Nichols plot

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title: "Nichols plot" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["plots-(graphics)", "signal-processing", "classical-control-theory"] description: "Chart of a transfer function's phase response vs. magnitude" topic_path: "engineering" source: "https://en.wikipedia.org/wiki/Nichols_plot" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Chart of a transfer function's phase response vs. magnitude ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/ba/Nichols_plot.svg" caption="A Nichols plot."] ::

The Nichols plot is a plot used in signal processing and control design, named after American engineer Nathaniel B. Nichols. It plots the phase response versus the response magnitude of a transfer function for any given frequency, and as such is useful in characterizing a system's frequency response.

Use in control design

Given a transfer function,

: G(s) = \frac{Y(s)}{X(s)}

with the closed-loop transfer function defined as,

: M(s) = \frac{G(s)}{1+G(s)}

the Nichols plots displays 20 \log_{10}(|G(s)|) versus \arg(G(s)). Loci of constant 20 \log_{10}(|M(s)|) and \arg(M(s)) are overlaid to allow the designer to obtain the closed loop transfer function directly from the open loop transfer function. Thus, the frequency \omega is the parameter along the curve. This plot may be compared to the Bode plot in which the two inter-related graphs - 20 \log_{10}(|G(s)|) versus \log_{10}(\omega) and \arg(G(s)) versus \log_{10}(\omega) ) - are plotted.

In feedback control design, the plot is useful for assessing the stability and robustness of a linear system. This application of the Nichols plot is central to the quantitative feedback theory (QFT) of Horowitz and Sidi, which is a well known method for robust control system design.

In most cases, \arg(G(s)) refers to the phase of the system's response. Although similar to a Nyquist plot, a Nichols plot is plotted in a Cartesian coordinate system while a Nyquist plot is plotted in a Polar coordinate system.

References

References

1. Isaac M. Howowitz, ''Synthesis of Feedback Systems'', Academic Press, 1963, Lib Congress 63-12033 p. 194-198
2. Boris J. Lurie and Paul J. Enright, ''Classical Feedback Control'', Marcel Dekker, 2000, {{ISBN. 0-8247-0370-7 p. 10
3. Allen Stubberud, Ivan Williams, and Joseph DeStefano, ''Shaums Outline Feedback and Control Systems'', McGraw-Hill, 1995, {{ISBN. 0-07-017052-5 ch. 17

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