Step response - Wikipedia

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class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Results</span> </div> </a> <ul id="toc-Results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Control_of_overshoot" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Control_of_overshoot"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Control of overshoot</span> </div> </a> <ul id="toc-Control_of_overshoot-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Control_of_settling_time" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Control_of_settling_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4</span> <span>Control of settling time</span> </div> </a> <ul id="toc-Control_of_settling_time-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-System_Identification_using_the_Step_Response:_System_with_two_real_poles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#System_Identification_using_the_Step_Response:_System_with_two_real_poles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.5</span> <span>System Identification using the Step Response: System with two real poles</span> </div> </a> <ul id="toc-System_Identification_using_the_Step_Response:_System_with_two_real_poles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Phase_margin" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Phase_margin"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.6</span> <span>Phase margin</span> </div> </a> <ul id="toc-Phase_margin-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References_and_notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References_and_notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References and notes</span> </div> </a> <ul id="toc-References_and_notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> 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searchaux" style="display:none">Time behavior of a system controlled by Heaviside step functions</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"step change" redirects here. For other uses, see <a href="/wiki/Step_change_(disambiguation)" class="mw-disambig" title="Step change (disambiguation)">step change (disambiguation)</a>.</div> <figure typeof="mw:File/Thumb"><a 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/300px-High_accuracy_settling_time_measurements_figure_1.png" decoding="async" width="300" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/High_accuracy_settling_time_measurements_figure_1.png/450px-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/600px-High_accuracy_settling_time_measurements_figure_1.png 2x" data-file-width="2942" data-file-height="2145" /></a><figcaption>A typical step response for a second order system, illustrating <a href="/wiki/Overshoot_(signal)" title="Overshoot (signal)">overshoot</a>, followed by <a href="/wiki/Ringing_(signal)" title="Ringing (signal)">ringing</a>, all subsiding within a <a href="/wiki/Settling_time" title="Settling time">settling time</a>.</figcaption></figure> <p>The <b>step response</b> of a system in a given initial state consists of the time evolution of its outputs when its control inputs are <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step functions</a>. In <a href="/wiki/Electronic_engineering" title="Electronic engineering">electronic engineering</a> and <a href="/wiki/Control_theory" title="Control theory">control theory</a>, step response is the time behaviour of the outputs of a general <a href="/wiki/System" title="System">system</a> when its inputs change from zero to one in a very short time. The concept can be extended to the abstract mathematical notion of a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a> using an <a href="/wiki/Dynamical_system_(definition)#General_definition" class="mw-redirect" title="Dynamical system (definition)">evolution parameter</a>. </p><p>From a practical standpoint, knowing how the system responds to a sudden input is important because large and possibly fast deviations from the long term steady state may have extreme effects on the component itself and on other portions of the overall system dependent on this component. In addition, the overall system cannot act until the component's output settles down to some vicinity of its final state, delaying the overall system response. Formally, knowing the step response of a dynamical system gives information on the <a href="/wiki/Stability_theory" title="Stability theory">stability</a> of such a system, and on its ability to reach one stationary state when starting from another. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_mathematical_description">Formal mathematical description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=1" title="Edit section: Formal mathematical description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Step_response.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/6/60/Step_response.jpg" decoding="async" width="340" height="100" class="mw-file-element" data-file-width="340" data-file-height="100" /></a><figcaption>Figure 4: Black box representation of a dynamical system, its input and its step response.</figcaption></figure> <p>This section provides a formal mathematical definition of step response in terms of the abstract mathematical concept of a <a href="/wiki/Dynamical_system_(definition)" class="mw-redirect" title="Dynamical system (definition)">dynamical system</a> <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 {\mathfrak {S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91d521f12ee3c00ee2fc7ab16af9ea17d915a750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {S}}}"></span>: all notations and assumptions required for the following description are listed here. </p> <ul><li><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 t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}"></span> is the <a href="/wiki/Dynamical_system_(definition)" class="mw-redirect" title="Dynamical system (definition)">evolution parameter</a> of the system, called "<a href="/wiki/Time" title="Time">time</a>" for the sake of simplicity,</li> <li><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 {\boldsymbol {x}}|_{t}\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}|_{t}\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90a1b72fd7f8336d3cfa8a2e1c88a9cd6ce6546" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.287ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {x}}|_{t}\in M}"></span> is the <a href="/wiki/Dynamical_system_(definition)" class="mw-redirect" title="Dynamical system (definition)">state</a> of the system at time <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, called "output" for the sake of simplicity,</li> <li><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 \Phi :T\times M\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <mi>T</mi> <mo>×</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi :T\times M\to M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/973105e3b60bb20cb925057979e0b2698afbd3b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.59ex; height:2.176ex;" alt="{\displaystyle \Phi :T\times M\to M}"></span> is the dynamical system <a href="/wiki/Dynamical_system_(definition)" class="mw-redirect" title="Dynamical system (definition)">evolution function</a>,</li> <li><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 \Phi (0,{\boldsymbol {x}})={\boldsymbol {x}}_{0}\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (0,{\boldsymbol {x}})={\boldsymbol {x}}_{0}\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab9b25232b5401a9ae5fe5cf3692b089f9aedd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.183ex; height:2.843ex;" alt="{\displaystyle \Phi (0,{\boldsymbol {x}})={\boldsymbol {x}}_{0}\in M}"></span> is the dynamical system <a href="/wiki/Dynamical_system_(definition)" class="mw-redirect" title="Dynamical system (definition)">initial state</a>,</li> <li><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 H(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1b6c8837aed2794e7b52afa88ad371f1d275fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.713ex; height:2.843ex;" alt="{\displaystyle H(t)}"></span> is the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Nonlinear_dynamical_system">Nonlinear dynamical system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=2" title="Edit section: Nonlinear dynamical system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a general dynamical system, the step response is defined as follows: </p> <dl><dd><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 {\boldsymbol {x}}|_{t}=\Phi _{\{H(t)\}}\left(t,{{\boldsymbol {x}}_{0}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}|_{t}=\Phi _{\{H(t)\}}\left(t,{{\boldsymbol {x}}_{0}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad5756ae237ebbc3fdd1b70db44e2a058bc3f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.679ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {x}}|_{t}=\Phi _{\{H(t)\}}\left(t,{{\boldsymbol {x}}_{0}}\right).