BUILT-IN MATHEMATICA SYMBOL

NyquistPlot

NyquistPlot[g]
gives the Nyquist plot of a rational function g in one complex variable.

NyquistPlot[sys]
gives the Nyquist plot of a TransferFunctionModel or StateSpaceModel object sys.

NyquistPlot

gives the plot for frequencies from

to

.

MORE INFORMATION

NyquistPlot gives the complex-plane plot of the transfer function as the Nyquist contour is traversed.

NyquistPlot[TransferFunctionModel[g, var]] is equivalent to NyquistPlot[g].

Frequencies are given in radians per time unit.

If the frequency range is not specified, the entire Nyquist contour is traversed.

NyquistPlot has the same options as Graphics, with the following additions and changes:

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	True	frequencies to exclude
ExclusionsStyle	Automatic	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	Automatic	how to shade regions between mesh points
MeshStyle	Automatic	the style for mesh divisions
NyquistGridLines	None	the Nyquist grid lines to draw
PerformanceGoal	$PerformanceGoal	aspects of performance to try to optimize
PlotPoints	Automatic	iniitial number of sample frequencies
PlotRange	Automatic	range of magnitude and phase values to include
PlotStyle	Automatic	graphics directives to specify the style of the plot
RegionFunction	Automatic	how to determine whether a point should be included
SamplingPeriod	None	the sampling period
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

The setting Exclusions->True excludes frequencies, including resonant frequencies, where the sinusoidal transfer function is discontinuous.

Exclusions excludes specific frequencies .

ExclusionsStyle->s specifies that style s should be used to render the curve joining opposite ends of each excluded point.

Points corresponding to exclusions at resonant frequencies are joined by semicircles at infinity.

For continuous-time systems, the Nyquist contour encloses the entire right half-plane, and excludes poles on the imaginary axis. It is traversed in a clockwise direction.

For discrete-time systems, the Nyquist contour is the unit circle, and it encloses poles on the unit circle. It is traversed in a counterclockwise direction.

The Nyquist contours:

The Nyquist stability criterion for negative feedback systems is , where is the number of unstable poles of the closed-loop system, is the number of clockwise encirclements of , and is the number of unstable poles of the open-loop system. The open-loop poles on the imaginary axis for continuous-time systems and those on the unit circle for discrete-time systems are considered stable.

For positive feedback systems, the stability criterion is still , but is the number of clockwise encirclements of .

The arrows on the NyquistPlot show the direction of the plot as the Nyquist contour is traversed.

EXAMPLES

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Basic Examples (6)

A Nyquist plot of a rational function:

A Nyquist plot of a transfer-function model:

A Nyquist plot of a system with resonant frequencies:

A Nyquist plot of a discrete-time system:

A discrete-time system with resonant frequencies:

Another discrete-time system with resonant frequencies:

A Nyquist plot of a rational function:

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A Nyquist plot of a transfer-function model:

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A Nyquist plot of a system with resonant frequencies:

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A Nyquist plot of a discrete-time system:

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A discrete-time system with resonant frequencies:

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Another discrete-time system with resonant frequencies:

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Scope (9)

Nyquist plot of a continuous-time system:

Nyquist plot of another continuous-time system:

Specify the frequency range:

Nyquist plot of a discrete-time system:

Nyquist plot of a continuous-time, transfer-function model:

Discrete-time, transfer-function model:

Nyquist plot of a state-space model:

The Nyquist plots of systems with resonant frequencies have encirclements at infinity:

An improper system with resonant frequencies:

Options (17)

Specify the aspect ratio:

Label the axes:

Obtain magnitude and phase (in degrees) by selecting the graphic and typing a period (.):

By default there are no exclusions for a system with no resonant frequencies:

Exclude the point corresponding to 0.75 radians per time unit:

Exclude multiple frequencies:

Resonant frequencies correspond to semicircles of infinite radius:

Exclude only one (5 radians per time unit) of the resonant frequencies:

Specify the style of the exclusions:

A Nyquist plot without the infinite encirclements:

Specify the range of coordinates to include in a plot:

Points at infinity are shown in the region specified by PlotRangePadding:

Specify the plot range explicitly:

Automatically chosen values of closed-loop magnitude and phase:

Draw specific contours:

Show the stability margins:

Stability margins for a system with resonant frequencies:

Specify the style of stability margins:

Applications (2)

Compute gain and phase margins:

With no encirclements of

and no poles in the right half-plane, the closed-loop with unity negative feedback is stable (Nyquist stability criterion):

Properties & Relations (1)

The sinusoidal transfer function of a discrete-time system is periodic: