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Bode plot

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In <a href="Electrical_engineering" title="Electrical engineering">electrical engineering</a> and <a href="Control_theory" title="Control theory">control theory</a>, a Bode plot is a <a href="Plot_(graphics)" title="Plot (graphics)">graph</a> of the <a href="Frequency_response" title="Frequency response">frequency response</a> of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude (usually in <a href="Decibel" title="Decibel">decibels</a>) of the frequency response, and a Bode phase plot, expressing the <a href="Phase_(waves)" title="Phase (waves)">phase shift</a>.

As originally conceived by <a href="Hendrik_Wade_Bode" title="Hendrik Wade Bode">Hendrik Wade Bode</a> in the 1930s, the plot is an <a href="Asymptotic" class="mw-redirect" title="Asymptotic">asymptotic</a> <a href="Approximation" title="Approximation">approximation</a> of the frequency response, <a href="Piecewise_linear_function" title="Piecewise linear function">using straight line segments</a>.<a href="#cite_note-Yarlagadda2010-1">[1]</a>

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

Among his several important contributions to <a href="Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">circuit theory</a> and <a href="Control_theory" title="Control theory">control theory</a>, engineer <a href="Hendrik_Wade_Bode" title="Hendrik Wade Bode">Hendrik Wade Bode</a>, while working at <a href="Bell_Labs" title="Bell Labs">Bell Labs</a> in the 1930s, devised a simple but accurate method for graphing <a href="Gain_(electronics)" title="Gain (electronics)">gain</a> and phase-shift plots. These bear his name, Bode gain plot and Bode phase plot. "Bode" is often pronounced in English as /ˈboʊdi/ BOH-dee, whereas in Dutch it is usually [ˈboːdə], closer to English /ˈboʊdə/ BOH-də, which is preferred by his family, but less common among researchers.<a href="#cite_note-Van_Valkenburg-2">[2]</a><a href="#cite_note-3">[3]</a>

Bode was faced with the problem of designing stable <a href="Amplifier" title="Amplifier">amplifiers</a> with <a href="Feedback" title="Feedback">feedback</a> for use in telephone networks. He developed the graphical design technique of the Bode plots to show the <a href="Gain_margin" class="mw-redirect" title="Gain margin">gain margin</a> and <a href="Phase_margin" title="Phase margin">phase margin</a> required to maintain stability under variations in circuit characteristics caused during manufacture or during operation.<a href="#cite_note-4">[4]</a> The principles developed were applied to design problems of <a href="Servomechanism" title="Servomechanism">servomechanisms</a> and other feedback control systems. The Bode plot is an example of analysis in the <a href="Frequency_domain" title="Frequency domain">frequency domain</a>.

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

The Bode plot for a <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">linear, time-invariant</a> system with <a href="Transfer_function" title="Transfer function">transfer function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)}">

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<annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation>

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</math><img src="./f1b91390324fc9c33ec00fe57e3924ad7118cc1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> being the complex frequency in the <a href="Laplace_domain" class="mw-redirect" title="Laplace domain">Laplace domain</a>) consists of a magnitude plot and a phase plot.

The Bode magnitude plot is the graph of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H(s=j\omega )|}">

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<annotation encoding="application/x-tex">{\displaystyle |H(s=j\omega )|}</annotation>

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</math><img src="./7bf64778fb62ea4a075d779e54a4b4f3c9f6c86d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle |H(s=j\omega )|}" loading="lazy"> of frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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

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</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> being the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>). The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">-axis of the magnitude plot is logarithmic and the magnitude is given in <a href="Decibel" title="Decibel">decibels</a>, i.e., a value for the magnitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H|}">

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

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</math><img src="./31c3c97e64ad558dcc09e9ff232ee0f4d447bac1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.357ex; height:2.843ex;" alt="{\displaystyle |H|}" loading="lazy"> is plotted on the axis at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 20\log _{10}|H|}">

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<annotation encoding="application/x-tex">{\displaystyle 20\log _{10}|H|}</annotation>

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The Bode phase plot is the graph of the <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">phase</a>, commonly expressed in degrees, of the <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">argument function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arg \left(H(s=j\omega )\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \arg \left(H(s=j\omega )\right)}</annotation>

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</math><img src="./8d112a919c55d770d5f8e9672a8fe5c1b231b46f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.512ex; height:2.843ex;" alt="{\displaystyle \arg \left(H(s=j\omega )\right)}" loading="lazy"> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">. The phase is plotted on the same logarithmic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">-axis as the magnitude plot, but the value for the phase is plotted on a linear vertical axis.

<div class="mw-heading mw-heading2"><h2 id="Frequency_response">Frequency response</h2></div>

This section illustrates that a Bode plot is a visualization of the frequency response of a system.

Consider a <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">linear, time-invariant</a> system with transfer function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)}">

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<annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation>

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</math><img src="./f1b91390324fc9c33ec00fe57e3924ad7118cc1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}" loading="lazy">. Assume that the system is subject to a sinusoidal input with frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle u(t)=\sin(\omega t),}</annotation>

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</math><img src="./e67a95bb6470fac2c993c92fdacc349c1d0b66a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.674ex; height:2.843ex;" alt="{\displaystyle u(t)=\sin(\omega t),}" loading="lazy"></dd></dl>

that is applied persistently, i.e. from a time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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

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</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"> to a time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy">. The response will be of the form

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<annotation encoding="application/x-tex">{\displaystyle y(t)=y_{0}\sin(\omega t+\varphi ),}</annotation>

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</math><img src="./f07bb0c779f964d87b4e6e8d1997298f4db4622a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.441ex; height:2.843ex;" alt="{\displaystyle y(t)=y_{0}\sin(\omega t+\varphi ),}" loading="lazy"></dd></dl>

i.e., also a sinusoidal signal with amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{0}}</annotation>

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</math><img src="./6d943dbbb0b56ca750c4d62c5b54b4ae29a773da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" loading="lazy"> shifted by a phase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> with respect to the input.