}"></span></dd></dl> <p>It is the <a href="/wiki/Dynamical_system_(definition)" class="mw-redirect" title="Dynamical system (definition)">evolution function</a> when the control inputs (or <a href="/wiki/Linear_differential_equation" title="Linear differential equation">source term</a>, or <a href="/w/index.php?title=Forcing_input&action=edit&redlink=1" class="new" title="Forcing input (page does not exist)">forcing inputs</a>) are Heaviside functions: the notation emphasizes this concept showing <i>H</i>(<i>t</i>) as a subscript. </p> <div class="mw-heading mw-heading3"><h3 id="Linear_dynamical_system">Linear dynamical system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=3" title="Edit section: Linear dynamical system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Linear_system" title="Linear system">linear</a> <a href="/wiki/Time-invariant_system" title="Time-invariant system">time-invariant</a> (LTI) black box, let <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 {\mathfrak {S}}\equiv S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mo>≡</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}\equiv S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44ae88415b3393ca1926fe04761ba25fd119031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.524ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {S}}\equiv S}"></span> for notational convenience: the step response can be obtained by <a href="/wiki/Convolution" title="Convolution">convolution</a> of the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> control and the <a href="/wiki/Impulse_response" title="Impulse response">impulse response</a> <i>h</i>(<i>t</i>) of the system itself </p> <dl><dd><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 a(t)=(h*H)(t)=\int _{-\infty }^{+\infty }h(\tau )H(t-\tau )\,d\tau =\int _{-\infty }^{t}h(\tau )\,d\tau .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∗</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>h</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>h</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(t)=(h*H)(t)=\int _{-\infty }^{+\infty }h(\tau )H(t-\tau )\,d\tau =\int _{-\infty }^{t}h(\tau )\,d\tau .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636e81723f21093f602497468383fdcca6cdf95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.78ex; height:6.343ex;" alt="{\displaystyle a(t)=(h*H)(t)=\int _{-\infty }^{+\infty }h(\tau )H(t-\tau )\,d\tau =\int _{-\infty }^{t}h(\tau )\,d\tau .}"></span></dd></dl> <p>which for an LTI system is equivalent to just integrating the latter. Conversely, for an LTI system, the derivative of the step response yields the impulse response: </p> <dl><dd><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 h(t)={\frac {d}{dt}}\,a(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>a</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t)={\frac {d}{dt}}\,a(t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7b640f212e28fd1a5b491061ef046e120419d2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.891ex; height:5.509ex;" alt="{\displaystyle h(t)={\frac {d}{dt}}\,a(t).}"></span></dd></dl> <p>However, these simple relations are not true for a non-linear or <a href="/wiki/Time-variant_system" title="Time-variant system">time-variant system</a>.<sup id="cite_ref-Shmaliy2007_1-0" class="reference"><a href="#cite_note-Shmaliy2007-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Time_domain_versus_frequency_domain">Time domain versus frequency domain</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=4" title="Edit section: Time domain versus frequency domain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Instead of frequency response, system performance may be specified in terms of parameters describing time-dependence of response. The step response can be described by the following quantities related to its <i>time behavior</i>, </p> <ul><li><a href="/wiki/Overshoot_(signal)" title="Overshoot (signal)">overshoot</a></li> <li><a href="/wiki/Rise_time" title="Rise time">rise time</a></li> <li><a href="/wiki/Settling_time" title="Settling time">settling time</a></li> <li><a href="/wiki/Ringing_(signal)" title="Ringing (signal)">ringing</a></li></ul> <p>In the case of <a href="/wiki/Linear" class="mw-redirect" title="Linear">linear</a> dynamic systems, much can be inferred about the system from these characteristics. <a class="mw-selflink-fragment" href="#Results">Below</a> the step response of a simple two-pole amplifier is presented, and some of these terms are illustrated. </p><p>In LTI systems, the function that has the steepest slew rate that doesn't create overshoot or ringing is the Gaussian function. This is because it is the only function whose Fourier transform has the same shape. </p> <div class="mw-heading mw-heading2"><h2 id="Feedback_amplifiers">Feedback amplifiers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=5" title="Edit section: Feedback amplifiers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Block_Diagram_for_Feedback.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Block_Diagram_for_Feedback.svg/220px-Block_Diagram_for_Feedback.svg.png" decoding="async" width="220" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Block_Diagram_for_Feedback.svg/330px-Block_Diagram_for_Feedback.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Block_Diagram_for_Feedback.svg/440px-Block_Diagram_for_Feedback.svg.png 2x" data-file-width="690" data-file-height="320" /></a><figcaption>Figure 1: Ideal negative feedback model; open loop gain is <i>A</i><sub>OL</sub> and feedback factor is β.</figcaption></figure> <p>This section describes the step response of a simple <a href="/wiki/Negative_feedback_amplifier" class="mw-redirect" title="Negative feedback amplifier">negative feedback amplifier</a> shown in Figure 1. The feedback amplifier consists of a main <b>open-loop</b> amplifier of gain <i>A</i><sub>OL</sub> and a feedback loop governed by a <b>feedback factor</b> β. This feedback amplifier is analyzed to determine how its step response depends upon the time constants governing the response of the main amplifier, and upon the amount of feedback used. </p><p>A negative-feedback amplifier has gain given by (see <a href="/wiki/Negative_feedback_amplifier" class="mw-redirect" title="Negative feedback amplifier">negative feedback amplifier</a>): </p> <dl><dd><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 A_{FB}={\frac {A_{OL}}{1+\beta A_{OL}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mi>L</mi> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mi>L</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{FB}={\frac {A_{OL}}{1+\beta A_{OL}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abbe5784d0bf7e33c179a91b3f608f2a5da3956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.718ex; height:5.843ex;" alt="{\displaystyle A_{FB}={\frac {A_{OL}}{1+\beta A_{OL}}},}"></span></dd></dl> <p>where <i>A</i><sub>OL</sub> = <b>open-loop</b> gain, <i>A</i><sub>FB</sub> = <b>closed-loop</b> gain (the gain with negative feedback present) and <i>β</i> = <b>feedback factor</b>. </p> <div class="mw-heading mw-heading3"><h3 id="With_one_dominant_pole">With one dominant pole</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=6" title="Edit section: With one dominant pole"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In many cases, the forward amplifier can be sufficiently well modeled in terms of a single dominant pole of time constant τ, that it, as an open-loop gain given by: </p> <dl><dd><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 A_{OL}={\frac {A_{0}}{1+j\omega \tau }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> <mi>ω</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{OL}={\frac {A_{0}}{1+j\omega \tau }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74599ac745ead599f3b3ca05db01a5bbd4ab69c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.539ex; height:5.843ex;" alt="{\displaystyle