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href="/wiki/File:High_accuracy_settling_time_measurements_figure_1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/High_accuracy_settling_time_measurements_figure_1.png/220px-High_accuracy_settling_time_measurements_figure_1.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/High_accuracy_settling_time_measurements_figure_1.png/330px-High_accuracy_settling_time_measurements_figure_1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/High_accuracy_settling_time_measurements_figure_1.png/440px-High_accuracy_settling_time_measurements_figure_1.png 2x" data-file-width="2942" data-file-height="2145" /></a><figcaption>Settling time is the time required for an output to reach and remain within a given error band following some input stimulus.</figcaption></figure> <p>In <a href="/wiki/Control_theory" title="Control theory">control theory</a> the <b>settling time</b> of a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a> such as an <a href="/wiki/Amplifier" title="Amplifier">amplifier</a> or other output device is the time elapsed from the application of an ideal instantaneous step input to the time at which the amplifier output has entered and remained within a specified error band. </p><p>Settling time includes a <a href="/wiki/Propagation_delay" title="Propagation delay">propagation delay</a>, plus the time required for the output to slew to the vicinity of the final value, recover from the overload condition associated with <a href="/wiki/Slew_rate" title="Slew rate">slew</a>, and finally settle to within the specified error. </p><p>Systems with energy storage cannot respond instantaneously and will exhibit transient responses when they are subjected to inputs or disturbances.<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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Settling_time&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tay, Mareels and Moore (1998) defined settling time as "the time required for the response curve to reach and stay within a range of certain percentage (usually 5% or 2%) of the final value."<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> <div class="mw-heading mw-heading2"><h2 id="Mathematical_detail">Mathematical detail</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Settling_time&action=edit&section=2" title="Edit section: Mathematical detail"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Settling time depends on the system response and <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequency</a>. </p><p>The settling time for a <a href="/wiki/Order_(differential_equation)" class="mw-redirect" title="Order (differential equation)">second order</a>, <a href="/wiki/Underdamped" class="mw-redirect" title="Underdamped">underdamped</a> system responding to a <a href="/wiki/Step_response" title="Step response">step response</a> can be approximated if the damping ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta \ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta \ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c1464329d67ff5e2ea6d519d3a3775f7c601f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.872ex; height:2.509ex;" alt="{\displaystyle \zeta \ll 1}"></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{s}=-{\frac {\ln({\text{tolerance fraction}})}{{\text{damping ratio}}\times {\text{natural freq}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>tolerance fraction</mtext> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>damping ratio</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>natural freq</mtext> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{s}=-{\frac {\ln({\text{tolerance fraction}})}{{\text{damping ratio}}\times {\text{natural freq}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6aa603cdab1a8fe4945308bbf2ddca766c233ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.935ex; height:6.176ex;" alt="{\displaystyle T_{s}=-{\frac {\ln({\text{tolerance fraction}})}{{\text{damping ratio}}\times {\text{natural freq}}}}}"></span> </p><p>A general form is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{s}=-{\frac {\ln({\text{tolerance fraction}}\times {\sqrt {1-\zeta ^{2}}})}{{\text{damping ratio}}\times {\text{natural freq}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>tolerance fraction</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>damping ratio</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>natural freq</mtext> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{s}=-{\frac {\ln({\text{tolerance fraction}}\times {\sqrt {1-\zeta ^{2}}})}{{\text{damping ratio}}\times {\text{natural freq}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184a4adf302c470128bd751119f4f56869ce1dd8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.775ex; height:6.676ex;" alt="{\displaystyle T_{s}=-{\frac {\ln({\text{tolerance fraction}}\times {\sqrt {1-\zeta ^{2}}})}{{\text{damping ratio}}\times {\text{natural freq}}}}}"></span> </p><p>Thus, if the damping ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta \ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta \ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c1464329d67ff5e2ea6d519d3a3775f7c601f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.872ex; height:2.509ex;" alt="{\displaystyle \zeta \ll 1}"></span>, settling time to within 2% = 0.02 is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{s}=-{\frac {\ln(0.02)}{\zeta \omega _{n}}}\approx {\frac {3.9}{\zeta \omega _{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0.02</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3.9</mn> <mrow> <mi>ζ</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{s}=-{\frac {\ln(0.02)}{\zeta \omega _{n}}}\approx {\frac {3.9}{\zeta \omega _{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b32f5bde8a4d19df6cd13152bd244b0285a2717" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.68ex; height:6.176ex;" alt="{\displaystyle T_{s}=-{\frac {\ln(0.02)}{\zeta \omega _{n}}}\approx {\frac {3.9}{\zeta \omega _{n}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Settling_time&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Rise_time" title="Rise time">Rise time</a></li> <li><a href="/wiki/Time_constant" title="Time constant">Time constant</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Settling_time&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Modern Control Engineering (5th Edition), Katsuhiko Ogata, p.160</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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFTayIven_MareelsJohn_B._Moore1998" class="citation book cs1">Tay, Teng-Tiow; Iven Mareels; John B. Moore (1998). <i>High performance control</i>. Birkhäuser. p. 93. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4004-0" title="Special:BookSources/978-0-8176-4004-0"><bdi>978-0-8176-4004-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=High+performance+control&rft.pages=93&rft.pub=Birkh%C3%A4user&rft.date=1998&rft.isbn=978-0-8176-4004-0&rft.aulast=Tay&rft.aufirst=Teng-Tiow&rft.au=Iven+Mareels&rft.au=John+B.+Moore&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASettling+time" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Settling_time&action=edit&section=5" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has a paper on settling time measurements.: <div style="margin-left:1.5em;font-weight:bold;font-style:italic;"><a href="https://en.wikisource.org/wiki/High_accuracy_settling_time_measurements" class="extiw" title="wikisource:High accuracy settling time measurements">Wikisource:High accuracy settling time measurements</a></div></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20040626102542/http://bass.gmu.edu/~gbeale/ece_421/second_order_04.html">Second-Order System Example</a></li> <li><a rel="nofollow" class="external text" href="http://www.analog.com/static/imported-files/application_notes/466359863287538299597392756AN359.pdf">Op Amp Settling Time</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070705124107/http://www.facstaff.bucknell.edu/mastascu/eControlHTML/SysDyn/SysDyn5.html">Graphical tutorial</a> of Settling time and Risetime</li> <li><a rel="nofollow" class="external text" href="http://www.mathworks.com/help/toolbox/ident/ref/stepinfo.html">MATLAB function</a> for computing settling time, rise time, and other step response characteristics</li> <li><a rel="nofollow" class="external text" href="https://3roam.com/settling-time-calculator/">Settling Time Calculator</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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