reflected waves). However its <i>phase</i> will be shifted by an amount dependent on the <a href="Electrical_length" title="Electrical length">electrical distance</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 \phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>
</semantics>
</math></span><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> from the load. If the coefficient is measured at a point <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 L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> meters from the load, so the <a href="Electrical_length" title="Electrical length">electrical distance</a> from the load 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 \phi =2\pi L/\lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo>=</mo>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>λ<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi =2\pi L/\lambda }</annotation>
</semantics>
</math></span><img src="./cde8bd52d1a570006f3737088dfd5329555bb8ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.079ex; height:2.843ex;" alt="{\displaystyle \phi =2\pi L/\lambda }" loading="lazy"></span> radians, the coefficient <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 \Gamma '}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma '}</annotation>
</semantics>
</math></span><img src="./cc1c460d7474b68828ae8281cad517b61348df7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.137ex; height:2.509ex;" alt="{\displaystyle \Gamma '}" loading="lazy"></span> at that point will be
</p>
<dl><dd><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 \Gamma '=\Gamma e^{-i\,2\phi }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>i</mi>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>ϕ<!-- ϕ --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma '=\Gamma e^{-i\,2\phi }}</annotation>
</semantics>
</math></span><img src="./e6dd66c336f1880f236f355d00aea81ddd32e0be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.039ex; height:2.676ex;" alt="{\displaystyle \Gamma '=\Gamma e^{-i\,2\phi }}" loading="lazy"></span></dd></dl>
<p>Note that the phase of the reflection coefficient is changed by <i>twice</i> the phase length of the attached transmission line. That is to take into account not only the phase delay of the reflected wave, but the phase shift that had first been applied to the forward wave, with the reflection coefficient being the quotient of these. The reflection coefficient so measured, <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 \Gamma '}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mo>′</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma '}</annotation>
</semantics>
</math></span><img src="./cc1c460d7474b68828ae8281cad517b61348df7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.137ex; height:2.509ex;" alt="{\displaystyle \Gamma '}" loading="lazy"></span>, corresponds to an impedance which is generally dissimilar to <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 Z_{L}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{L}}</annotation>
</semantics>
</math></span><img src="./8fb811eeda6b543aafa2f13aa8bbeb80fd77a0c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.939ex; height:2.509ex;" alt="{\displaystyle Z_{L}}" loading="lazy"></span> present at the far side of the transmission line.
</p><p>The complex reflection coefficient (in the region <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 |\Gamma |\leq 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Gamma |\leq 1}</annotation>
</semantics>
</math></span><img src="./c57049754fc4cd30f4a176be4bc248ae843a38a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.007ex; height:2.843ex;" alt="{\displaystyle |\Gamma |\leq 1}" loading="lazy"></span>, corresponding to passive loads) may be displayed graphically using a <a href="Smith_chart" title="Smith chart">Smith chart</a>. The Smith chart is a polar plot 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 \Gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>
</semantics>
</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span>, therefore the magnitude 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 \Gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>
</semantics>
</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> is given directly by the distance of a point to the center (with the edge of the Smith chart corresponding to <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 |\Gamma |=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Gamma |=1}</annotation>
</semantics>
</math></span><img src="./f35a0b80be6e7e0c631f7112ef6740b4009c0423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.007ex; height:2.843ex;" alt="{\displaystyle |\Gamma |=1}" loading="lazy"></span>). Its evolution along a transmission line is likewise described by a rotation 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 2\phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2\phi }</annotation>
</semantics>
</math></span><img src="./913feff4f1627646eea7f2a496a94ffee3c1c11b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.509ex;" alt="{\displaystyle 2\phi }" loading="lazy"></span> around the chart's center. Using the scales on a Smith chart, the resulting impedance (normalized to <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 Z_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{0}}</annotation>
</semantics>
</math></span><img src="./bfcd49ab63d30163ac54e60a8e24ff9ccd7bcd44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{0}}" loading="lazy"></span>) can directly be read. Before the advent of modern electronic computers, the Smith chart was of particular use as a sort of <a href="Nomogram" title="Nomogram">analog computer</a> for this purpose.