It can be shown<a href="#cite_note-multivar_fb_control-5">[5]</a> that the magnitude of the response is

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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}=|H(\mathrm {j} \omega )|}">

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<annotation encoding="application/x-tex">{\displaystyle y_{0}=|H(\mathrm {j} \omega )|}</annotation>

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</math><img src="./b4716d842877f57bc23bd17dd60753f086435369.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.616ex; height:2.843ex;" alt="{\displaystyle y_{0}=|H(\mathrm {j} \omega )|}" loading="lazy"></td> <td></td> <td class="nowrap">1</td></tr></tbody></table>

and that the phase shift is

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<annotation encoding="application/x-tex">{\displaystyle \varphi =\arg H(\mathrm {j} \omega ).}</annotation>

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</math><img src="./961f65b88002da697740f89fddfbe38ee8a1bd4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.92ex; height:2.843ex;" alt="{\displaystyle \varphi =\arg H(\mathrm {j} \omega ).}" loading="lazy"></td> <td></td> <td class="nowrap">2</td></tr></tbody></table>

In summary, subjected to an input with frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">, the system responds at the same frequency with an output that is amplified by a factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H(\mathrm {j} \omega )|}">

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<annotation encoding="application/x-tex">{\displaystyle |H(\mathrm {j} \omega )|}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \arg H(\mathrm {j} \omega )}</annotation>

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</math><img src="./52a45d56c8f4a7dd9d179533962e6cd9aeb2133a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \arg H(\mathrm {j} \omega )}" loading="lazy">. These quantities, thus, characterize the frequency response and are shown in the Bode plot.

<div class="mw-heading mw-heading2"><h2 id="Rules_for_handmade_Bode_plot">Rules for handmade Bode plot</h2></div>

For many practical problems, the detailed Bode plots can be approximated with straight-line segments that are <a href="Asymptote" title="Asymptote">asymptotes</a> of the precise response. The effect of each of the terms of a multiple element <a href="Transfer_function" title="Transfer function">transfer function</a> can be approximated by a set of straight lines on a Bode plot. This allows a graphical solution of the overall frequency response function. Before widespread availability of digital computers, graphical methods were extensively used to reduce the need for tedious calculation; a graphical solution could be used to identify feasible ranges of parameters for a new design.

The premise of a Bode plot is that one can consider the log of a function in the form

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<annotation encoding="application/x-tex">{\displaystyle f(x)=A\prod (x-c_{n})^{a_{n}}}</annotation>

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</math><img src="./f78ea3d99ae5d0a963ddab63f715157d5cde5424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.887ex; height:3.843ex;" alt="{\displaystyle f(x)=A\prod (x-c_{n})^{a_{n}}}" loading="lazy"></dd></dl>

as a sum of the logs of its <a href="Zeros_and_poles" title="Zeros and poles">zeros and poles</a>:

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<annotation encoding="application/x-tex">{\displaystyle \log(f(x))=\log(A)+\sum a_{n}\log(x-c_{n}).}</annotation>

</semantics>

</math><img src="./bbbfa63f1c2bb772c62ff2a4ff1b996a80f005ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:40.062ex; height:3.843ex;" alt="{\displaystyle \log(f(x))=\log(A)+\sum a_{n}\log(x-c_{n}).}" loading="lazy"></dd></dl>

This idea is used explicitly in the method for drawing phase diagrams. The method for drawing amplitude plots implicitly uses this idea, but since the log of the amplitude of each pole or zero always starts at zero and only has one asymptote change (the straight lines), the method can be simplified.

<div class="mw-heading mw-heading3"><h3 id="Straight-line_amplitude_plot">Straight-line amplitude plot</h3></div>

Amplitude decibels is usually done using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{dB}}=20\log _{10}(X)}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{dB}}=20\log _{10}(X)}</annotation>

</semantics>

</math><img src="./525c73fb95563fda4005183366c762535855a8b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.386ex; height:2.843ex;" alt="{\displaystyle {\text{dB}}=20\log _{10}(X)}" loading="lazy"> to define decibels. Given a transfer function in the form

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(s)=A\prod {\frac {(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}},}</annotation>

</semantics>

</math><img src="./91c245819beaa147615c657f7c09a9ef9c9b3d1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.387ex; height:6.509ex;" alt="{\displaystyle H(s)=A\prod {\frac {(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2c5fbb0c89590b028eba7239a8803fd0cd2e698e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.009ex;" alt="{\displaystyle y_{n}}" loading="lazy"> are constants, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\mathrm {j} \omega }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=\mathrm {j} \omega }</annotation>

</semantics>

</math><img src="./e196003429bb0d04d835ef38f49ff955f19cc7ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.347ex; height:2.509ex;" alt="{\displaystyle s=\mathrm {j} \omega }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n},b_{n}>0}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{n},b_{n}>0}</annotation>

</semantics>

</math><img src="./b6249f620970dc4a293000bfd4dd71ea9b40ac3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.959ex; height:2.509ex;" alt="{\displaystyle a_{n},b_{n}>0}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> is the transfer function:

<ul><li>At every value of s where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =x_{n}}</annotation>

</semantics>

</math><img src="./d48fe4daec794d6cc2332dd4ead18c1a8a0d2678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.092ex; height:2.009ex;" alt="{\displaystyle \omega =x_{n}}" loading="lazy"> (a zero), increase the slope of the line by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 20a_{n}\ {\text{dB}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 20a_{n}\ {\text{dB}}}</annotation>

</semantics>

</math><img src="./7daed3eeca5d602d60b69dfad85b8d73a25f79ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.292ex; height:2.509ex;" alt="{\displaystyle 20a_{n}\ {\text{dB}}}" loading="lazy"> per <a href="Decade_(log_scale)" title="Decade (log scale)">decade</a>.</li>

<li>At every value of s where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =y_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =y_{n}}</annotation>

</semantics>

</math><img src="./dded46b7d0cced3826e96542763b7c1522b1ca9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.902ex; height:2.009ex;" alt="{\displaystyle \omega =y_{n}}" loading="lazy"> (a pole), decrease the slope of the line by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 20b_{n}\ {\text{dB}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 20b_{n}\ {\text{dB}}}</annotation>

</semantics>

</math><img src="./d8a4707758b8b60cdd2aa2b97efce8fbd98d7fb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.06ex; height:2.509ex;" alt="{\displaystyle 20b_{n}\ {\text{dB}}}" loading="lazy"> per decade.</li>

