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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Sinusoid</span></h1>
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<p>Der <b>Sinusoid</b> (Adjektiv <i>sinusoidal</i>) ist eine <a href="Sinus" class="mw-redirect" title="Sinus">sinusförmige</a> <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a>, die aus der Sinusfunktion durch <a href="Skalarmultiplikation" title="Skalarmultiplikation">Skalierung</a> von <a href="Amplitude" title="Amplitude">Amplitude</a> und <a href="Frequenz" title="Frequenz">Frequenz</a> sowie <a href="Phasenverschiebung" title="Phasenverschiebung">Phasenverschiebung</a> gebildet wird. Er bildet die Grundlage der Darstellung im <a href="Frequenzspektrum" title="Frequenzspektrum">Frequenzbereich</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Der Sinusoid <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 x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> ist definiert durch
</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 x\left(t\right):=A\sin {\bigl (}\omega t+\varphi {\bigr )}+C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>:=</mo>
<mi>A</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="1.2em" minsize="1.2em">(</mo>
</mrow>
</mrow>
<mi>ω<!-- ω --></mi>
<mi>t</mi>
<mo>+</mo>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="1.2em" minsize="1.2em">)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\left(t\right):=A\sin {\bigl (}\omega t+\varphi {\bigr )}+C}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/606c474f1f0b3387bba2431ac83943f52d16f423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.866ex; height:3.176ex;" alt="{\displaystyle x\left(t\right):=A\sin {\bigl (}\omega t+\varphi {\bigr )}+C}" loading="lazy"></span>,</dd></dl>
<p>wobei
</p>
<ul><li><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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> die Skalierung der <a href="Amplitude" title="Amplitude">Amplitude</a>,</li>
<li><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 \omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> die Skalierung der <a href="Kreisfrequenz" title="Kreisfrequenz">Kreisfrequenz</a></li>
<li><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 \varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> die <a href="Phasenverschiebung" title="Phasenverschiebung">Phasenverschiebung</a> und</li>
<li><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 C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> das Absolutglied</li></ul>
<p>ist. In der <a href="Elektrotechnik" title="Elektrotechnik">Elektrotechnik</a> versteht man unter dem Absolutglied den zeitlich konstanten <a href="Gleichanteil" class="mw-redirect" title="Gleichanteil">Gleichanteil</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Kosinus">Kosinus</h2></div>
<p>Da <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 \cos \left(t\right)=\sin \left(t+{\tfrac {\pi }{2}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos \left(t\right)=\sin \left(t+{\tfrac {\pi }{2}}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ba4ec5a1874dfc47a15ffea74b2a8cdb9a35f82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.302ex; height:3.343ex;" alt="{\displaystyle \cos \left(t\right)=\sin \left(t+{\tfrac {\pi }{2}}\right)}" loading="lazy"></span> gilt, ist eine Funktion der Form <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 x\left(t\right)=A\cos \left(\omega t+\varphi \right)+C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>A</mi>
<mi>cos</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>ω<!-- ω --></mi>
<mi>t</mi>
<mo>+</mo>
<mi>φ<!-- φ --></mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\left(t\right)=A\cos \left(\omega t+\varphi \right)+C}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c9ecbf441f70a2ee74693b5eebb32a02e09e160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.767ex; height:2.843ex;" alt="{\displaystyle x\left(t\right)=A\cos \left(\omega t+\varphi \right)+C}" loading="lazy"></span> ebenfalls ein Sinusoid, die anderen <a href="Trigonometrische_Funktion" title="Trigonometrische Funktion">trigonometrischen Funktionen</a> jedoch nicht.
</p>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><cite class="lang" lang="en" dir="auto" style="font-style:italic">Sinusoid</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Sinusoid">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Sinusoid&rft.atitle=Sinusoid&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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