A_{OL}={\frac {A_{0}}{1+j\omega \tau }},}"></span></dd></dl> <p>with zero-frequency gain <i>A</i><sub>0</sub> and angular frequency ω = 2π<i>f</i>. This forward amplifier has unit step response </p> <dl><dd><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 S_{OL}(t)=A_{0}(1-e^{-t/\tau })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{OL}(t)=A_{0}(1-e^{-t/\tau })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c589654a0fe718f162ecdaa93fe3954ef6a2081b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.247ex; height:3.343ex;" alt="{\displaystyle S_{OL}(t)=A_{0}(1-e^{-t/\tau })}"></span>,</dd></dl> <p>an exponential approach from 0 toward the new equilibrium value of <i>A</i><sub>0</sub>. </p><p>The one-pole amplifier's transfer function leads to the closed-loop gain: </p> <dl><dd><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 A_{FB}={\frac {A_{0}}{1+\beta A_{0}}}\;\cdot \;\ {\frac {1}{1+j\omega {\frac {\tau }{1+\beta A_{0}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>⋅</mo> <mspace width="thickmathspace" /> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>τ</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{FB}={\frac {A_{0}}{1+\beta A_{0}}}\;\cdot \;\ {\frac {1}{1+j\omega {\frac {\tau }{1+\beta A_{0}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671c08003cbdd4d65bece33804bde3ed7849b4b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:33.903ex; height:7.009ex;" alt="{\displaystyle A_{FB}={\frac {A_{0}}{1+\beta A_{0}}}\;\cdot \;\ {\frac {1}{1+j\omega {\frac {\tau }{1+\beta A_{0}}}}}.}"></span></dd></dl> <p>This closed-loop gain is of the same form as the open-loop gain: a one-pole filter. Its step response is of the same form: an exponential decay toward the new equilibrium value. But the time constant of the closed-loop step function is <i>τ</i> / (1 + <i>β</i> <i>A</i><sub>0</sub>), so it is faster than the forward amplifier's response by a factor of 1 + <i>β</i> <i>A</i><sub>0</sub>: </p> <dl><dd><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 S_{FB}(t)={\frac {A_{0}}{1+\beta A_{0}}}\left(1-e^{-t(1+\beta A_{0})/\tau }\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{FB}(t)={\frac {A_{0}}{1+\beta A_{0}}}\left(1-e^{-t(1+\beta A_{0})/\tau }\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a63c1ffd57c7ff7f161de95e213f12d1665230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.296ex; height:5.843ex;" alt="{\displaystyle S_{FB}(t)={\frac {A_{0}}{1+\beta A_{0}}}\left(1-e^{-t(1+\beta A_{0})/\tau }\right),}"></span></dd></dl> <p>As the feedback factor <i>β</i> is increased, the step response will get faster, until the original assumption of one dominant pole is no longer accurate. If there is a second pole, then as the closed-loop time constant approaches the time constant of the second pole, a two-pole analysis is needed. </p> <div class="mw-heading mw-heading3"><h3 id="Two-pole_amplifiers">Two-pole amplifiers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=7" title="Edit section: Two-pole amplifiers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the case that the open-loop gain has two poles (two <a href="/wiki/Time_constant" title="Time constant">time constants</a>, <i>τ</i><sub>1</sub>, <i>τ</i><sub>2</sub>), the step response is a bit more complicated. The open-loop gain is given by: </p> <dl><dd><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 A_{OL}={\frac {A_{0}}{(1+j\omega \tau _{1})(1+j\omega \tau _{2})}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>j</mi> <mi>ω</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>j</mi> <mi>ω</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{OL}={\frac {A_{0}}{(1+j\omega \tau _{1})(1+j\omega \tau _{2})}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4a241406aa952914cec82ba16c1389f2e497716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.503ex; height:6.176ex;" alt="{\displaystyle A_{OL}={\frac {A_{0}}{(1+j\omega \tau _{1})(1+j\omega \tau _{2})}},}"></span></dd></dl> <p>with zero-frequency gain <i>A</i><sub>0</sub> and angular frequency <i>ω</i> = 2<i>πf</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Analysis">Analysis</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=8" title="Edit section: Analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The two-pole amplifier's transfer function leads to the closed-loop gain: </p> <dl><dd><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 A_{FB}={\frac {A_{0}}{1+\beta A_{0}}}\;\cdot \;\ {\frac {1}{1+j\omega {\frac {\tau _{1}+\tau _{2}}{1+\beta A_{0}}}+(j\omega )^{2}{\frac {\tau _{1}\tau _{2}}{1+\beta A_{0}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>⋅</mo> <mspace width="thickmathspace" /> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{FB}={\frac {A_{0}}{1+\beta A_{0}}}\;\cdot \;\ {\frac {1}{1+j\omega {\frac {\tau _{1}+\tau _{2}}{1+\beta A_{0}}}+(j\omega )^{2}{\frac {\tau _{1}\tau _{2}}{1+\beta A_{0}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0418c849143d12835ce5f8be815f19366ddc7297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:47.953ex; height:7.509ex;" alt="{\displaystyle A_{FB}={\frac {A_{0}}{1+\beta A_{0}}}\;\cdot \;\ {\frac {1}{1+j\omega {\frac {\tau _{1}+\tau _{2}}{1+\beta A_{0}}}+(j\omega )^{2}{\frac {\tau _{1}\tau _{2}}{1+\beta A_{0}}}}}.}"></span></dd></dl> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Conjugate_poles_in_s-plane.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Conjugate_poles_in_s-plane.svg/250px-Conjugate_poles_in_s-plane.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Conjugate_poles_in_s-plane.svg/375px-Conjugate_poles_in_s-plane.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Conjugate_poles_in_s-plane.svg/500px-Conjugate_poles_in_s-plane.svg.png 2x" data-file-width="450" data-file-height="450" /></a><figcaption>Figure 2: Conjugate pole locations for a two-pole feedback amplifier; Re(<i>s</i>) is the real axis and Im(<i>s</i>) is the imaginary axis.</figcaption></figure> <p>The time dependence of the amplifier is easy to discover by switching variables to <i>s</i> = <i>j</i>ω, whereupon the gain becomes: </p> <dl><dd><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 A_{FB}={\frac {A_{0}}{\tau _{1}\tau _{2}}}\;\cdot \;{\frac {1}{s^{2}+s\left({\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}\right)+{\frac {1+\beta A_{0}}{\tau _{1}\tau _{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>⋅</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>s</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{FB}={\frac {A_{0}}{\tau _{1}\tau _{2}}}\;\cdot \;{\frac {1}{s^{2}+s\left({\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}\right)+{\frac {1+\beta A_{0}}{\tau _{1}\tau _{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75168c21b89d972f630c09575d6c12d07eff8d37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:41.969ex; height:8.009ex;" alt="{\displaystyle A_{FB}={\frac {A_{0}}{\tau _{1}\tau _{2}}}\;\cdot \;{\frac {1}{s^{2}+s\left({\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}\right)+{\frac {1+\beta A_{0}}{\tau _{1}\tau _{2}}}}}}"></span></dd></dl> <p>The poles of this expression (that is, the zeros of the denominator) occur at: </p> <dl><dd><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 2s=-\left({\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}\right)\pm {\sqrt {\left({\frac {1}{\tau _{1}}}-{\frac {1}{\tau _{2}}}\right)^{2}-{\frac {4\beta A_{0}}{\tau _{1}\tau _{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2s=-\left({\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}\right)\pm {\sqrt {\left({\frac {1}{\tau _{1}}}-{\frac {1}{\tau _{2}}}\right)^{2}-{\frac {4\beta A_{0}}{\tau _{1}\tau _{2}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c57750efb21be5a786774cb1cd50cefc812b777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.529ex; height:7.676ex;" alt="{\displaystyle 2s=-\left({\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}\right)\pm {\sqrt {\left({\frac {1}{\tau _{1}}}-{\frac {1}{\tau _{2}}}\right)^{2}-{\frac {4\beta A_{0}}{\tau _{1}\tau _{2}}}}},}"></span></dd></dl> <p>which shows for large enough values of <i>βA</i><sub>0</sub> the square root becomes the square root of a negative number, that is the square root becomes imaginary, and the pole positions are complex conjugate numbers, either <i>s</i><sub>+</sub> or <i>s</i><sub>−</sub>; see Figure 2: </p> <dl><dd><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 