<li>The initial value of the graph depends on the boundaries. The initial point is found by putting the initial angular frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> into the function and finding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H(\mathrm {j} \omega )|}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |H(\mathrm {j} \omega )|}</annotation>

</semantics>

</math><img src="./1feca4b7e907b63822ce794ca0d3eb1ae9c9529a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.324ex; height:2.843ex;" alt="{\displaystyle |H(\mathrm {j} \omega )|}" loading="lazy">.</li>

<li>The initial slope of the function at the initial value depends on the number and order of zeros and poles that are at values below the initial value, and is found using the first two rules.</li></ul>

To handle irreducible 2nd-order polynomials, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c}</annotation>

</semantics>

</math><img src="./126c6935d3dd9f1c1da0c388ca2799be4f6f237c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.629ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c}" loading="lazy"> can, in many cases, be approximated as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\sqrt {a}}x+{\sqrt {c}})^{2}}">

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\sqrt {a}}x+{\sqrt {c}})^{2}}</annotation>

</semantics>

</math><img src="./a7026e7e209307791a8da1813d617dae744f8672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.142ex; height:3.343ex;" alt="{\displaystyle ({\sqrt {a}}x+{\sqrt {c}})^{2}}" loading="lazy">.

Note that zeros and poles happen when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> is equal to a certain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2c5fbb0c89590b028eba7239a8803fd0cd2e698e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.009ex;" alt="{\displaystyle y_{n}}" loading="lazy">. This is because the function in question is the magnitude of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(\mathrm {j} \omega )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(\mathrm {j} \omega )}</annotation>

</semantics>

</math><img src="./d62d6b957b52438e6bcd067e1968fe9a1df15e4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.031ex; height:2.843ex;" alt="{\displaystyle H(\mathrm {j} \omega )}" loading="lazy">, and since it is a complex function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H(\mathrm {j} \omega )|={\sqrt {H\cdot H^{*}}}}">

</mrow>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |H(\mathrm {j} \omega )|={\sqrt {H\cdot H^{*}}}}</annotation>

</semantics>

</math><img src="./440d9f2be38cc7576c356cf641e0264d923a6014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.259ex; height:3.176ex;" alt="{\displaystyle |H(\mathrm {j} \omega )|={\sqrt {H\cdot H^{*}}}}" loading="lazy">. Thus at any place where there is a zero or pole involving the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s+x_{n})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (s+x_{n})}</annotation>

</semantics>

</math><img src="./c907b776ab26692297a4f895eb9115401c5f1d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.288ex; height:2.843ex;" alt="{\displaystyle (s+x_{n})}" loading="lazy">, the magnitude of that term is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {(x_{n}+\mathrm {j} \omega )(x_{n}-\mathrm {j} \omega )}}={\sqrt {x_{n}^{2}+\omega ^{2}}}}">

<msqrt>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msqrt>

</mrow>

<msqrt>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {(x_{n}+\mathrm {j} \omega )(x_{n}-\mathrm {j} \omega )}}={\sqrt {x_{n}^{2}+\omega ^{2}}}}</annotation>

</semantics>

</math><img src="./3875d0f7803a10d21b8b3aa7425feca671a27f62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.346ex; height:5.176ex;" alt="{\displaystyle {\sqrt {(x_{n}+\mathrm {j} \omega )(x_{n}-\mathrm {j} \omega )}}={\sqrt {x_{n}^{2}+\omega ^{2}}}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Corrected_amplitude_plot">Corrected amplitude plot</h3></div>

To correct a straight-line amplitude plot:

<ul><li>At every zero, put a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3a_{n}\ {\text{dB}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3a_{n}\ {\text{dB}}}</annotation>

</semantics>

</math><img src="./0600a535d0c70df1405cc5bac5b1f013b440fdb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.129ex; height:2.509ex;" alt="{\displaystyle 3a_{n}\ {\text{dB}}}" loading="lazy"> above the line.</li>

<li>At every pole, put a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3b_{n}\ {\text{dB}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3b_{n}\ {\text{dB}}}</annotation>

</semantics>

</math><img src="./3c3fd02a2d3bf20788f727d47695e41978e35f7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.897ex; height:2.509ex;" alt="{\displaystyle 3b_{n}\ {\text{dB}}}" loading="lazy"> below the line.</li>

<li>Draw a smooth curve through those points using the straight lines as asymptotes (lines which the curve approaches).</li></ul>

Note that this correction method does not incorporate how to handle complex values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2c5fbb0c89590b028eba7239a8803fd0cd2e698e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.009ex;" alt="{\displaystyle y_{n}}" loading="lazy">. In the case of an <a href="Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomial</a>, the best way to correct the plot is to actually calculate the magnitude of the transfer function at the pole or zero corresponding to the irreducible polynomial, and put that dot over or under the line at that pole or zero.

<div class="mw-heading mw-heading3"><h3 id="Straight-line_phase_plot">Straight-line phase plot</h3></div>

Given a transfer function in the same form as above,

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(s)=A\prod {\frac {(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}},}</annotation>

</semantics>

</math><img src="./91c245819beaa147615c657f7c09a9ef9c9b3d1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.387ex; height:6.509ex;" alt="{\displaystyle H(s)=A\prod {\frac {(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}},}" loading="lazy"></dd></dl>

the idea is to draw separate plots for each pole and zero, then add them up. The actual phase curve is given by

<mi>arctan</mi>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (s)=-\arctan {\frac {\operatorname {Im} [H(s)]}{\operatorname {Re} [H(s)]}}.}</annotation>

</semantics>

</math><img src="./710a251acf7f8aece45a2fa9ffefa6e1cb20fbcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.082ex; height:6.509ex;" alt="{\displaystyle \varphi (s)=-\arctan {\frac {\operatorname {Im} [H(s)]}{\operatorname {Re} [H(s)]}}.}" loading="lazy"></dd></dl>

To draw the phase plot, for each pole and zero:

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is positive, start line (with zero slope) at 0°.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is negative, start line (with zero slope) at −180°.</li>

<li>If the sum of the number of unstable zeros and poles is odd, add 180° to that basis.</li>

<li>At every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =|x_{n}|}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =|x_{n}|}</annotation>

