--- title: "NicholsPlot" language: "en" type: "Symbol" summary: "NicholsPlot[lsys] generates a Nichols plot of the transfer function for the system lsys. NicholsPlot[lsys, {\\[Omega]min, \\[Omega]max}] plots for the frequency range \\[Omega]min to \\[Omega]max. NicholsPlot[expr, {\\[Omega], \\[Omega]min, \\[Omega]max}] plots expr using the variable \\[Omega]." keywords: - Nichols plot - frequency response characteristics - magnitude phase plot - sinusoidal transfer function - phase margin - gain margin - bandwidth - gain crossover frequency - phase crossover frequency - Nichols grid lines - Nichols chart - M-contours - constant magnitude contours - N-contours - constant phase contours - classical control - control systems - control theory - nichols - ngrid canonical_url: "https://reference.wolfram.com/language/ref/NicholsPlot.html" source: "Wolfram Language Documentation" related_guides: - title: "Classical Analysis and Design" link: "https://reference.wolfram.com/language/guide/ClassicalAnalysisAndDesign.en.md" - title: "Fourier Analysis" link: "https://reference.wolfram.com/language/guide/FourierAnalysis.en.md" - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" related_functions: - title: "NicholsGridLines" link: "https://reference.wolfram.com/language/ref/NicholsGridLines.en.md" - title: "NyquistPlot" link: "https://reference.wolfram.com/language/ref/NyquistPlot.en.md" - title: "BodePlot" link: "https://reference.wolfram.com/language/ref/BodePlot.en.md" - title: "GainPhaseMargins" link: "https://reference.wolfram.com/language/ref/GainPhaseMargins.en.md" - title: "TransferFunctionModel" link: "https://reference.wolfram.com/language/ref/TransferFunctionModel.en.md" --- # NicholsPlot NicholsPlot[lsys] generates a Nichols plot of the transfer function for the system lsys. NicholsPlot[lsys, {ωmin, ωmax}] plots for the frequency range ωmin to ωmax. NicholsPlot[expr, {ω, ωmin, ωmax}] plots expr using the variable ω. ## Details and Options * ``NicholsPlot`` gives the magnitude versus phase plot of the transfer function of ``lsys``. * The system ``lsys`` can be ``TransferFunctionModel`` or ``StateSpaceModel``, including descriptor and delay systems. * For a system ``lsys`` with the corresponding transfer function $g(s)$, the following expressions are plotted: | | | | --------------------------------------------------------- | -------------------------------------------------------------------------- | | $g(i \omega )$ | continuous-time system | | $g(\exp (i \tau \omega ))$ | discrete-time system with sample time $\tau$ | * ``NicholsPlot`` treats the variable ``ω`` as local, effectively using ``Block``. * ``NicholsPlot`` has the same options as ``Graphics``, with the following changes and additions: [] | | | | | ---------------------- | ----------------- | ----------------------------------------------------------------- | | Axes | True | whether to draw axes | | ColorFunction | Automatic | how to apply coloring to the curve | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | EvaluationMonitor | None | expression to evaluate at every evaluation | | Exclusions | Automatic | frequencies to exclude | | ExclusionsStyle | None | what to draw at excluded frequencies | | FeedbackType | "Negative" | the feedback type | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | Automatic | how many mesh divisions to draw | | MeshFunctions | {#3&} | how to determine the placement of mesh divisions | | MeshShading | None | how to shade regions between mesh points | | MeshStyle | Automatic | the style for mesh divisions | | NicholsGridLines | None | the Nichols grid lines to draw | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLegends | None | legends for curves | | PhaseRange | Automatic | range of phase values to use | | PlotPoints | Automatic | iniitial number of sample frequencies | | PlotRange | Automatic | range of magnitude and phase values to include | | PlotRangeClipping | True | whether to clip at the PlotRange | | PlotStyle | Automatic | graphics directives to specify the style of the plot | | PlotTheme | \$PlotTheme | overall theme for the plot | | RegionFunction | Automatic | how to determine whether a point should be included | | SamplingPeriod | None | the sampling period | | ScalingFunctions | {"Degree", "dB"} | the scaling functions | | StabilityMargins | False | whether to show the stability margins | | StabilityMarginsStyle | Automatic | graphics directives to specify the style of the stability margins | | WorkingPrecision | MachinePrecision | the precision used in internal computations | * ``ColorData["DefaultPlotColors"]`` gives the default sequence of colors used by ``PlotStyle``. * ``ScalingFunctions -> {phasescale, magscale}`` can be used to specify the scaling functions. * The phase scale ``phasescale`` can be ``"Degree"`` or ``"Radian"``. * The magnitude scale ``magscale`` can be ``"dB"`` or ``"Absolute"``, which correspond to the decibel and absolute values of the magnitude, respectively. ### List of all options | | | | | ---------------------- | ----------------- | ---------------------------------------------------------------------------------- | | AlignmentPoint | Center | the default point in the graphic to align with | | AspectRatio | Automatic | ratio of height to width | | Axes | True | whether to draw axes | | AxesLabel | None | axes labels | | AxesOrigin | Automatic | where axes should cross | | AxesStyle | {} | style specifications for the axes | | Background | None | background color for the plot | | BaselinePosition | Automatic | how to align with a surrounding text baseline | | BaseStyle | {} | base style specifications for the graphic | | ColorFunction | Automatic | how to apply coloring to the curve | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | ContentSelectable | Automatic | whether to allow contents to be selected | | CoordinatesToolOptions | Automatic | detailed behavior of the coordinates tool | | Epilog | {} | primitives rendered after the main plot | | EvaluationMonitor | None | expression to evaluate at every evaluation | | Exclusions | Automatic | frequencies to exclude | | ExclusionsStyle | None | what to draw at excluded frequencies | | FeedbackType | "Negative" | the feedback type | | FormatType | TraditionalForm | the default format type for text | | Frame | False | whether to put a frame around the plot | | FrameLabel | None | frame labels | | FrameStyle | {} | style specifications for the frame | | FrameTicks | Automatic | frame ticks | | FrameTicksStyle | {} | style specifications for frame ticks | | GridLines | None | grid lines to draw | | GridLinesStyle | {} | style specifications for grid lines | | ImageMargins | 0. | the margins to leave around the graphic | | ImagePadding | All | what extra padding to allow for labels etc. | | ImageSize | Automatic | the absolute size at which to render the graphic | | LabelStyle | {} | style specifications for labels | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | Automatic | how many mesh divisions to draw | | MeshFunctions | {#3&} | how to determine the placement of mesh divisions | | MeshShading | None | how to shade regions between mesh points | | MeshStyle | Automatic | the style for mesh divisions | | Method | Automatic | details of graphics methods to use | | NicholsGridLines | None | the Nichols grid lines to draw | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PhaseRange | Automatic | range of phase values to use | | PlotLabel | None | an overall label for the plot | | PlotLegends | None | legends for curves | | PlotPoints | Automatic | iniitial number of sample frequencies | | PlotRange | Automatic | range of magnitude and phase values to include | | PlotRangeClipping | True | whether to clip at the PlotRange | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotRegion | Automatic | the final display region to be filled | | PlotStyle | Automatic | graphics directives to specify the style of the plot | | PlotTheme | \$PlotTheme | overall theme for the plot | | PreserveImageOptions | Automatic | whether to preserve image options when displaying new versions of the same graphic | | Prolog | {} | primitives rendered before the main plot | | RegionFunction | Automatic | how to determine whether a point should be included | | RotateLabel | True | whether to rotate y labels on the frame | | SamplingPeriod | None | the sampling period | | ScalingFunctions | {"Degree", "dB"} | the scaling functions | | StabilityMargins | False | whether to show the