s_{\pm }=-\rho \pm j\mu ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>ρ</mi> <mo>±</mo> <mi>j</mi> <mi>μ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\pm }=-\rho \pm j\mu ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3078e2e5842ae30d38c9329492c52cbfe028cb0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.557ex; height:2.676ex;" alt="{\displaystyle s_{\pm }=-\rho \pm j\mu ,}"></span></dd></dl> <p>with </p> <dl><dd><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 \rho ={\frac {1}{2}}\left({\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {1}{2}}\left({\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5046d91c4ec46905b0c615089eaea6fad4a8a52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.795ex; height:6.176ex;" alt="{\displaystyle \rho ={\frac {1}{2}}\left({\frac {1}{\tau _{1}}}+{\frac {1}{\tau _{2}}}\right),}"></span></dd></dl> <p>and </p> <dl><dd><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 \mu ={\frac {1}{2}}{\sqrt {{\frac {4\beta A_{0}}{\tau _{1}\tau _{2}}}-\left({\frac {1}{\tau _{1}}}-{\frac {1}{\tau _{2}}}\right)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ={\frac {1}{2}}{\sqrt {{\frac {4\beta A_{0}}{\tau _{1}\tau _{2}}}-\left({\frac {1}{\tau _{1}}}-{\frac {1}{\tau _{2}}}\right)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/562564c8f0d5627bd85ea609775038c4f224b4c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.566ex; height:7.676ex;" alt="{\displaystyle \mu ={\frac {1}{2}}{\sqrt {{\frac {4\beta A_{0}}{\tau _{1}\tau _{2}}}-\left({\frac {1}{\tau _{1}}}-{\frac {1}{\tau _{2}}}\right)^{2}}}.}"></span></dd></dl> <p>Using polar coordinates with the magnitude of the radius to the roots given by |<i>s</i>| (Figure 2): </p> <dl><dd><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 |s|=|s_{\pm }|={\sqrt {\rho ^{2}+\mu ^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |s|=|s_{\pm }|={\sqrt {\rho ^{2}+\mu ^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1fc8a8c4cdc7e3a9ccbca49dd03758f5800d9d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:22.999ex; height:4.843ex;" alt="{\displaystyle |s|=|s_{\pm }|={\sqrt {\rho ^{2}+\mu ^{2}}},}"></span></dd></dl> <p>and the angular coordinate φ is given by: </p> <dl><dd><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 \cos \phi ={\frac {\rho }{|s|}},\sin \phi ={\frac {\mu }{|s|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \phi ={\frac {\rho }{|s|}},\sin \phi ={\frac {\mu }{|s|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2791e2a09d2a14092caaf99d5ef84d40d8cd962d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.83ex; height:5.676ex;" alt="{\displaystyle \cos \phi ={\frac {\rho }{|s|}},\sin \phi ={\frac {\mu }{|s|}}.}"></span></dd></dl> <p>Tables of <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transforms</a> show that the time response of such a system is composed of combinations of the two functions: </p> <dl><dd><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 e^{-\rho t}\sin(\mu t)\quad {\text{and}}\quad e^{-\rho t}\cos(\mu t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ρ</mi> <mi>t</mi> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ρ</mi> <mi>t</mi> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\rho t}\sin(\mu t)\quad {\text{and}}\quad e^{-\rho t}\cos(\mu t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d1c869d70c6782a003c1d28497453fe546776e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.957ex; height:3.009ex;" alt="{\displaystyle e^{-\rho t}\sin(\mu t)\quad {\text{and}}\quad e^{-\rho t}\cos(\mu t),}"></span></dd></dl> <p>which is to say, the solutions are damped oscillations in time. In particular, the unit step response of the system is:<sup id="cite_ref-Kuo_2-0" class="reference"><a href="#cite_note-Kuo-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 S(t)=\left({\frac {A_{0}}{1+\beta A_{0}}}\right)\left(1-e^{-\rho t}\ {\frac {\sin \left(\mu t+\phi \right)}{\sin \phi }}\right)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ρ</mi> <mi>t</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(t)=\left({\frac {A_{0}}{1+\beta A_{0}}}\right)\left(1-e^{-\rho t}\ {\frac {\sin \left(\mu t+\phi \right)}{\sin \phi }}\right)\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28018fc1c4f0ad24bc4fbc26b520536b5d54faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.649ex; height:6.343ex;" alt="{\displaystyle S(t)=\left({\frac {A_{0}}{1+\beta A_{0}}}\right)\left(1-e^{-\rho t}\ {\frac {\sin \left(\mu t+\phi \right)}{\sin \phi }}\right)\ ,}"></span></dd></dl> <p>which simplifies to </p> <dl><dd><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 S(t)=1-e^{-\rho t}\ {\frac {\sin \left(\mu t+\phi \right)}{\sin \phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ρ</mi> <mi>t</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(t)=1-e^{-\rho t}\ {\frac {\sin \left(\mu t+\phi \right)}{\sin \phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8ed70b2602e439498a72117f18b5abbdf847b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.836ex; height:6.176ex;" alt="{\displaystyle S(t)=1-e^{-\rho t}\ {\frac {\sin \left(\mu t+\phi \right)}{\sin \phi }}}"></span></dd></dl> <p>when <i>A</i><sub>0</sub> tends to infinity and the feedback factor <i>β</i> is one. </p><p>Notice that the damping of the response is set by ρ, that is, by the time constants of the open-loop amplifier. In contrast, the frequency of oscillation is set by μ, that is, by the feedback parameter through β<i>A</i><sub>0</sub>. Because ρ is a sum of reciprocals of time constants, it is interesting to notice that ρ is dominated by the <i>shorter</i> of the two. </p> <div class="mw-heading mw-heading4"><h4 id="Results">Results</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=9" title="Edit section: Results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Step_response_for_two-pole_feedback_amplifier.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Step_response_for_two-pole_feedback_amplifier.PNG/350px-Step_response_for_two-pole_feedback_amplifier.PNG" decoding="async" width="350" height="373" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/4f/Step_response_for_two-pole_feedback_amplifier.PNG 1.5x" data-file-width="465" data-file-height="496" /></a><figcaption>Figure 3: Step-response of a linear two-pole feedback amplifier; time is in units of 1/<i>ρ</i>, that is, in terms of the time constants of <i>A</i><sub>OL</sub>; curves are plotted for three values of <i>mu</i> = <i>μ</i>, which is controlled by <i>β</i>.</figcaption></figure> <p>Figure 3 shows the time response to a unit step input for three values of the parameter μ. It can be seen that the frequency of oscillation increases with μ, but the oscillations are contained between the two asymptotes set by the exponentials [ 1 − exp(−<i>ρt</i>) ] and [ 1 + exp(−ρt) ]. These asymptotes are determined by ρ and therefore by the time constants of the open-loop amplifier, independent of feedback. </p><p>The phenomenon of oscillation about the final value is called <b><a href="/wiki/Ringing_(signal)" title="Ringing (signal)">ringing</a></b>. The <b><a href="/wiki/Overshoot_(signal)" title="Overshoot (signal)">overshoot</a></b> is the maximum swing above final value, and clearly increases with μ. Likewise, the <b>undershoot</b> is the minimum swing below final value, again increasing with μ. The <b><a href="/wiki/Settling_time" title="Settling time">settling time</a></b> is the time for departures from final value to sink below some specified level, say 10% of final value. </p><p>The dependence of settling time upon μ is not obvious, and the approximation of a two-pole system probably is not accurate enough to make any real-world conclusions about feedback dependence of settling time. However, the asymptotes [ 1 − exp(−<i>ρt</i>) ] and [ 1 + exp (−<i>ρt</i>) ] clearly impact settling time, and they are controlled by the time constants of the open-loop amplifier, particularly the shorter of the two time constants. That suggests that a specification on settling time must be met by appropriate design of the open-loop amplifier. </p><p>The two major conclusions from this analysis are: </p> <ol><li>Feedback controls the amplitude of oscillation about final value for a given open-loop amplifier and given values of open-loop time constants, τ<sub>1</sub> and τ<sub>2</sub>.</li> <li>The open-loop amplifier decides settling time. It sets the time scale of Figure 3, and the faster the open-loop amplifier, the faster this time scale.