</semantics>

</math><img src="./40194eaaac889ffd4402af94e60ba9ea9e0c9e23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle \omega =|x_{n}|}" loading="lazy"> (for stable zeros <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\operatorname {Re} (z)<0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\operatorname {Re} (z)<0}</annotation>

</semantics>

</math><img src="./1e115a5e7f09519dfefb0610345d624d0773916d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.097ex; height:2.843ex;" alt="{\displaystyle -\operatorname {Re} (z)<0}" loading="lazy">), increase the slope by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 45a_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 45a_{n}}</annotation>

</semantics>

</math><img src="./1cfc2704937442c28af3993cb4e4b34ec0f8a773.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.773ex; height:2.509ex;" alt="{\displaystyle 45a_{n}}" loading="lazy"> degrees per decade, beginning one decade before <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =|x_{n}|}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =|x_{n}|}</annotation>

</semantics>

</math><img src="./40194eaaac889ffd4402af94e60ba9ea9e0c9e23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle \omega =|x_{n}|}" loading="lazy"> (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{n}|/10}">

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</math><img src="./754c46e346af45e23d949503cbb728fbc00739e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.329ex; height:2.843ex;" alt="{\displaystyle |x_{n}|/10}" loading="lazy">).</li>

<li>At every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =|y_{n}|}">

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</math><img src="./678771b99d245655f426e0a057214385001dd3dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.196ex; height:2.843ex;" alt="{\displaystyle \omega =|y_{n}|}" loading="lazy"> (for stable poles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\operatorname {Re} (p)<0}">

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<annotation encoding="application/x-tex">{\displaystyle -\operatorname {Re} (p)<0}</annotation>

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</math><img src="./a6af81bef0ef095e4a32e5ae3fb7cd40ed3aa608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.178ex; height:2.843ex;" alt="{\displaystyle -\operatorname {Re} (p)<0}" loading="lazy">), decrease the slope by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 45b_{n}}">

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</math><img src="./678771b99d245655f426e0a057214385001dd3dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.196ex; height:2.843ex;" alt="{\displaystyle \omega =|y_{n}|}" loading="lazy"> (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |y_{n}|/10}">

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<annotation encoding="application/x-tex">{\displaystyle |y_{n}|/10}</annotation>

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</math><img src="./2597ee7b7a8be8a472f7cfe2ee26d10ab42a25da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.139ex; height:2.843ex;" alt="{\displaystyle |y_{n}|/10}" loading="lazy">).</li>

<li>"Unstable" (right half-plane) poles and zeros (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (s)>0}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (s)>0}</annotation>

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</math><img src="./ce0992e3b29be0b1bb2feec2e4682f43fe61c38b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (s)>0}" loading="lazy">) have opposite behavior.</li>

<li>Flatten the slope again when the phase has changed by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90a_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle 90a_{n}}</annotation>

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</math><img src="./25324eb545809489dfa8bfcbf47deb816429b333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.773ex; height:2.509ex;" alt="{\displaystyle 90a_{n}}" loading="lazy"> degrees (for a zero) or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90b_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle 90b_{n}}</annotation>

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</math><img src="./5768aa08450aebd2ac46a50aaf9b6eae6d92b977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle 90b_{n}}" loading="lazy"> degrees (for a pole).</li>

<li>After plotting one line for each pole or zero, add the lines together to obtain the final phase plot; that is, the final phase plot is the superposition of each earlier phase plot.</li></ul>

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

To create a straight-line plot for a first-order (one-pole) low-pass filter, one considers the normalized form of the transfer function in terms of the <a href="Angular_frequency" title="Angular frequency">angular frequency</a>:

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<annotation encoding="application/x-tex">{\displaystyle H_{\text{lp}}(\mathrm {j} \omega )={\frac {1}{1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}}}.}</annotation>

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</math><img src="./180c83d91fc57550d46f2b1c53c55cdb8e10c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.413ex; height:6.509ex;" alt="{\displaystyle H_{\text{lp}}(\mathrm {j} \omega )={\frac {1}{1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}}}.}" loading="lazy"></dd></dl>

The Bode plot is shown in Figure 1(b) above, and construction of the straight-line approximation is discussed next.

<div class="mw-heading mw-heading3"><h3 id="Magnitude_plot">Magnitude plot</h3></div>

The magnitude (in <a href="Decibel" title="Decibel">decibels</a>) of the transfer function above (normalized and converted to angular-frequency form), given by the decibel gain expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\text{vdB}}}">

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

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</math><img src="./b24c3cc4f10df39100c5ef96c9385aca3771e034.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.921ex; height:2.509ex;" alt="{\displaystyle A_{\text{vdB}}}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A_{\text{vdB}}&=20\log |H_{\text{lp}}(\mathrm {j} \omega )|\\&=20\log {\frac {1}{\left|1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}\right|}}\\&=-20\log \left|1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}\right|\\&=-10\log \left(1+{\frac {\omega ^{2}}{\omega _{\text{c}}^{2}}}\right).\end{aligned}}}</annotation>

</semantics>

</math><img src="./2140d738df3adb93d9c6d71ccf7211dfadca85e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:27.67ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}A_{\text{vdB}}&=20\log |H_{\text{lp}}(\mathrm {j} \omega )|\\&=20\log {\frac {1}{\left|1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}\right|}}\\&=-20\log \left|1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}\right|\\&=-10\log \left(1+{\frac {\omega ^{2}}{\omega _{\text{c}}^{2}}}\right).\end{aligned}}}" loading="lazy"></dd></dl>

Then plotted versus input frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> on a logarithmic scale, can be approximated by two lines, forming the asymptotic (approximate) magnitude Bode plot of the transfer function:

<ul><li>The first line for angular frequencies below <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\text{c}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{\text{c}}}</annotation>

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</math><img src="./3cca75b1b68b41fbc378508edf81f6bb5524194a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.408ex; height:2.009ex;" alt="{\displaystyle \omega _{\text{c}}}" loading="lazy"> is a horizontal line at 0 dB, since at low frequencies the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega /\omega _{\text{c}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega /\omega _{\text{c}}}</annotation>