</p><p>The reflected power in terms of the reflection coefficient is:
</p>
<dl><dd><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_{reflected}=P_{incident}|\Gamma |^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mi>e</mi>
<mi>f</mi>
<mi>l</mi>
<mi>e</mi>
<mi>c</mi>
<mi>t</mi>
<mi>e</mi>
<mi>d</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>n</mi>
<mi>c</mi>
<mi>i</mi>
<mi>d</mi>
<mi>e</mi>
<mi>n</mi>
<mi>t</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P_{reflected}=P_{incident}|\Gamma |^{2}}</annotation>
</semantics>
</math></span><img src="./bbfd3695d13480fc43a887e1cbf106d2011acdb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.987ex; height:3.509ex;" alt="{\displaystyle P_{reflected}=P_{incident}|\Gamma |^{2}}" loading="lazy"></span> .</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Standing_wave_ratio">Standing wave ratio</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Standing_wave_ratio" title="Standing wave ratio">Standing wave ratio</a></div>
<p>The <a href="Standing_wave_ratio" title="Standing wave ratio">standing wave ratio</a> (SWR) is determined solely by the <i>magnitude</i> of the reflection coefficient:
</p>
<dl><dd><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 SWR={1+|\Gamma | \over 1-|\Gamma |}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mi>W</mi>
<mi>R</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle SWR={1+|\Gamma | \over 1-|\Gamma |}.}</annotation>
</semantics>
</math></span><img src="./3c9ac17652556b677e8a727ef7db280411520e8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.029ex; height:6.509ex;" alt="{\displaystyle SWR={1+|\Gamma | \over 1-|\Gamma |}.}" loading="lazy"></span></dd></dl>
<p>Along a lossless transmission line of characteristic impedance <i>Z</i><sub>0</sub>, the SWR signifies the ratio of the voltage (or current) maxima to minima (or what it would be if the transmission line were long enough to produce them). The above calculation assumes that <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 \Gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>
</semantics>
</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> has been calculated using <i>Z</i><sub>0</sub> as the reference impedance. Since it uses only the <i>magnitude</i> 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 \Gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>
</semantics>
</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span>, the SWR intentionally ignores the specific value of the load impedance <i>Z<sub>L</sub></i> responsible for it, but only the magnitude of the resulting <a href="Impedance_mismatch" class="mw-redirect" title="Impedance mismatch">impedance mismatch</a>. That SWR remains the same wherever measured along a transmission line (looking towards the load) since the addition of a transmission line length to a load <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 Z_{L}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z_{L}}</annotation>
</semantics>
</math></span><img src="./8fb811eeda6b543aafa2f13aa8bbeb80fd77a0c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.939ex; height:2.509ex;" alt="{\displaystyle Z_{L}}" loading="lazy"></span> only changes the phase, not magnitude 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 \Gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>
</semantics>
</math></span><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span>. While having a one-to-one correspondence with reflection coefficient, SWR is the most commonly used figure of merit in describing the mismatch affecting a <a href="Radio_antenna" class="mw-redirect" title="Radio antenna">radio antenna</a> or antenna system. It is most often <a href="SWR_meter" title="SWR meter">measured</a> at the transmitter side of a transmission line, but having, as explained, the same value as would be measured at the antenna (load) itself.
</p>
<div class="mw-heading mw-heading2"><h2 id="Electrical_networks">Electrical networks</h2></div>
<p>A transmission line is an example of a <a href="Two-port_network" title="Two-port network">2-port</a> <a href="Electrical_network" title="Electrical network">electrical network</a>, but reflection coefficients are useful in the analysis of any electrical networks. A reflection coefficient for each port in the same way as for the boundary of a transmission line. It will, however, also depend on the properties of connections at other ports and so is not a property intrinsic to the network itself. For a 2-port network with the 2x2 <a href="Scattering_parameters" title="Scattering parameters">scattering matrix</a> <i>S</i>, and with a source and load connected to its input and output, where the reflections off the source back into the input are <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 \Gamma _{S}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma _{S}}</annotation>
</semantics>
</math></span><img src="./974863df213a910130845c98a9f45b4521ca1437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.745ex; height:2.509ex;" alt="{\displaystyle \Gamma _{S}}" loading="lazy"></span> and the reflections off the load back into the output are <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 \Gamma _{L}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma _{L}}</annotation>
</semantics>
</math></span><img src="./eb9e1cecff63c9ffb5dd2fb41603122b2ca9d89c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\displaystyle \Gamma _{L}}" loading="lazy"></span>, then the reflection coefficients at the input and output are given by:<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>
</p>
<dl><dd><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 |\Gamma _{\mathrm {in} }|=\left|S_{11}+{\frac {S_{12}S_{21}\Gamma _{L}}{1-S_{22}\Gamma _{L}}}\right|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
<mi mathvariant="normal">n</mi>
</mrow>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mrow>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>11</mn>
</mrow>