</li></ol> <p>As an aside, it may be noted that real-world departures from this linear two-pole model occur due to two major complications: first, real amplifiers have more than two poles, as well as zeros; and second, real amplifiers are nonlinear, so their step response changes with signal amplitude. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Overshoot_control.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Overshoot_control.PNG/300px-Overshoot_control.PNG" decoding="async" width="300" height="421" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Overshoot_control.PNG/450px-Overshoot_control.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Overshoot_control.PNG/600px-Overshoot_control.PNG 2x" data-file-width="645" data-file-height="906" /></a><figcaption>Figure 4: Step response for three values of α. Top: α  = 4; Center: α = 2; Bottom: α = 0.5. As α is reduced the pole separation reduces, and the overshoot increases.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Control_of_overshoot">Control of overshoot</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=10" title="Edit section: Control of overshoot"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>How overshoot may be controlled by appropriate parameter choices is discussed next. </p><p>Using the equations above, the amount of overshoot can be found by differentiating the step response and finding its maximum value. The result for maximum step response <i>S</i><sub>max</sub> is:<sup id="cite_ref-Kuo2_3-0" class="reference"><a href="#cite_note-Kuo2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 S_{\max }=1+\exp \left(-\pi {\frac {\rho }{\mu }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mi>μ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\max }=1+\exp \left(-\pi {\frac {\rho }{\mu }}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/950cabc49d544c9fb6d5ff5437e145f7c528ccdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.816ex; height:6.176ex;" alt="{\displaystyle S_{\max }=1+\exp \left(-\pi {\frac {\rho }{\mu }}\right).}"></span></dd></dl> <p>The final value of the step response is 1, so the exponential is the actual overshoot itself. It is clear the overshoot is zero if <i>μ</i> = 0, which is the condition: </p> <dl><dd><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 {\frac {4\beta A_{0}}{\tau _{1}\tau _{2}}}=\left({\frac {1}{\tau _{1}}}-{\frac {1}{\tau _{2}}}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4\beta A_{0}}{\tau _{1}\tau _{2}}}=\left({\frac {1}{\tau _{1}}}-{\frac {1}{\tau _{2}}}\right)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5570fe41cd2d5e69a92b2e01562b45451768d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.002ex; height:6.509ex;" alt="{\displaystyle {\frac {4\beta A_{0}}{\tau _{1}\tau _{2}}}=\left({\frac {1}{\tau _{1}}}-{\frac {1}{\tau _{2}}}\right)^{2}.}"></span></dd></dl> <p>This quadratic is solved for the ratio of time constants by setting <i>x</i> = (<i>τ</i><sub>1</sub> / <i>τ</i><sub>2</sub>)<sup>1/2</sup> with the result </p> <dl><dd><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 x={\sqrt {\beta A_{0}}}+{\sqrt {\beta A_{0}+1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\sqrt {\beta A_{0}}}+{\sqrt {\beta A_{0}+1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc04c60349c330d2de48cb07db724c2be0ed260f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.824ex; height:3.509ex;" alt="{\displaystyle x={\sqrt {\beta A_{0}}}+{\sqrt {\beta A_{0}+1}}.}"></span></dd></dl> <p>Because β <i>A</i><sub>0</sub> ≫ 1, the 1 in the square root can be dropped, and the result is </p> <dl><dd><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 {\frac {\tau _{1}}{\tau _{2}}}=4\beta A_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\tau _{1}}{\tau _{2}}}=4\beta A_{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74643b0d95e94705bfffc7bf563e1cc737eae401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.944ex; height:5.009ex;" alt="{\displaystyle {\frac {\tau _{1}}{\tau _{2}}}=4\beta A_{0}.}"></span></dd></dl> <p>In words, the first time constant must be much larger than the second. To be more adventurous than a design allowing for no overshoot we can introduce a factor <i>α</i> in the above relation: </p> <dl><dd><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 {\frac {\tau _{1}}{\tau _{2}}}=\alpha \beta A_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>α</mi> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\tau _{1}}{\tau _{2}}}=\alpha \beta A_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b7dba617be0a3d98430002b71cff81001dc36e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.269ex; height:5.009ex;" alt="{\displaystyle {\frac {\tau _{1}}{\tau _{2}}}=\alpha \beta A_{0},}"></span></dd></dl> <p>and let α be set by the amount of overshoot that is acceptable. </p><p>Figure 4 illustrates the procedure. Comparing the top panel (α = 4) with the lower panel (α = 0.5) shows lower values for α increase the rate of response, but increase overshoot. The case α = 2 (center panel) is the <a href="/wiki/Butterworth_filter#Maximal_flatness" title="Butterworth filter"><i>maximally flat</i></a> design that shows no peaking in the <a href="/wiki/Bode_plot" title="Bode plot">Bode gain vs. frequency plot</a>. That design has the <a href="/wiki/Rule_of_thumb" title="Rule of thumb">rule of thumb</a> built-in safety margin to deal with non-ideal realities like multiple poles (or zeros), nonlinearity (signal amplitude dependence) and manufacturing variations, any of which can lead to too much overshoot. The adjustment of the pole separation (that is, setting α) is the subject of <a href="/wiki/Frequency_compensation" title="Frequency compensation">frequency compensation</a>, and one such method is <a href="/wiki/Pole_splitting" title="Pole splitting">pole splitting</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Control_of_settling_time">Control of settling time</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=11" title="Edit section: Control of settling time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The amplitude of ringing in the step response in Figure 3 is governed by the damping factor exp(−<i>ρt</i>). That is, if we specify some acceptable step response deviation from final value, say Δ, that is: </p> <dl><dd><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 S(t)\leq 1+\Delta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>1</mn> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(t)\leq 1+\Delta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d81aee6a1dbc6a37f47875ba4dfe7bdbd3a7c47d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.832ex; height:2.843ex;" alt="{\displaystyle S(t)\leq 1+\Delta ,}"></span></dd></dl> <p>this condition is satisfied regardless of the value of β <i>A</i><sub>OL</sub> provided the time is longer than the settling time, say <i>t</i><sub>S</sub>, given by:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 \Delta =e^{-\rho t_{S}}{\text{ or }}t_{S}={\frac {\ln {\frac {1}{\Delta }}}{\rho }}=\tau _{2}{\frac {2\ln {\frac {1}{\Delta }}}{1+{\frac {\tau _{2}}{\tau _{1}}}}}\approx 2\tau _{2}\ln {\frac {1}{\Delta }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ρ</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi mathvariant="normal">Δ</mi> </mfrac> </mrow> </mrow> <mi>ρ</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi mathvariant="normal">Δ</mi> </mfrac> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>2</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi mathvariant="normal">Δ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =e^{-\rho t_{S}}{\text{ or }}t_{S}={\frac {\ln {\frac {1}{\Delta }}}{\rho }}=\tau _{2}{\frac {2\ln {\frac {1}{\Delta }}}{1+{\frac {\tau _{2}}{\tau _{1}}}}}\approx 2\tau _{2}\ln {\frac {1}{\Delta }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/371f5d9fde697825de239bf1a907346b2a1383d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:48.788ex; height:8.176ex;" alt="{\displaystyle \Delta =e^{-\rho t_{S}}{\text{ or }}t_{S}={\frac {\ln {\frac {1}{\Delta }}}{\rho }}=\tau _{2}{\frac {2\ln {\frac {1}{\Delta }}}{1+{\frac {\tau _{2}}{\tau _{1}}}}}\approx 2\tau _{2}\ln {\frac {1}{\Delta }},}"></span></dd></dl> <p>where the τ<sub>1</sub> ≫ τ<sub>2</sub> is applicable because of the overshoot control condition, which makes <i>τ</i><sub>1</sub> = <i>αβA</i><sub>OL</sub> τ<sub>2</sub>. Often the settling time condition is referred to by saying the settling period is inversely proportional to the unity gain bandwidth, because 1/(2<i>π</i> <i>τ</i><sub>2</sub>) is close to this bandwidth for an amplifier with typical <a href="/wiki/Frequency_compensation#Dominant-pole_compensation" title="Frequency compensation">dominant pole compensation</a>. However, this result is more precise than this <a href="/wiki/Rule_of_thumb" title="Rule of thumb">rule of thumb</a>. As an example of this formula, if <span class="nowrap">Δ = 1/e<sup>4</sup> = 1.8 %,</span> the settling time condition is <i>t</i><sub>S</sub> = 8 <i>τ</i><sub>2</sub>. </p><p>In general, control of overshoot sets the time constant ratio, and settling time <i>t</i><sub>S</sub> sets τ<sub>2</sub>.