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</math><img src="./602c0d8bd83563cc277948b2f73fe72c033579e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.016ex; height:2.843ex;" alt="{\displaystyle \omega /\omega _{\text{c}}}" loading="lazy"> term is small and can be neglected, making the decibel gain equation above equal to zero.</li>

<li>The second line for angular frequencies above <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\text{c}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{\text{c}}}</annotation>

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</math><img src="./3cca75b1b68b41fbc378508edf81f6bb5524194a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.408ex; height:2.009ex;" alt="{\displaystyle \omega _{\text{c}}}" loading="lazy"> is a line with a slope of −20 dB per decade, since at high frequencies the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega /\omega _{\text{c}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega /\omega _{\text{c}}}</annotation>

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</math><img src="./602c0d8bd83563cc277948b2f73fe72c033579e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.016ex; height:2.843ex;" alt="{\displaystyle \omega /\omega _{\text{c}}}" loading="lazy"> term dominates, and the decibel gain expression above simplifies to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -20\log(\omega /\omega _{\text{c}})}">

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<annotation encoding="application/x-tex">{\displaystyle -20\log(\omega /\omega _{\text{c}})}</annotation>

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</math><img src="./23c67e0f8388a41a866c5600e1a824d01958c683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.318ex; height:2.843ex;" alt="{\displaystyle -20\log(\omega /\omega _{\text{c}})}" loading="lazy">, which is a straight line with a slope of −20 dB per decade.</li></ul>

These two lines meet at the <a href="Corner_frequency" class="mw-redirect" title="Corner frequency">corner frequency</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\text{c}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{\text{c}}}</annotation>

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</math><img src="./3cca75b1b68b41fbc378508edf81f6bb5524194a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.408ex; height:2.009ex;" alt="{\displaystyle \omega _{\text{c}}}" loading="lazy">. From the plot, it can be seen that for frequencies well below the corner frequency, the circuit has an attenuation of 0 dB, corresponding to a unity pass-band gain, i.e. the amplitude of the filter output equals the amplitude of the input. Frequencies above the corner frequency are attenuated – the higher the frequency, the higher the <a href="Attenuation" title="Attenuation">attenuation</a>.

<div class="mw-heading mw-heading3"><h3 id="Phase_plot">Phase plot</h3></div>

The phase Bode plot is obtained by plotting the phase angle of the transfer function given by

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<annotation encoding="application/x-tex">{\displaystyle \arg H_{\text{lp}}(\mathrm {j} \omega )=-\tan ^{-1}{\frac {\omega }{\omega _{\text{c}}}}}</annotation>

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</math><img src="./92e93df207054bb0723683a805987d67a13162a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.743ex; height:5.009ex;" alt="{\displaystyle \arg H_{\text{lp}}(\mathrm {j} \omega )=-\tan ^{-1}{\frac {\omega }{\omega _{\text{c}}}}}" loading="lazy"></dd></dl>

versus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\text{c}}}">

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</math><img src="./3cca75b1b68b41fbc378508edf81f6bb5524194a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.408ex; height:2.009ex;" alt="{\displaystyle \omega _{\text{c}}}" loading="lazy"> are the input and cutoff angular frequencies respectively. For input frequencies much lower than corner, the ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega /\omega _{\text{c}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega /\omega _{\text{c}}}</annotation>

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</math><img src="./602c0d8bd83563cc277948b2f73fe72c033579e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.016ex; height:2.843ex;" alt="{\displaystyle \omega /\omega _{\text{c}}}" loading="lazy"> is small, and therefore the phase angle is close to zero. As the ratio increases, the absolute value of the phase increases and becomes −45° when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\omega _{\text{c}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega =\omega _{\text{c}}}</annotation>

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</math><img src="./b1a4d72545c6934ff1914f4f7a6bc49ec5a07101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.952ex; height:2.009ex;" alt="{\displaystyle \omega =\omega _{\text{c}}}" loading="lazy">. As the ratio increases for input frequencies much greater than the corner frequency, the phase angle asymptotically approaches −90°. The frequency scale for the phase plot is logarithmic.

<div class="mw-heading mw-heading3"><h3 id="Normalized_plot">Normalized plot</h3></div>

The horizontal frequency axis, in both the magnitude and phase plots, can be replaced by the normalized (nondimensional) frequency ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega /\omega _{\text{c}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega /\omega _{\text{c}}}</annotation>

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</math><img src="./602c0d8bd83563cc277948b2f73fe72c033579e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.016ex; height:2.843ex;" alt="{\displaystyle \omega /\omega _{\text{c}}}" loading="lazy">. In such a case the plot is said to be normalized, and units of the frequencies are no longer used, since all input frequencies are now expressed as multiples of the cutoff frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\text{c}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{\text{c}}}</annotation>

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</math><img src="./3cca75b1b68b41fbc378508edf81f6bb5524194a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.408ex; height:2.009ex;" alt="{\displaystyle \omega _{\text{c}}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="An_example_with_zero_and_pole">An example with zero and pole</h2></div>

Figures 2–5 further illustrate construction of Bode plots. This example with both a pole and a zero shows how to use superposition. To begin, the components are presented separately.

Figure 2 shows the Bode magnitude plot for a zero and a low-pass pole, and compares the two with the Bode straight line plots. The straight-line plots are horizontal up to the pole (zero) location and then drop (rise) at 20 dB/decade. The second Figure 3 does the same for the phase. The phase plots are horizontal up to a frequency factor of ten below the pole (zero) location and then drop (rise) at 45°/decade until the frequency is ten times higher than the pole (zero) location. The plots then are again horizontal at higher frequencies at a final, total phase change of 90°.

Figure 4 and Figure 5 show how superposition (simple addition) of a pole and zero plot is done. The Bode straight line plots again are compared with the exact plots. The zero has been moved to higher frequency than the pole to make a more interesting example. Notice in Figure 4 that the 20 dB/decade drop of the pole is arrested by the 20 dB/decade rise of the zero resulting in a horizontal magnitude plot for frequencies above the zero location. Notice in Figure 5 in the phase plot that the straight-line approximation is pretty approximate in the region where both pole and zero affect the phase. Notice also in Figure 5 that the range of frequencies where the phase changes in the straight line plot is limited to frequencies a factor of ten above and below the pole (zero) location. Where the phase of the pole and the zero both are present, the straight-line phase plot is horizontal because the 45°/decade drop of the pole is arrested by the overlapping 45°/decade rise of the zero in the limited range of frequencies where both are active contributors to the phase.

