--- title: "GainPhaseMargins" language: "en" type: "Symbol" summary: "GainPhaseMargins[lsys] gives the gain and phase margins of the linear time-invariant system lsys." keywords: - gain phase margins - gain margin - phase margin - gain crossover frequency - phase crossover frequency - stability margins - rolloff - classical control - control systems - control theory - margin canonical_url: "https://reference.wolfram.com/language/ref/GainPhaseMargins.html" source: "Wolfram Language Documentation" related_guides: - title: "Classical Analysis and Design" link: "https://reference.wolfram.com/language/guide/ClassicalAnalysisAndDesign.en.md" related_functions: - title: "GainMargins" link: "https://reference.wolfram.com/language/ref/GainMargins.en.md" - title: "PhaseMargins" link: "https://reference.wolfram.com/language/ref/PhaseMargins.en.md" - title: "StabilityMargins" link: "https://reference.wolfram.com/language/ref/StabilityMargins.en.md" - title: "BodePlot" link: "https://reference.wolfram.com/language/ref/BodePlot.en.md" - title: "NyquistPlot" link: "https://reference.wolfram.com/language/ref/NyquistPlot.en.md" - title: "NicholsPlot" link: "https://reference.wolfram.com/language/ref/NicholsPlot.en.md" - title: "TransferFunctionModel" link: "https://reference.wolfram.com/language/ref/TransferFunctionModel.en.md" --- # GainPhaseMargins GainPhaseMargins[lsys] gives the gain and phase margins of the linear time-invariant system lsys. ## Details and Options * The system ``lsys`` can be a ``TransferFunctionModel`` or a ``StateSpaceModel``. * ``GainPhaseMargins`` returns ``{{{ωp1, g1}, {ωp2, g2}, …}, {{ωg1, p1}, {ωg2, p2}, …}}``, where the ``ωpi`` are the phase crossover frequencies, the ``gi`` are the gain margins, the ``ωgi`` are the gain crossover frequencies, and the ``pi`` are the phase margins. * The gain margins ``gi`` are absolute values and the phase margins ``pi`` are in radians. * The gain margins are the reciprocals of the magnitude of ``lsys`` at the phase crossover frequencies. * At the phase crossover frequencies, ``lsys`` has phase $-180 {}^{\circ}$. * The phase margins are phase lags needed to make the phase $-180 {}^{\circ}$ at the gain crossover frequencies. * At the gain crossover frequencies, the gain of ``lsys`` is unity. * The following options can be given: | | | | | -------------- | ---------- | ------------------- | | FeedbackType | "Negative" | the feedback type | | Method | Automatic | method to use | | SamplingPeriod | None | the sampling period | * Explicit settings for the ``Method`` option include ``"Solve"`` and ``"NSolve"``. In each case the methods of ``Solve`` or ``NSolve`` can be specified as suboptions. The default setting of ``Automatic`` switches between these methods, depending on whether ``lsys`` is exact or inexact. * ``GainPhaseMargins`` has the attribute ``Listable``. --- ## Examples (9) ### Basic Examples (1) The gain and phase margins of a system: ```wl In[1]:= GainPhaseMargins[TransferFunctionModel[{{{10}}, (1 + s)*(2 + s)*(3 + s)}, s]] Out[1]= {{{Sqrt[11], 6}}, {{1, (π/2)}}} ``` ### Scope (3) A system with multiple crossover frequencies: ```wl In[1]:= tfm = TransferFunctionModel[ {{{85*(43.25 + 45.25*s + 3*s^2 + s^3)}}, 8282*s^2 + 366*s^3 + 187*s^4 + 4*s^5 + s^6}, s]; In[2]:= {gm, pm} = GainPhaseMargins[tfm] Out[2]= {{{10.3431, 1.26245}}, {{0.743648, 0.641179}, {9.83883, 0.682599}, {9.45112, 1.25977}}} ``` The number of gain crossover frequencies: ```wl In[3]:= Length[pm] Out[3]= 3 ``` --- A discrete-time system: ```wl In[1]:= GainPhaseMargins[TransferFunctionModel[{{{2 + z}}, 0.88 - z + z^2}, z, SamplingPeriod -> 2]] Out[1]= {{{0.540753, 0.06}}, {{1.00754, -1.43336}}} ``` --- Margins of a time-delay system: ```wl