<sup id="cite_ref-Johns_5-0" class="reference"><a href="#cite_note-Johns-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Sansen_6-0" class="reference"><a href="#cite_note-Sansen-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="System_Identification_using_the_Step_Response:_System_with_two_real_poles">System Identification using the Step Response: System with two real poles</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=12" title="Edit section: System Identification using the Step Response: System with two real poles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:PT2_System_Step-Response_Diagram_with_required_Measurements_(2018).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/PT2_System_Step-Response_Diagram_with_required_Measurements_%282018%29.png/340px-PT2_System_Step-Response_Diagram_with_required_Measurements_%282018%29.png" decoding="async" width="340" height="279" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/PT2_System_Step-Response_Diagram_with_required_Measurements_%282018%29.png/510px-PT2_System_Step-Response_Diagram_with_required_Measurements_%282018%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/PT2_System_Step-Response_Diagram_with_required_Measurements_%282018%29.png/680px-PT2_System_Step-Response_Diagram_with_required_Measurements_%282018%29.png 2x" data-file-width="910" data-file-height="746" /></a><figcaption>Step response of the system with <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 x(t)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3cb65fd2c87c18f1c6377bd108226c96d28631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.24ex; height:2.843ex;" alt="{\displaystyle x(t)=1}"></span>. Measure the significant point <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>, <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 t_{25}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{25}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59fb64b0bb5fe83b079e45ea255fdf9fa215ae37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.343ex;" alt="{\displaystyle t_{25}}"></span>and <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 t_{75}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>75</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{75}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c09d7cb90efa9c74178b704862d74a4a64453a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.343ex;" alt="{\displaystyle t_{75}}"></span>.</figcaption></figure> <p>This method uses significant points of the step response. There is no need to guess tangents to the measured Signal. The equations are derived using numerical simulations, determining some significant ratios and fitting parameters of nonlinear equations. See also.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Here the steps: </p> <ul><li>Measure the system step-response <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 y(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397de1edef5bf2ee15c020f325d7d781a3aa7f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}"></span>of the system with an input step signal <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 x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"></span>.</li> <li>Determine the time-spans <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 t_{25}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{25}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59fb64b0bb5fe83b079e45ea255fdf9fa215ae37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.343ex;" alt="{\displaystyle t_{25}}"></span>and <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 t_{75}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>75</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{75}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c09d7cb90efa9c74178b704862d74a4a64453a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.343ex;" alt="{\displaystyle t_{75}}"></span>where the step response reaches 25% and 75% of the steady state output value.</li> <li>Determine the system steady-state gain <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 k=A_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=A_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7458a7bc388adaf10b649183c908b56aa367002f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.107ex; height:2.509ex;" alt="{\displaystyle k=A_{0}}"></span>with <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 k=\lim _{t\to \infty }{\dfrac {y(t)}{x(t)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\lim _{t\to \infty }{\dfrac {y(t)}{x(t)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188079c4900f81448e9454c83a475a8ef3ace2df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.391ex; height:6.509ex;" alt="{\displaystyle k=\lim _{t\to \infty }{\dfrac {y(t)}{x(t)}}}"></span></li> <li>Calculate <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 r={\dfrac {t_{25}}{t_{75}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>75</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\dfrac {t_{25}}{t_{75}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65606672ad13e4de3319dd32c3bf9355653d16cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.699ex; height:5.509ex;" alt="{\displaystyle r={\dfrac {t_{25}}{t_{75}}}}"></span> <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 P=-18.56075\,r+{\dfrac {0.57311}{r-0.20747}}+4.16423}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>18.56075</mn> <mspace width="thinmathspace" /> <mi>r</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>0.57311</mn> <mrow> <mi>r</mi> <mo>−</mo> <mn>0.20747</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <mn>4.16423</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=-18.56075\,r+{\dfrac {0.57311}{r-0.20747}}+4.16423}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb2a32112b10a0914deee0950dd85b4e7b16671" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:42.521ex; height:5.343ex;" alt="{\displaystyle P=-18.56075\,r+{\dfrac {0.57311}{r-0.20747}}+4.16423}"></span> <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 X=14.2797\,r^{3}-9.3891\,r^{2}+0.25437\,r+1.32148}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mn>14.2797</mn> <mspace width="thinmathspace" /> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>9.3891</mn> <mspace width="thinmathspace" /> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0.25437</mn> <mspace width="thinmathspace" /> <mi>r</mi> <mo>+</mo> <mn>1.32148</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=14.2797\,r^{3}-9.3891\,r^{2}+0.25437\,r+1.32148}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b08b7296afcfb3294a759e7561f989cd11d98659" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:49.339ex; height:2.843ex;" alt="{\displaystyle X=14.2797\,r^{3}-9.3891\,r^{2}+0.25437\,r+1.32148}"></span></li> <li>Determine the two time constants <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 \tau _{2}=T_{2}={\dfrac {t_{75}-t_{25}}{X\,(1+1/P)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>75</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msub> </mrow> <mrow> <mi>X</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{2}=T_{2}={\dfrac {t_{75}-t_{25}}{X\,(1+1/P)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23500b84863b2615c80a68aa91582e102c300977" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.765ex; height:6.009ex;" alt="{\displaystyle \tau _{2}=T_{2}={\dfrac {t_{75}-t_{25}}{X\,(1+1/P)}}}"></span> <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 \tau _{1}=T_{1}={\dfrac {T_{2}}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>P</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{1}=T_{1}={\dfrac {T_{2}}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f8c362b2c41ec1165c21608f773236337b7b2e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.927ex; height:5.176ex;" alt="{\displaystyle \tau _{1}=T_{1}={\dfrac {T_{2}}{P}}}"></span></li> <li>Calculate the transfer function of the identified system within the Laplace-domain <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 G(s)={\dfrac {k}{(1+s\,T_{1})\cdot (1+s\,T_{2})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>k</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>s</mi> <mspace width="thinmathspace" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>s</mi> <mspace width="thinmathspace" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(s)={\dfrac {k}{(1+s\,T_{1})\cdot (1+s\,T_{2})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493eff64301aff704e4e4eb5ae8f867a46875d44" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.743ex; height:6.176ex;" alt="{\displaystyle G(s)={\dfrac {k}{(1+s\,T_{1})\cdot (1+s\,T_{2})}}}"></span></li></ul> <div class="mw-heading mw-heading4"><h4 id="Phase_margin">Phase margin</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=13" title="Edit section: Phase margin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Phase_for_Step_Response.