<li class="gallerycaption">Example with pole and zero</li>

<div class="gallerytext">Figure 2: Bode magnitude plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots</div>

</li>

<div class="gallerytext">Figure 3: Bode phase plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots</div>

</li>

<div class="gallerytext">Figure 4: Bode magnitude plot for pole-zero combination; the location of the zero is ten times higher than in Figures 2 and 3; curves labeled "Bode" are the straight-line Bode plots</div>

</li>

<div class="gallerytext">Figure 5: Bode phase plot for pole-zero combination; the location of the zero is ten times higher than in Figures 2 and 3; curves labeled "Bode" are the straight-line Bode plots</div>

</li>

</ul>

<div class="mw-heading mw-heading2"><h2 id="Gain_margin_and_phase_margin">Gain margin and phase margin</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Phase_margin" title="Phase margin">Phase margin</a></div>

Bode plots are used to assess the stability of <a href="Negative-feedback_amplifier" title="Negative-feedback amplifier">negative-feedback amplifiers</a> by finding the gain and <a href="Phase_margin" title="Phase margin">phase margins</a> of an amplifier. The notion of gain and phase margin is based upon the gain expression for a negative feedback amplifier given by

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<annotation encoding="application/x-tex">{\displaystyle A_{\text{FB}}={\frac {A_{\text{OL}}}{1+\beta A_{\text{OL}}}},}</annotation>

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</math><img src="./e60769b0a896b35c0592a3794e811db1b14d3ec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.41ex; height:5.843ex;" alt="{\displaystyle A_{\text{FB}}={\frac {A_{\text{OL}}}{1+\beta A_{\text{OL}}}},}" loading="lazy"></dd></dl>

where AFB is the gain of the amplifier with feedback (the closed-loop gain), β is the feedback factor, and AOL is the gain without feedback (the open-loop gain). The gain AOL is a complex function of frequency, with both magnitude and phase.<a href="#cite_note-6">[note 1]</a> Examination of this relation shows the possibility of infinite gain (interpreted as instability) if the product βAOL = −1 (that is, the magnitude of βAOL is unity and its phase is −180°, the so-called <a href="Barkhausen_stability_criterion" title="Barkhausen stability criterion">Barkhausen stability criterion</a>). Bode plots are used to determine just how close an amplifier comes to satisfying this condition.

Key to this determination are two frequencies. The first, labeled here as f180, is the frequency where the <a href="Open-loop_gain" title="Open-loop gain">open-loop gain</a> flips sign. The second, labeled here f0 dB, is the frequency where the magnitude of the product |βAOL| = 1 = 0 dB. That is, frequency f180 is determined by the condition

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<annotation encoding="application/x-tex">{\displaystyle \beta A_{\text{OL}}(f_{180})=-|\beta A_{\text{OL}}(f_{180})|=-|\beta A_{\text{OL}}|_{180},}</annotation>

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</math><img src="./30372e0f0813fec4661fa7ebdef7657cb7c1e57f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.879ex; height:3.009ex;" alt="{\displaystyle \beta A_{\text{OL}}(f_{180})=-|\beta A_{\text{OL}}(f_{180})|=-|\beta A_{\text{OL}}|_{180},}" loading="lazy"></dd></dl>

where vertical bars denote the <a href="Absolute_value#Complex_numbers" title="Absolute value">magnitude of a complex number</a>, and frequency f0 dB is determined by the condition

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<annotation encoding="application/x-tex">{\displaystyle |\beta A_{\text{OL}}(f_{\text{0 dB}})|=1.}</annotation>

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</math><img src="./da6cccf36f3bfb7eb49f5b46fec10595a78c1529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.476ex; height:2.843ex;" alt="{\displaystyle |\beta A_{\text{OL}}(f_{\text{0 dB}})|=1.}" loading="lazy"></dd></dl>

One measure of proximity to instability is the gain margin. The Bode phase plot locates the frequency where the phase of βAOL reaches −180°, denoted here as frequency f180. Using this frequency, the Bode magnitude plot finds the magnitude of βAOL. If |βAOL|180 ≥ 1, the amplifier is unstable, as mentioned. If |βAOL|180 < 1, instability does not occur, and the separation in dB of the magnitude of |βAOL|180 from |βAOL| = 1 is called the gain margin. Because a magnitude of 1 is 0 dB, the gain margin is simply one of the equivalent forms: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 20\log _{10}|\beta A_{\text{OL}}|_{180}=20\log _{10}|A_{\text{OL}}|-20\log _{10}\beta ^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle 20\log _{10}|\beta A_{\text{OL}}|_{180}=20\log _{10}|A_{\text{OL}}|-20\log _{10}\beta ^{-1}}</annotation>

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</math><img src="./fdc34a5bc6ea45eac16c75505370e0c092a663f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.63ex; height:3.343ex;" alt="{\displaystyle 20\log _{10}|\beta A_{\text{OL}}|_{180}=20\log _{10}|A_{\text{OL}}|-20\log _{10}\beta ^{-1}}" loading="lazy">.

Another equivalent measure of proximity to instability is the <a href="Phase_margin" title="Phase margin">phase margin</a>. The Bode magnitude plot locates the frequency where the magnitude of |βAOL| reaches unity, denoted here as frequency f0 dB. Using this frequency, the Bode phase plot finds the phase of βAOL. If the phase of βAOL(f0 dB) > −180°, the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when f = f180), and the distance of the phase at f0 dB in degrees above −180° is called the phase margin.

