BodePlot

BodePlot[lsys]

generates a Bode plot of a linear time-invariant system lsys.

BodePlot[lsys,{ωmin,ωmax}]

plots for the frequency range ωmin to ωmax.

BodePlot[expr,{ω,ωmin,ωmax}]

plots expr using the variable ω.

Details and Options

  • BodePlot plots the magnitude and phase 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 , the following expressions are plotted:
  • continuous-time system
    discrete-time system with sample time
  • BodePlot treats the variable ω as local, effectively using Block.
  • BodePlot has the same options as Plot, with the following changes and additions:
  • ExclusionsNonefrequencies to exclude
    FeedbackType"Negative"the feedback type
    FrameTruewhether to draw a frame around each plot
    MeshFunctions{{#1&},{#1&}}how to determine the placement of mesh divisions
    PhaseRangeAutomaticrange of phase values to use
    PlotLayout"VerticalGrid"the layout to be used
    PlotRange{{Full,Automatic},{Full,Automatic}}range of values to include
    SamplingPeriodNonethe sampling period
    ScalingFunctions{{"Log10","dB"},{"Log10","Degree"}}the scaling functions
    StabilityMarginsFalsewhether to show the stability margins
    StabilityMarginsStyleAutomaticgraphics directives to specify the style of the stability margins
  • Possible explicit settings for the option PlotLayout are "VerticalGrid" and "List".
  • The option PlotLayout accepts the following values:
  • "VerticalGrid"magnitude and phase plot in a vertical grid
    "List"magnitude and phase plot in a list
    "Magnitude"magnitude plot only
    "Phase"phase plot only
  • The other options of BodePlot can be specified as a list of two elements, with the first element corresponding to the magnitude plot and the second to the phase plot.
  • Option specifications include:
  • opt->valuse val for both the magnitude and the phase plot
    opt->{val}use val for the magnitude plot and the default for the phase plot
    opt->{val1,val2}use val1 for the magnitude plot and val2 for the phase plot
    opt->{Automatic,val}use the default for the magnitude plot and val for the phase plot
  • With the PlotLayout settings "Magnitude" or "Phase", a single setting refers to the plot requested.
  • The scaling functions can be specified as ScalingFunctions->{{magfreqscale,magscale},{phasefreqscale,phasescale}}.
  • The frequency scales magfreqscale and phasefreqscale can be "Log10" or "Linear", which correspond to the base-10 logarithmic scale and linear scale, respectively.
  • The magnitude scale magscale can be "dB" or "Absolute", which correspond to the decibel and absolute values of the magnitude, respectively.
  • The phase scale phasescale can be "Degree" or "Radian".

Examples

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Basic Examples  (3)

The Bode plot of a system:

A second-order system:

Plot over an explicit frequency range:

Bode plot of a state-space model:

Scope  (13)

Bode plot of a constant-gain system:

Bode plot of an integrator:

Bode plot of a differentiator:

Bode plot of a first-order lag:

Bode plot of a first-order lead:

Bode plot of a second-order system:

A higher-order TransferFunctionModel:

Bode plot of a time-delay system:

A discrete-time system:

Specify the frequency range:

The Bode plot of a state-space model:

A system specified using its sinusoidal transfer function:

The Bode plot of a multiple-input, multiple-output system:

Generalizations & Extensions  (1)

BodePlot[TransferFunctionModel[g,var]] is equivalent to BodePlot[g]:

Options  (21)

CoordinatesToolOptions  (5)

To obtain coordinates, select the graphics and press the period key:

If frequency is in rad/s, obtain coordinate frequency values in hertz by dividing it by :

Coordinate frequency values in different units for each plot:

Frequency values in original units, magnitude in absolute units, and phase in radians:

Specify radians as both the displayed and copied value of phase:

GridLines  (3)

Show grid lines:

Show grid lines only on the magnitude plot:

Show specific grid lines:

GridLinesStyle  (2)

Specify grid lines style:

Specify different grid lines styles:

PhaseRange  (1)

The phase is typically plotted as a continuous function:

Specify a phase range:

PlotLayout  (1)

By default, the magnitude plot is placed vertically above the phase plot:

Obtain the result in a list:

Only the magnitude plot:

Only the phase plot:

PlotTheme  (2)

Use a theme with simple ticks and grid lines in a bright color scheme:

Change the color scheme:

SamplingPeriod  (2)

Specify the sampling period in the system description:

Specify the sampling period in the BodePlot function:

A smaller sampling period results in a higher bandwidth:

ScalingFunctions  (2)

Show absolute values of magnitude and phase in radians:

Plot frequency in the linear scale:

StabilityMargins  (2)

Show stability margins:

Show the phase margin only:

StabilityMarginsStyle  (1)

Specify the stability margins style:

Applications  (7)

The static position error constant of a type 0 system is the magnitude at steady state:

Discrete-time type 0 system:

The static velocity error constant of a type 1 system is approximately the intersection of the initial dB/decade segment (or its extension) with the 0 dB line:

Discrete-time type 1 system:

The square root of the static acceleration error constant of a type 2 system is approximately the intersection of the initial dB/decade segment (or its extension) with the 0 dB line:

Discrete-time type 2 system:

Visualize the improvement in phase margin by using a proportional-integral (PI) compensator:

Visualize the frequency response of a zero-order hold with sampling period 1:

Obtain the Bode plot with frequency in Hertz, when the Laplace variable is in radians/second:

For continuous-time systems, the same result can be obtained by scaling the Laplace variable:

The plot in Hertz for a discrete-time system with the Z-transform variable in radians/second:

Properties & Relations  (1)

SingularValuePlot generalizes the Bode magnitude plot to MIMO systems:

Introduced in 2010
 (8.0)
 |
Updated in 2012
 (9.0)
2014
 (10.0)