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Phase_for_Step_Response.PNG/280px-Phase_for_Step_Response.PNG" decoding="async" width="280" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Phase_for_Step_Response.PNG/420px-Phase_for_Step_Response.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Phase_for_Step_Response.PNG/560px-Phase_for_Step_Response.PNG 2x" data-file-width="637" data-file-height="555" /></a><figcaption>Figure 5: Bode gain plot to find phase margin; scales are logarithmic, so labeled separations are multiplicative factors. For example, <span class="nowrap"><i>f</i><sub>0 dB</sub> = <i>βA</i><sub>0</sub> × <i>f</i><sub>1</sub>.</span></figcaption></figure> <p>Next, the choice of pole ratio <i>τ</i><sub>1</sub>/<i>τ</i><sub>2</sub> is related to the phase margin of the feedback amplifier.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The procedure outlined in the <a href="/wiki/Bode_plot#Examples_using_Bode_plots" title="Bode plot">Bode plot</a> article is followed. Figure 5 is the Bode gain plot for the two-pole amplifier in the range of frequencies up to the second pole position. The assumption behind Figure 5 is that the frequency <i>f</i><sub>0 dB</sub> lies between the lowest pole at <i>f</i><sub>1</sub> = 1/(2πτ<sub>1</sub>) and the second pole at <i>f</i><sub>2</sub> = 1/(2πτ<sub>2</sub>). As indicated in Figure 5, this condition is satisfied for values of α ≥ 1. </p><p>Using Figure 5 the frequency (denoted by <i>f</i><sub>0 dB</sub>) is found where the loop gain β<i>A</i><sub>0</sub> satisfies the unity gain or 0 dB condition, as defined by: </p> <dl><dd><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 |\beta A_{\text{OL}}(f_{\text{0 db}})|=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>OL</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>0 db</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\beta A_{\text{OL}}(f_{\text{0 db}})|=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180306ba360692670bd858f5c4cb4fbe6e82a618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.226ex; height:2.843ex;" alt="{\displaystyle |\beta A_{\text{OL}}(f_{\text{0 db}})|=1.}"></span></dd></dl> <p>The slope of the downward leg of the gain plot is (20 dB/decade); for every factor of ten increase in frequency, the gain drops by the same factor: </p> <dl><dd><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 f_{\text{0 dB}}=\beta A_{0}f_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>0 dB</mtext> </mrow> </msub> <mo>=</mo> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\text{0 dB}}=\beta A_{0}f_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1ff942a5409fd71f6d400572a17e6da215e977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.92ex; height:2.509ex;" alt="{\displaystyle f_{\text{0 dB}}=\beta A_{0}f_{1}.}"></span></dd></dl> <p>The phase margin is the departure of the phase at <i>f</i><sub>0 dB</sub> from −180°. Thus, the margin is: </p> <dl><dd><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 \phi _{m}=180^{\circ }-\arctan(f_{\text{0 dB}}/f_{1})-\arctan(f_{\text{0 dB}}/f_{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>0 dB</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>0 dB</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{m}=180^{\circ }-\arctan(f_{\text{0 dB}}/f_{1})-\arctan(f_{\text{0 dB}}/f_{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f7bc79cbab74e79ad7a32d11e42823f51a66a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.994ex; height:2.843ex;" alt="{\displaystyle \phi _{m}=180^{\circ }-\arctan(f_{\text{0 dB}}/f_{1})-\arctan(f_{\text{0 dB}}/f_{2}).}"></span></dd></dl> <p>Because <i>f</i><sub>0 dB</sub> / <i>f</i><sub>1</sub> = <i>βA</i><sub>0</sub> ≫ 1, the term in <i>f</i><sub>1</sub> is 90°. That makes the phase margin: </p> <dl><dd><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 {\begin{aligned}\phi _{m}&=90^{\circ }-\arctan(f_{\text{0 dB}}/f_{2})\\&=90^{\circ }-\arctan {\frac {\beta A_{0}f_{1}}{\alpha \beta A_{0}f_{1}}}\\&=90^{\circ }-\arctan {\frac {1}{\alpha }}=\arctan \alpha \,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>0 dB</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>α</mi> <mi>β</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mi>α</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\phi _{m}&=90^{\circ }-\arctan(f_{\text{0 dB}}/f_{2})\\&=90^{\circ }-\arctan {\frac {\beta A_{0}f_{1}}{\alpha \beta A_{0}f_{1}}}\\&=90^{\circ }-\arctan {\frac {1}{\alpha }}=\arctan \alpha \,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf3e3f0dbcca7dbda9a86393aa41aa94231bbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:34.78ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\phi _{m}&=90^{\circ }-\arctan(f_{\text{0 dB}}/f_{2})\\&=90^{\circ }-\arctan {\frac {\beta A_{0}f_{1}}{\alpha \beta A_{0}f_{1}}}\\&=90^{\circ }-\arctan {\frac {1}{\alpha }}=\arctan \alpha \,.\end{aligned}}}"></span></dd></dl> <p>In particular, for case <i>α</i> = 1, <i>φ</i><sub>m</sub> = 45°, and for <i>α</i> = 2, <i>φ</i><sub>m</sub> = 63.4°. Sansen<sup id="cite_ref-Sansen3_10-0" class="reference"><a href="#cite_note-Sansen3-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> recommends <i>α</i> = 3, <i>φ</i><sub>m</sub> = 71.6° as a "good safety position to start with". </p><p>If α is increased by shortening <i>τ</i><sub>2</sub>, the settling time <i>t</i><sub>S</sub> also is shortened. If <i>α</i> is increased by lengthening <i>τ</i><sub>1</sub>, the settling time <i>t</i><sub>S</sub> is little altered. More commonly, both <i>τ</i><sub>1</sub> <i>and</i> <i>τ</i><sub>2</sub> change, for example if the technique of <a href="/wiki/Pole_splitting" title="Pole splitting">pole splitting</a> is used. </p><p>As an aside, for an amplifier with more than two poles, the diagram of Figure 5 still may be made to fit the Bode plots by making <i>f</i><sub>2</sub> a fitting parameter, referred to as an "equivalent second pole" position.<sup id="cite_ref-Palumbo_11-0" class="reference"><a href="#cite_note-Palumbo-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </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=Step_response&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Impulse_response" title="Impulse response">Impulse response</a></li> <li><a href="/wiki/Overshoot_(signal)" title="Overshoot (signal)">Overshoot (signal)</a></li> <li><a href="/wiki/Pole_splitting" title="Pole splitting">Pole splitting</a></li> <li><a href="/wiki/Rise_time" title="Rise time">Rise time</a></li> <li><a href="/wiki/Settling_time" title="Settling time">Settling 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_and_notes">References and notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=15" title="Edit section: References and notes"><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 reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-Shmaliy2007-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Shmaliy2007_1-0">^</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="CITEREFYuriy_Shmaliy2007" class="citation book cs1">Yuriy Shmaliy (2007). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/continuoustimesy00shma"><i>Continuous-Time Systems</i></a></span>. Springer Science & Business Media. p. <a rel="nofollow" class="external text" href="https://archive.org/details/continuoustimesy00shma/page/n61">46</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-6272-8" title="Special:BookSources/978-1-4020-6272-8"><bdi>978-1-4020-6272-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Continuous-Time+Systems&rft.pages=46&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=978-1-4020-6272-8&rft.au=Yuriy+Shmaliy&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcontinuoustimesy00shma&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStep+response" class="Z3988"></span></span> </li> <li id="cite_note-Kuo-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kuo_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenjamin_C_Kuo_&_Golnaraghi_F2003" class="citation book cs1">Benjamin C Kuo & Golnaraghi F (2003). <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-471-13476-7"><i>Automatic control systems</i></a> (Eighth ed.). New York: Wiley. p. 253. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-13476-7" title="Special:BookSources/0-471-13476-7"><bdi>0-471-13476-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Automatic+control+systems&rft.place=New+York&rft.pages=253&rft.edition=Eighth&rft.pub=Wiley&rft.date=2003&rft.isbn=0-471-13476-7&rft.au=Benjamin+C+Kuo+%26+Golnaraghi+F&rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-471-13476-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStep+response" class="Z3988"></span></span> </li> <li id="cite_note-Kuo2-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kuo2_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenjamin_C_Kuo_&_Golnaraghi_F2003" class="citation book cs1">Benjamin C Kuo & Golnaraghi F (2003). <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-471-13476-7"><i>p. 259</i></a>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-13476-7" title="Special:BookSources/0-471-13476-7"><bdi>0-471-13476-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=p.+259&rft.pub=Wiley&rft.date=2003&rft.isbn=0-471-13476-7&rft.au=Benjamin+C+Kuo+%26+Golnaraghi+F&rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-471-13476-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStep+response" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">This estimate is a bit conservative (long) because the factor 1 /sin(φ) in the overshoot contribution to <i>S</i> (<i>t</i>) has been replaced by 1 /sin(<i>φ</i>) ≈ 1.</span> </li> <li id="cite_note-Johns-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Johns_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_A._Johns_&_Martin_K_W1997" class="citation book cs1">David A. Johns & Martin K W (1997). <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-471-14448-7"><i>Analog integrated circuit design</i></a>. New York: Wiley. pp. 234–235. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-14448-7" title="Special:BookSources/0-471-14448-7"><bdi>0-471-14448-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analog+integrated+circuit+design&rft.place=New+York&rft.pages=234-235&rft.pub=Wiley&rft.date=1997&rft.isbn=0-471-14448-7&rft.au=David+A.+Johns+%26+Martin+K+W&rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-471-14448-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStep+response" class="Z3988"></span></span> </li> <li id="cite_note-Sansen-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sansen_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilly_M_C_Sansen2006" class="citation book cs1">Willy M C Sansen (2006). <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-387-25746-2"><i>Analog design essentials</i></a>. Dordrecht, The Netherlands: Springer. p. §0528 p. 163. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-25746-2" title="Special:BookSources/0-387-25746-2"><bdi>0-387-25746-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analog+design+essentials&rft.place=Dordrecht%2C+The+Netherlands&rft.pages=%C2%A70528+p.+163&rft.pub=Springer&rft.date=2006&rft.isbn=0-387-25746-2&rft.au=Willy+M+C+Sansen&rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-387-25746-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStep+response" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">According to Johns and Martin, <i>op. cit.</i>, settling time is significant in <a href="/wiki/Switched_capacitor" title="Switched capacitor">switched-capacitor circuits</a>, for example, where an op amp settling time must be less than half a clock period for sufficiently rapid charge transfer.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://hackaday.io/page/4829-identification-of-a-damped-pt2-system">"Identification of a damped PT2 system | Hackaday.io"</a>. <i>hackaday.io</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-08-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=hackaday.io&rft.atitle=Identification+of+a+damped+PT2+system+%7C+Hackaday.io&rft_id=https%3A%2F%2Fhackaday.io%2Fpage%2F4829-identification-of-a-damped-pt2-system&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStep+response" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">The gain margin of the amplifier cannot be found using a two-pole model, because gain margin requires determination of the frequency <i>f</i><sub>180</sub> where the gain flips sign, and this never happens in a two-pole system. If we know <i>f</i><sub>180</sub> for the amplifier at hand, the gain margin can be found approximately, but <i>f</i><sub>180</sub> then depends on the third and higher pole positions, as does the gain margin, unlike the estimate of phase margin, which is a two-pole estimate.</span> </li> <li id="cite_note-Sansen3-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sansen3_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilly_M_C_Sansen2006" class="citation book cs1">Willy M C Sansen (2006-11-30). <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-387-25746-2"><i>§0526 p. 162</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-25746-2" title="Special:BookSources/0-387-25746-2"><bdi>0-387-25746-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C2%A70526+p.+162&rft.pub=Springer&rft.date=2006-11-30&rft.isbn=0-387-25746-2&rft.au=Willy+M+C+Sansen&rft_id=http%3A%2F%2Fworldcat.org%2Fisbn%2F0-387-25746-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStep+response" class="Z3988"></span></span> </li> <li id="cite_note-Palumbo-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Palumbo_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGaetano_Palumbo_&_Pennisi_S2002" class="citation book cs1">Gaetano Palumbo & Pennisi S (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Xb0W1VsQFe0C&q=%22equivalent+two-pole+amplifier%22&pg=PA98"><i>Feedback amplifiers: theory and design</i></a>. Boston/Dordrecht/London: Kluwer Academic Press. pp. § 4.4 pp. 97–98. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-7643-9" title="Special:BookSources/0-7923-7643-9"><bdi>0-7923-7643-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feedback+amplifiers%3A+theory+and+design&rft.place=Boston%2FDordrecht%2FLondon&rft.pages=%C2%A7+4.4+pp.+97-98&rft.pub=Kluwer+Academic+Press&rft.date=2002&rft.isbn=0-7923-7643-9&rft.au=Gaetano+Palumbo+%26+Pennisi+S&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXb0W1VsQFe0C%26q%3D%2522equivalent%2Btwo-pole%2Bamplifier%2522%26pg%3DPA98&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStep+response" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Step_response&action=edit&section=16" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li>Robert I. Demrow <i>Settling time of operational amplifiers</i> <a rel="nofollow" class="external autonumber" href="http://www.analog.com/static/imported-files/application_notes/466359863287538299597392756AN359.pdf">[1]</a></li> <li>Cezmi Kayabasi <i>Settling time measurement techniques achieving high precision at high speeds</i> <a rel="nofollow" class="external autonumber" href="http://www.wpi.edu/Pubs/ETD/Available/etd-050505-140358/unrestricted/ckayabasi.pdf">[2]</a></li> <li>Vladimir Igorevic Arnol'd "Ordinary differential equations", various editions from MIT Press and from Springer Verlag, chapter 1 "Fundamental concepts"</li></ul> </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=Step_response&action=edit&section=17" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://bcs.wiley.com/he-bcs/Books?action=resource&bcsId=2357&itemId=0471134767&resourceId=5596">Kuo power point slides; Chapter 7 especially</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist 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