If a simple yes or no on the stability issue is all that is needed, the amplifier is stable if f0 dB < f180. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions (<a href="Minimum_phase" title="Minimum phase">minimum phase</a> systems). Although these restrictions usually are met, if they are not, then another method must be used, such as the <a href="Nyquist_plot" class="mw-redirect" title="Nyquist plot">Nyquist plot</a>.<a href="#cite_note-Lee-7">[6]</a><a href="#cite_note-Levine-8">[7]</a>

Optimal gain and phase margins may be computed using <a href="Nevanlinna%E2%80%93Pick_interpolation" title="Nevanlinna–Pick interpolation">Nevanlinna–Pick interpolation</a> theory.<a href="#cite_note-Tannenbaum-9">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Examples_using_Bode_plots">Examples using Bode plots</h3></div>

Figures 6 and 7 illustrate the gain behavior and terminology. For a three-pole amplifier, Figure 6 compares the Bode plot for the gain without feedback (the open-loop gain) AOL with the gain with feedback AFB (the closed-loop gain). See <a href="Negative_feedback_amplifier" class="mw-redirect" title="Negative feedback amplifier">negative feedback amplifier</a> for more detail.

In this example, AOL = 100 dB at low frequencies, and 1 / β = 58 dB. At low frequencies, AFB ≈ 58 dB as well.

Because the open-loop gain AOL is plotted and not the product β AOL, the condition AOL = 1 / β decides f0 dB. The feedback gain at low frequencies and for large AOL is AFB ≈ 1 / β (look at the formula for the feedback gain at the beginning of this section for the case of large gain AOL), so an equivalent way to find f0 dB is to look where the feedback gain intersects the open-loop gain. (Frequency f0 dB is needed later to find the phase margin.)

Near this crossover of the two gains at f0 dB, the Barkhausen criteria are almost satisfied in this example, and the feedback amplifier exhibits a massive peak in gain (it would be infinity if β AOL = −1). Beyond the unity gain frequency f0 dB, the open-loop gain is sufficiently small that AFB ≈ AOL (examine the formula at the beginning of this section for the case of small AOL).

Figure 7 shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency f180 where the open-loop gain has a phase of −180°. In this vicinity, the phase of the feedback amplifier plunges abruptly downward to become almost the same as the phase of the open-loop amplifier. (Recall, AFB ≈ AOL for small AOL.)

Comparing the labeled points in Figure 6 and Figure 7, it is seen that the unity gain frequency f0 dB and the phase-flip frequency f180 are very nearly equal in this amplifier, f180 ≈ f0 dB ≈ 3.332 kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable.

Figures 8 and 9 illustrate the gain margin and phase margin for a different amount of feedback β. The feedback factor is chosen smaller than in Figure 6 or 7, moving the condition | β AOL | = 1 to lower frequency. In this example, 1 / β = 77 dB, and at low frequencies AFB ≈ 77 dB as well.

Figure 8 shows the gain plot. From Figure 8, the intersection of 1 / β and AOL occurs at f0 dB = 1 kHz. Notice that the peak in the gain AFB near f0 dB is almost gone.<a href="#cite_note-10">[note 2]</a><a href="#cite_note-Sansen-11">[9]</a>

Figure 9 is the phase plot. Using the value of f0 dB = 1 kHz found above from the magnitude plot of Figure 8, the open-loop phase at f0 dB is −135°, which is a phase margin of 45° above −180°.

Using Figure 9, for a phase of −180° the value of f180 = 3.332 kHz (the same result as found earlier, of course<a href="#cite_note-12">[note 3]</a>). The open-loop gain from Figure 8 at f180 is 58 dB, and 1 / β = 77 dB, so the gain margin is 19 dB.

Stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good <a href="Step_response#Step_response_of_feedback_amplifiers" title="Step response">step response</a>. As a <a href="Rule_of_thumb" title="Rule of thumb">rule of thumb</a>, good step response requires a phase margin of at least 45°, and often a margin of over 70° is advocated, particularly where component variation due to manufacturing tolerances is an issue.<a href="#cite_note-Sansen-11">[9]</a> See also the discussion of phase margin in the <a href="Step_response#Phase_margin" title="Step response">step response</a> article.

<li class="gallerycaption">Examples</li>

<div class="gallerytext">Figure 6: Gain of feedback amplifier AFB in dB and corresponding open-loop amplifier AOL. Parameter 1/β = 58 dB, and at low frequencies AFB ≈ 58 dB as well. The gain margin in this amplifier is nearly zero because | βAOL| = 1 occurs at almost f = f180°.</div>

</li>

<div class="gallerytext">Figure 7: Phase of feedback amplifier °AFB in degrees and corresponding open-loop amplifier °AOL. The phase margin in this amplifier is nearly zero because the phase-flip occurs at almost the unity gain frequency f = f0 dB where | βAOL| = 1.</div>

</li>

<div class="gallerytext">Figure 8: Gain of feedback amplifier AFB in dB and corresponding open-loop amplifier AOL. In this example, 1 / β = 77 dB. The gain margin in this amplifier is 19 dB.</div>

</li>

<div class="gallerytext">Figure 9: Phase of feedback amplifier AFB in degrees and corresponding open-loop amplifier AOL. The phase margin in this amplifier is 45°.</div>

</li>

</ul>

<div class="mw-heading mw-heading2"><h2 id="Bode_plotter">Bode plotter</h2></div>

The Bode plotter is an electronic instrument resembling an <a href="Oscilloscope" title="Oscilloscope">oscilloscope</a>, which produces a Bode diagram, or a graph, of a circuit's voltage gain or phase shift plotted against <a href="Frequency" title="Frequency">frequency</a> in a feedback control system or a filter. An example of this is shown in Figure 10. It is extremely useful for analyzing and testing filters and the stability of <a href="Feedback" title="Feedback">feedback</a> control systems, through the measurement of corner (cutoff) frequencies and gain and phase margins.

This is identical to the function performed by a <a href="Vector_network_analyzer" class="mw-redirect" title="Vector network analyzer">vector network analyzer</a>, but the network analyzer is typically used at much higher frequencies.

For education and research purposes, plotting Bode diagrams for given transfer functions facilitates better understanding and getting faster results (see external links).

<div class="mw-heading mw-heading2"><h2 id="Related_plots">Related plots</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Nyquist_plot" class="mw-redirect" title="Nyquist plot">Nyquist plot</a> and <a href="Nichols_plot" title="Nichols plot">Nichols plot</a></div>

Two related plots that display the same data in different <a href="Coordinate_systems" class="mw-redirect" title="Coordinate systems">coordinate systems</a> are the <a href="Nyquist_plot" class="mw-redirect" title="Nyquist plot">Nyquist plot</a> and the <a href="Nichols_plot" title="Nichols plot">Nichols plot</a>. These are <a href="Parametric_plots" class="mw-redirect" title="Parametric plots">parametric plots</a>, with frequency as the input and magnitude and phase of the frequency response as the output. The Nyquist plot displays these in <a href="Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a>, with magnitude mapping to radius and phase to argument (angle). The Nichols plot displays these in rectangular coordinates, on the <a href="Log_scale" class="mw-redirect" title="Log scale">log scale</a>.