In[1]:= GainPhaseMargins[TransferFunctionModel[{{{-6}}, E^s*(-1 + s^2)}, s]] ``` Mod::indet: Indeterminate expression Mod[0,0] encountered. ```wl Out[1]= {{{All, Indeterminate}}, {{None, ∞}}} ``` ### Generalizations & Extensions (1) ``GainPhaseMargins[TransferFunctionModel[g, var]]`` equals ``GainPhaseMargins[g]`` : ```wl In[1]:= g = (6/6 + 3 s + 4 s^2 + s^3); In[2]:= GainPhaseMargins[TransferFunctionModel[g, s]] == GainPhaseMargins[g] Out[2]= True ``` ### Options (2) #### FeedbackType (2) The system is assumed to be the loop transfer function of a negative-feedback system: ```wl In[1]:= lsys = N[TransferFunctionModel[{{{6}}, (2 + s)*(2 + 2*s + s^2)}, s]]; In[2]:= GainPhaseMargins[lsys] Out[2]= {{{2.44949, 3.33333}}, {{1.25284, 1.18129}}} ``` Specify the system as part of a positive-feedback system: ```wl In[3]:= GainPhaseMargins[SystemsModelSeriesConnect[TransferFunctionModel[{{{-1}}, 1}, s], lsys], FeedbackType -> "Positive"] Out[3]= {{{2.44949, 3.33333}}, {{1.25284, 1.18129}}} ``` Specify the system as a closed-loop system: ```wl In[4]:= GainPhaseMargins[SystemsModelFeedbackConnect[lsys], FeedbackType -> None] Out[4]= {{{2.44949, 3.33333}}, {{1.25284, 1.18129}}} ``` --- A positive-feedback system: ```wl In[1]:= GainPhaseMargins[N[TransferFunctionModel[{{{-10000}}, (-5 + s)*(20 + s)* (50 + s)}, s]], FeedbackType -> "Positive"] Out[1]= {{{0., 0.5}, {25.4951, 4.725}}, {{7.74164, 0.474405}}} ``` ### Properties & Relations (2) The gain and phase margins can be visualized on all the frequency response plots: ```wl In[1]:= tfm = TransferFunctionModel[{{{10.}}, (1 + s)*(2 + s)*(3 + s)}, s]; In[2]:= {gm, pm} = GainPhaseMargins[tfm] Out[2]= {{{3.31662, 6.}}, {{1., 1.5708}}} ``` The Bode plot: ```wl In[3]:= BodePlot[tfm, StabilityMargins -> {gm, pm}, StabilityMarginsStyle -> {Red, Green}, ImageSize -> Small] Out[3]= | | | ------- | | [image] | | [image] | ``` The Nyquist plot: ```wl In[4]:= NyquistPlot[tfm, StabilityMargins -> {gm, pm}, StabilityMarginsStyle -> {Directive[Red, Thick], Directive[Green, Thick]}] Out[4]= [image] ``` The Nichols plot: ```wl In[5]:= NicholsPlot[tfm, StabilityMargins -> {gm, pm}, StabilityMarginsStyle -> {Red, Green}] Out[5]= [image] ``` The gain margins in decibels: ```wl In[6]:= gm /. {f_, m_} :> {f, 20 Log10[m]} Out[6]= {{3.31662, 15.563}} ``` The phase margins in degrees: ```wl In[7]:= pm /. {f_, m_} :> {f, (m/Degree)} Out[7]= {{1., 90.}} ``` If the crossover frequencies are in radians/second, they can be converted to hertz as follows: ```wl In[8]:= Replace[{gm, pm}, {f_, m_} :> {(f/2π), m}, {2}] Out[8]= {{{0.527857, 6.}}, {{0.159155, 1.5708}}} ``` --- The setting ``StabilityMargins -> True`` computes and draws the gain and phase margins: ```wl In[1]:= BodePlot[TransferFunctionModel[{{{-10}}, (1 + 0.2*s)*(1 + 0.5*s)*s^2* (1 + s)*(5 + s)}, s], {0.5, 10}, StabilityMargins -> True, ImageSize -> Small] Out[1]= | | | ------- | | [image] | | [image] | ``` ## See Also * [`GainMargins`](https://reference.wolfram.com/language/ref/GainMargins.en.md) * [`PhaseMargins`](https://reference.wolfram.com/language/ref/PhaseMargins.en.md) * [`StabilityMargins`](https://reference.wolfram.com/language/ref/StabilityMargins.en.md) * [`BodePlot`](https://reference.wolfram.com/language/ref/BodePlot.en.md) * [`NyquistPlot`](https://reference.wolfram.com/language/ref/NyquistPlot.en.md) * [`NicholsPlot`](https://reference.wolfram.com/language/ref/NicholsPlot.en.md) * [`TransferFunctionModel`](https://reference.wolfram.com/language/ref/TransferFunctionModel.en.md) ## Related Guides * [Classical Analysis and Design](https://reference.wolfram.com/language/guide/ClassicalAnalysisAndDesign.en.md) ## History * [Introduced in 2010 (8.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn80.en.md)