</msub>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>12</mn>
</mrow>
</msub>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>21</mn>
</mrow>
</msub>
<msub>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>22</mn>
</mrow>
</msub>
<msub>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>L</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Gamma _{\mathrm {in} }|=\left|S_{11}+{\frac {S_{12}S_{21}\Gamma _{L}}{1-S_{22}\Gamma _{L}}}\right|}</annotation>
</semantics>
</math></span><img src="./7c4da8d742dc8096152e61438619e03a995830c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.828ex; height:5.843ex;" alt="{\displaystyle |\Gamma _{\mathrm {in} }|=\left|S_{11}+{\frac {S_{12}S_{21}\Gamma _{L}}{1-S_{22}\Gamma _{L}}}\right|}" 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 |\Gamma _{\mathrm {out} }|=\left|S_{22}+{\frac {S_{12}S_{21}\Gamma _{S}}{1-S_{11}\Gamma _{S}}}\right|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">o</mi>
<mi mathvariant="normal">u</mi>
<mi mathvariant="normal">t</mi>
</mrow>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mrow>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>22</mn>
</mrow>
</msub>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>12</mn>
</mrow>
</msub>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>21</mn>
</mrow>
</msub>
<msub>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>11</mn>
</mrow>
</msub>
<msub>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>S</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Gamma _{\mathrm {out} }|=\left|S_{22}+{\frac {S_{12}S_{21}\Gamma _{S}}{1-S_{11}\Gamma _{S}}}\right|}</annotation>
</semantics>
</math></span><img src="./2caa0e6504995c7e8c269408728b1f7610557ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.773ex; height:5.843ex;" alt="{\displaystyle |\Gamma _{\mathrm {out} }|=\left|S_{22}+{\frac {S_{12}S_{21}\Gamma _{S}}{1-S_{11}\Gamma _{S}}}\right|}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Seismology">Seismology</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Reflection_seismology" title="Reflection seismology">Reflection seismology</a></div>
<p>Reflection coefficient is used in feeder testing for reliability of medium.
</p>
<div class="mw-heading mw-heading2"><h2 id="Optics_and_microwaves">Optics and microwaves</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Fresnel_equations" title="Fresnel equations">Fresnel equations</a></div>
<p>In <a href="Optics" title="Optics">optics</a> and electromagnetics in general, <i>reflection coefficient</i> can refer to either the amplitude reflection coefficient described here, or the <a href="Reflectance" title="Reflectance">reflectance</a>, depending on context. Typically, the reflectance is represented by a capital <i>R</i>, while the amplitude reflection coefficient is represented by a lower-case <i>r</i>. These related concepts are covered by <a href="Fresnel_equations" title="Fresnel equations">Fresnel equations</a> in <a href="Optics#Classical_optics" title="Optics">classical optics</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Acoustics">Acoustics</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Acoustic_wave#Reflection" title="Acoustic wave">Acoustic wave § Reflection</a></div>
<p>Acousticians use reflection coefficients to understand the effect of different materials on their acoustic environments. The field properties used to define the reflection coefficient are typically the acoustic pressure and velocity in the incident and reflected <a href="Acoustics" title="Acoustics">acoustic</a> waves.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Microwave" title="Microwave">Microwave</a></li>
<li><a href="Mismatch_loss" title="Mismatch loss">Mismatch loss</a></li>
<li><a href="Reflections_of_signals_on_conducting_lines" title="Reflections of signals on conducting lines">Reflections of signals on conducting lines</a></li>
<li><a href="Scattering_parameters" title="Scattering parameters">Scattering parameters</a></li>
<li><a href="Transmission_coefficient" title="Transmission coefficient">Transmission coefficient</a></li>
<li><a href="Target_strength" title="Target strength">Target strength</a></li>
<li><a href="Hagen%E2%80%93Rubens_relation" title="Hagen–Rubens relation">Hagen–Rubens relation</a></li>
<li><a href="Reflection_phase_change" title="Reflection phase change">Reflection phase change</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Pozar, David M. (2012); p. 29.</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">Pozar, David M. (2012); p. 197.</span>
</li>
</ol></div></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1041539562">
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</style><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></li>
<li><cite id="CITEREFBogatin2004" class="citation book cs1">Bogatin, Eric (2004). <i>Signal Integrity - Simplified</i>. Upper Saddle River, New Jersey: Pearson Education, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-066946-6</bdi>.</cite> Figure 8-2 and Eqn. 8-1 Pg. 279</li>
<li><cite id="CITEREFPozar2012" class="citation book cs1">Pozar, David M. (2012). <i>Microwave Electronics</i> (Fourth ed.). John Wiley & Sons Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118213636</bdi>.</cite></li></ul>
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<ul><li><a rel="nofollow" class="external text" href="http://www.fourier-series.com/rf-concepts/reflection.html">Flash tutorial for understanding reflection</a> A flash program that shows how a reflected wave is generated, the reflection coefficient and VSWR</li>
<li><a rel="nofollow" class="external text" href="https://poynting.herokuapp.com">Application for drawing standing wave diagrams including the reflection coefficient, input impedance, SWR, etc.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201125200215/https://poynting.herokuapp.com/">Archived</a> 2020-11-25 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul>
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