<div class="gallerytext">Figure 11: A <a href="Nyquist_plot" class="mw-redirect" title="Nyquist plot">Nyquist plot</a>.</div>

</li>

<div class="gallerytext">Figure 12: A <a href="Nichols_plot" title="Nichols plot">Nichols plot</a> of the same response from Figure 11.</div>

</li>

</ul>

<ul><li><a href="Analog_signal_processing" title="Analog signal processing">Analog signal processing</a></li>

<li><a href="Phase_margin" title="Phase margin">Phase margin</a></li>

<li><a href="Bode's_sensitivity_integral" title="Bode's sensitivity integral">Bode's sensitivity integral</a></li>

<li><a href="Kramers%E2%80%93Kronig_relations#Magnitude_(gain)–phase_relation" title="Kramers–Kronig relations">Bode's magnitude (gain)–phase relation</a></li>

<li><a href="Dielectric_spectroscopy" title="Dielectric spectroscopy">Dielectric spectroscopy</a></li></ul>

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

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> Ordinarily, as frequency increases, the magnitude of the gain drops, and the phase becomes more negative, although these are only trends and may be reversed in particular frequency ranges. Unusual gain behavior can render the concepts of gain and phase margin inapplicable. Then other methods such as the <a href="Nyquist_plot" class="mw-redirect" title="Nyquist plot">Nyquist plot</a> have to be used to assess stability.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> The critical amount of feedback where the peak in the gain just disappears altogether is the maximally flat or <a href="Butterworth_filter#Maximal_flatness" title="Butterworth filter">Butterworth</a> design.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> The frequency where the open-loop gain flips sign f180 does not change with a change in feedback factor; it is a property of the open-loop gain. The value of the gain at f180 also does not change with a change in β. Therefore, we could use the previous values from Figures 6 and 7. However, for clarity the procedure is described using only Figures 8 and 9.

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</ol></div></div>

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

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</style><cite id="CITEREFR._K._Rao_Yarlagadda2010" class="citation book cs1">R. K. Rao Yarlagadda (2010). <a rel="nofollow" class="external text" href="https://archive.org/details/analogdigitalsig00yarl_849">Analog and Digital Signals and Systems</a>. Springer Science & Business Media. p. <a rel="nofollow" class="external text" href="https://archive.org/details/analogdigitalsig00yarl_849/page/n271">243</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-0034-0</bdi>.</cite>

</li>

<li id="cite_note-Van_Valkenburg-2"><a href="#cite_ref-Van_Valkenburg_2-0">^</a> Van Valkenburg, M. E. University of Illinois at Urbana-Champaign, "In memoriam: Hendrik W. Bode (1905-1982)", <a href="IEEE" class="mw-redirect" title="IEEE">IEEE</a> Transactions on Automatic Control, Vol. AC-29, No 3., March 1984, pp. 193–194. Quote: "Something should be said about his name. To his colleagues at Bell Laboratories and the generations of engineers that have followed, the pronunciation is boh-dee. The Bode family preferred that the original Dutch be used as boh-dah."

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mijnwoordenboek.nl/vertalen.php?s1=&s2=NL+%3E+EN&s3=NL+%3E+EN&woord=postbode">"Vertaling van postbode, NL>EN"</a>. mijnwoordenboek.nl. Retrieved 2013-10-07.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> David A. Mindell Between Human and Machine: Feedback, Control, and Computing Before Cybernetics JHU Press, 2004, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0801880572</bdi>, pp. 127–131.

</li>

<li id="cite_note-multivar_fb_control-5"><a href="#cite_ref-multivar_fb_control_5-0">^</a> <cite id="CITEREFSkogestadPostlewaite2005" class="citation book cs1">Skogestad, Sigurd; Postlewaite, Ian (2005). Multivariable Feedback Control. Chichester, West Sussex, England: John Wiley & Sons, Ltd. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-470-01167-X</bdi>.</cite>

</li>

<cite id="CITEREFThomas_H._Lee2004" class="citation book cs1">Thomas H. Lee (2004). "§14.6. Gain and Phase Margins as Stability Measures". <a rel="nofollow" class="external text" href="http://worldcat.org/isbn/0-521-83539-9">The design of CMOS radio-frequency integrated circuits</a> (2nd ed.). Cambridge UK: Cambridge University Press. pp. 451–453. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-83539-9</bdi>.</cite>

</li>

<cite id="CITEREFWilliam_S._Levine1996" class="citation book cs1">William S. Levine (1996). "§10.1. Specifications of Control System". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2WQP5JGaJOgC&q=stability+%22minimum+phase%22&pg=RA1-PA163">The control handbook: the electrical engineering handbook series</a> (2nd ed.). Boca Raton FL: CRC Press/IEEE Press. p. 163. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8493-8570-9</bdi>.</cite>

</li>

<cite id="CITEREFAllen_Tannenbaum1981" class="citation book cs1"><a href="Allen_Tannenbaum" title="Allen Tannenbaum">Allen Tannenbaum</a> (February 1981). Invariance and Systems Theory: Algebraic and Geometric Aspects. New York, NY: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783540105657</bdi>.</cite>

</li>

<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">Analog design essentials</a>. Dordrecht, The Netherlands: Springer. pp. 157–163. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-25746-2</bdi>.</cite>

</li>

</ol></div></div>

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

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Bode_plots" class="extiw external" title="commons:Category:Bode plots">Bode plots</a>.</div></div>

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<ul><li><a rel="nofollow" class="external text" href="http://lpsa.swarthmore.edu/Bode/BodeHow.html">How to draw piecewise asymptotic Bode plots</a></li>

<li>Gnuplot code for generating Bode plot: DIN-A4 printing template (pdf)</li></ul></div><div><div class="zim-footer">

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