stability margins | | StabilityMarginsStyle | Automatic | graphics directives to specify the style of the stability margins | | Ticks | Automatic | axes ticks | | TicksStyle | {} | style specifications for axes ticks | | WorkingPrecision | MachinePrecision | the precision used in internal computations | --- ## Examples (69) ### Basic Examples (4) Nichols plot of a transfer-function model: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{10*(0.5 + s)*(1 + 10*s)}}, s*(2 + s)}, s]] Out[1]= [image] ``` --- Specify the frequency range: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4 + s}}, 1 + 3*s + s^2}, s], {0.2, 10}] Out[1]= [image] ``` --- Nichols plot of a discrete-time system: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{2}}, 0.3125 - z + z^2}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` --- Use legends for multiple systems: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{1 + s, 1 + s}}, {{1 + 2*s + s^2, 1 - 2*s + s^2}}}, s], PlotLegends -> "Expressions"] Out[1]= [image] ``` ### Scope (8) The Nichols plot of a continuous-time system: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{15}}, s*(1 + s + s^2)}, s]] Out[1]= [image] ``` --- The Nichols plot of a discrete-time system: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{Rational[1, 2]*(0.5 + z)*(1 + z)}}, (-1 + z)*(-0.5 + z)}, z, SamplingPeriod -> 2]] Out[1]= [image] ``` --- The system can also be specified as an expression: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{2*(0.25 + s)}}, s^2*(3 + 3*s + s^2)}, s]] Out[1]= [image] ``` --- A discrete-time system with sampling period 1, specified as an expression: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4*(-0.5 + z)*(-0.3 + z)}}, (-0.8 + z)*(0.5 + z)}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` --- A system with a time delay: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{1.}}, 0.8/E^(5.*s) + s}, s]] Out[1]= [image] ``` --- Specify the frequency range: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{1}}, -0.1 + 0.5*z - 0.8*z^2 - 0.6*z^3 + z^4}, z, SamplingPeriod -> 1], {0.5, π}] Out[1]= [image] ``` --- The Nichols plot of a state-space model: ```wl In[1]:= NicholsPlot[StateSpaceModel[{{{-90.5, -162.5}, {45.5, 81.5}}, {{4}, {-2}}, {{1.5, 2.5}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]] Out[1]= [image] ``` --- A system specified as a sinusoidal transfer function: ```wl In[1]:= NicholsPlot[(1/E^I ω + 0.8), ω, SamplingPeriod -> 1] Out[1]= [image] ``` ### Generalizations & Extensions (1) ``NicholsPlot[TransferFunctionModel[g, var]]`` is equivalent to ``NicholsPlot[g]`` : ```wl In[1]:= g = (4(s + 0.5)/s^2(s^2 + 2 s + 4)); In[2]:= {NicholsPlot[g], NicholsPlot[TransferFunctionModel[g, s]]} Out[2]= {[image], [image]} ``` ### Options (56) #### AspectRatio (1) Specify the aspect ratio: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AspectRatio -> (1/2)] Out[1]= [image] ``` #### Axes (3) By default, axes are drawn: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s]] Out[1]= [image] ``` --- Use ``Axes -> False`` to turn off axes: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], Axes -> False] Out[1]= [image] ``` --- Turn each axis on individually: ```wl In[1]:= {NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], Axes -> {True, False}], NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], Axes -> {False, True}]} Out[1]= {[image], [image]} ``` #### AxesLabel (4) No axes labels are drawn by default: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s]] Out[1]= [image] ``` --- Place a label on the $y$ axis: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AxesLabel -> label] Out[1]= [image] ``` --- Specify axes labels: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{3*s}}, 2 + 5*s + s^2}, s], AxesLabel -> {"Phase angle (deg)", "Magnitude (dB)"}] Out[1]= [image] ``` --- Use labels based on variables specified in ``NicholsPlot`` : ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AxesLabel -> Automatic] Out[1]= [image] ``` #### AxesOrigin (2) The position of the axes is determined automatically: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s]] Out[1]= [image] ``` --- Specify an explicit origin for the axes: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AxesOrigin -> {100, 0}] Out[1]= [image] ``` #### AxesStyle (4) Change the style for the axes: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AxesStyle -> Red] Out[1]= [image] ``` --- Specify the style of each axis: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AxesStyle -> {{Thick, Red}, {Thick, Blue}}] Out[1]= [image] ``` --- Use different styles for the ticks and the axes: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AxesStyle -> Green, TicksStyle -> Black] Out[1]= [image] ``` --- Use different styles for the labels and the axes: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AxesStyle -> Green, LabelStyle -> Black] Out[1]= [image] ``` #### ColorFunction (3) Color the curve by scaled frequency values: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{10}}, s*(2 + s)*(4 + s)}, s], ColorFunction -> Function[{x, y, f}, Hue[f]]] Out[1]= [image] ``` --- Use a named color gradient: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{10 + s}}, (1 + s)*(20 + s)* (40 + s)}, s], ColorFunction -> "SolarColors"] Out[1]= [image] ``` --- Use red when the sensitivity function is less than 1, and black otherwise: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{1}}, (1 + s)*(2 + s)}, s], ColorFunction -> Function[{x, y, f}, With[{mag = 10^(y/20), phase = x Degree}, If[Abs[(1/1 + mag Exp[I phase])] < 1, Red, Black]]], ColorFunctionScaling -> False ] Out[1]= [image] ``` #### ColorFunctionScaling (1) Scale the frequency to be between 0 and 1: ```wl In[1]:= NicholsPlot[TransferFunctionModel[ {{{2*(1 + 0.001*s)*(0.05 + s)*(0.5 + s)}}, s*(2.1 + s)*(4.2 + s)}, s], {0.05, 10}, ColorFunction -> Function[{x, y, f}, Hue[f]], ColorFunctionScaling -> True] Out[1]= [image] ``` ``ColorFunctionScaling -> False`` uses absolute values: ```wl In[2]:= NicholsPlot[TransferFunctionModel[ {{{2*(1 + 0.001*s)*(0.05 + s)*(0.5 + s)}}, s*(2.1 + s)*(4.2 + s)}, s], {0.05, 10}, ColorFunction -> Function[{x, y, f}, Hue[f]], ColorFunctionScaling -> False] Out[2]= [image] ``` Specify the scaling manually: ```wl In[3]:= NicholsPlot[TransferFunctionModel[ {{{2*(1 + 0.001*s)*(0.05 + s)*(0.5 + s)}}, s*(2.1 + s)*(4.2 + s)}, s], {0.1, 10}, ColorFunction -> Function[{x, y, f}, Hue[(f/10)]], ColorFunctionScaling -> False] Out[3]= [image] ``` #### CoordinatesToolOptions (1) Display the tooltip coordinates in radians and absolute magnitude: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{2 + s}}, 4 + 2*s + s^2}, s], CoordinatesToolOptions -> {"DisplayFunction" -> Function[pt, {pt[[1]] Degree, 10^(pt[[2]]/20)}]}] Out[1]= [image] ``` #### Exclusions (1) By default the singular frequencies are excluded: ```wl In[1]:= Table[NicholsPlot[TransferFunctionModel[{{{10}}, (3 + s)*(16 + s^2)}, s], {0.1, 6}, Exclusions -> excl], {excl, {Automatic, None}}] Out[1]= {[image], [image]} ``` #### ImageSize (7) Use named sizes such as ``Tiny``, ``Small``, ``Medium`` and ``Large`` : ```wl In[1]:= {NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], ImageSize -> Tiny], NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], ImageSize -> Small]} Out[1]= {[image], [image]} ``` --- Specify the width of the plot: ```wl In[1]:= {NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], ImageSize -> 150], NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AspectRatio -> 1.5, ImageSize -> 150]} Out[1]= {[image], [image]} ``` Specify the height of the plot: ```wl In[2]:= {NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], ImageSize -> {Automatic, 150}], NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AspectRatio -> 2, ImageSize -> {Automatic, 150}]} Out[2]= {[image], [image]} ``` --- Allow the width and height to be up to a certain size: ```wl In[1]:= {NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], ImageSize -> UpTo[200]], NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AspectRatio -> 2, ImageSize -> UpTo[200]]} Out[1]= {[image], [image]} ``` --- Specify the width and height for a graphic, padding with space if necessary: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], ImageSize -> {200, 300}, Background -> LightBlue] Out[1]= [image] ``` Setting ``AspectRatio -> Full`` will fill the available space: ```wl In[2]:= NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AspectRatio -> Full, ImageSize -> {200, 300}, Background -> LightBlue] Out[2]= [image] ``` --- Use maximum sizes for the width and height: ```wl In[1]:= {NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], ImageSize -> {UpTo[150], UpTo[100]}], NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AspectRatio -> 2, ImageSize -> {UpTo[150], UpTo[100]}]} Out[1]= {[image], [image]} ``` --- Use ``ImageSize -> Full`` to fill the available space in an object: ```wl In[1]:= Framed[Pane[NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], ImageSize -> Full, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` --- Specify the image size as a fraction of the available space: ```wl In[1]:= Framed[Pane[NicholsPlot[TransferFunctionModel[{{{s^2}}, 4 + 2*s + s^2}, s], AspectRatio -> Full, ImageSize -> {Scaled[0.5], Scaled[0.5]}, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` #### Mesh (4) Show equally spaced frequency locations: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{1 + s}}, (2 + s)* (1 + s + s^2)}, s], Mesh -> 5] Out[1]= [image] ``` --- Show the coordinates at 1 radian per time unit: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{0.09*(0.005 + s)*(1 + 10*s)}}, s^3*(2 + s)}, s], {0.05, 20}, Mesh -> {{1}}] Out[1]= [image] ``` --- Show the coordinates at several frequencies: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{0.09*(0.005 + s)*(1 + 10*s)}}, s^3*(2 + s)}, s], {0.05, 20}, Mesh -> {{1, 2, 4, 8}}] Out[1]= [image] ``` --- Specify the graphics directives: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{0.09*(0.005 + s)*(1 + 10*s)}}, s^3*(2 + s)}, s], {0.05, 20}, Mesh -> {Table[{2^i, Directive[{PointSize[Medium], Hue[(i/10)]}]}, {i, 0, 3}]}] Out[1]= [image] ``` #### MeshFunctions (1) By default the mesh is located at evenly spaced frequencies: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{10*s}}, 3 + s + 2*s^2 + s^3}, s], Mesh -> 5, MeshStyle -> Red] Out[1]= [image] ``` Specify a mesh with evenly spaced log-10 frequency values: ```wl In[2]:= NicholsPlot[TransferFunctionModel[{{{10*s}}, 3 + s + 2*s^2 + s^3}, s], Mesh -> 5, MeshStyle -> Red, MeshFunctions -> Function[{x, y, f}, Log10[f]]] Out[2]= [image] ``` #### MeshStyle (1) Specify the mesh style: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{(0.09*(0.005 + s))*(1 + 10*s)}}, s^3*(2 + s)}, s], {0.05, 20}, Mesh -> {{1, 2, 4, 8}}, MeshStyle -> Red] Out[1]= [image] ``` #### NicholsGridLines (2) Use automatically chosen values of closed-loop magnitude and phase: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s}}, 4 + s}, s], NicholsGridLines -> Automatic] Out[1]= [image] ``` --- Draw specific contours: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s}}, 4 + s}, s], NicholsGridLines -> {Range[0.02, 0.5, 0.02], Range[10 Degree, 80 Degree, 10 Degree]}] Out[1]= [image] ``` #### PhaseRange (1) The phase is typically plotted as a continuous function: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{1}}, (1 + s)^12}, s]] Out[1]= [image] ``` Specify a phase range: ```wl In[2]:= NicholsPlot[TransferFunctionModel[{{{1}}, (1 + s)^12}, s], PhaseRange -> {-2 π, 0}] Out[2]= [image] ``` #### PlotLegends (4) Use automatic legends for multiple systems: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s, -1 + s}}, 4 + s}, s], PlotLegends -> Automatic] Out[1]= [image] ``` --- Use named legends: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s, -1 + s}}, {{-1 + s, s}}}, s], PlotLegends -> "Expressions"] Out[1]= [image] ``` --- Use ``LineLegend`` to add the label for overall legend: ```wl In[1]:= NicholsPlot[Table[TransferFunctionModel[{{{(1 + s)*ω}}, 1 + 2*s + s^2}, s], {ω, 0.2, 0.5, 0.1}], PlotLegends -> LineLegend[Range[0.2, 0.5, 0.1], LegendLabel -> "ω"]] Out[1]= [image] ``` --- Place the legend above the plot: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s, -1 + s}}, 4 + s}, s], PlotLegends -> Placed["Expressions", Above]] Out[1]= [image] ``` #### PlotPoints (1) Use more initial points to get a smoother curve: ```wl In[1]:= Table[NicholsPlot[TransferFunctionModel[{{{0.01283 - 0.0094*z - 0.6326*z^2}}, 0.000335 - 0.05686*z + 1.05652*z^2 - z^3}, z, SamplingPeriod -> 1], PlotPoints -> pp], {pp, {500, 7000}}] Out[1]= {[image], [image]} ``` #### PlotTheme (2) Use a theme with a frame and grid lines: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{s, -1 + s}}, 4 + s}, s], PlotTheme -> "Detailed"] Out[1]= [image] ``` --- Change the style of the grid lines: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{s, -1 + s}}, 4 + s}, s], PlotTheme -> "Detailed", GridLinesStyle -> LightGray] Out[1]= [image] ``` #### ScalingFunctions (1) Show the phase in radians: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{1 + s}}, s*(1 + s + 0.25*s^2)}, s], ScalingFunctions -> {"Radian", Automatic}] Out[1]= [image] ``` Show absolute values of magnitude: ```wl In[2]:= NicholsPlot[TransferFunctionModel[{{{1}}, 1 + s}, s], ScalingFunctions -> {Automatic, "Absolute"}] Out[2]= [image] ``` #### StabilityMargins (3) Show stability margins: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{1}}, 1 + 2*s + 4*s^2 + 2*s^3}, s], StabilityMargins -> True] Out[1]= [image] ``` --- Show only the gain margin: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{1}}, 1 + 2*s + 4*s^2 + 2*s^3}, s], StabilityMargins -> {True, False}] Out[1]= [image] ``` --- Only margins and crossover frequencies with numerical values are shown: ```wl In[1]:= lsys = TransferFunctionModel[{{{4*(1 + s)}}, s*(4 + 4*s + s^2)}, s]; In[2]:= NicholsPlot[lsys, StabilityMargins -> True] Out[2]= [image] In[3]:= GainPhaseMargins[lsys//N] Out[3]= {{{None, ∞}}, {{1.14494, 1.3838}}} ``` #### StabilityMarginsStyle (1) Specify stability margins style: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{6}}, 3*(1 + Rational[1, 3]*s)*s* (1 + s)}, s], StabilityMargins -> True, StabilityMarginsStyle -> {Red, Black}] Out[1]= [image] ``` #### Ticks (4) Ticks are placed automatically in each plot: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4 + s}}, 1 + 3*s + s^2}, s], {0.2, 10}] Out[1]= [image] ``` --- Use ``Ticks -> None`` to not draw any tick marks: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4 + s}}, 1 + 3*s + s^2}, s], {0.2, 10}, Ticks -> None] Out[1]= [image] ``` --- Place tick marks at specific positions: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4 + s}}, 1 + 3*s + s^2}, s], {0.2, 10}, Ticks -> {{-90, -60, -30}, {10, 0, -17}}] Out[1]= [image] ``` --- Draw tick marks at the specified positions with the specified labels: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4 + s}}, 1 + 3*s + s^2}, s], {0.2, 10}, Ticks -> {{{-90, a}, {-60, b}, {-30, c}}, {{10, d}, {0, e}, {-17, f}}}] Out[1]= [image] ``` #### TicksStyle (4) Specify overall ticks style, including the tick labels: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4 + s}}, 1 + 3*s + s^2}, s], {0.2, 10}, TicksStyle -> Directive[Bold, Red]] Out[1]= [image] ``` --- Specify tick style for each of the axes: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4 + s}}, 1 + 3*s + s^2}, s], {0.2, 10}, TicksStyle -> {Directive[Black, Bold], Directive[Bold, Red]}] Out[1]= [image] ``` --- Specify tick marks with scaled lengths: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4 + s}}, 1 + 3*s + s^2}, s], {0.2, 10}, Ticks -> {{{-90, a, .5}, {-60, b, .83}, {-30, c, .95}}, {{10, d, .8}, {0, e, .2}, {-17, f}}}] Out[1]= [image] ``` --- Customize each tick with position, length, labeling and styling: ```wl In[1]:= NicholsPlot[TransferFunctionModel[{{{4 + s}}, 1 + 3*s + s^2}, s], {0.2, 10}, Ticks -> {{{-90, a, .5, Directive[Red, Thick]}, {-60, b, .83, Directive[Red]}, {-30, c, .95, Directive[Darker@Red, Thick, Dashed]}}, {{10, d, .8, Directive[Blue]}, {0, e, .2, Directive[Blue]}, {-17, f, Automatic, Directive[Thick, Blue]}}}] Out[1]= [image] ``` ## See Also * [`NicholsGridLines`](https://reference.wolfram.com/language/ref/NicholsGridLines.en.md) * [`NyquistPlot`](https://reference.wolfram.com/language/ref/NyquistPlot.en.md) * [`BodePlot`](https://reference.wolfram.com/language/ref/BodePlot.en.md) * [`GainPhaseMargins`](https://reference.wolfram.com/language/ref/GainPhaseMargins.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) * [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.en.md) * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) ## History * [Introduced in 2010 (8.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn80.en.md) \| [Updated in 2012 (9.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn90.en.md) ▪ [2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100.en.md)