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GainPhaseMargins
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GainPhaseMargins

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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 .
  • The phase margins are phase lags needed to make the phase 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
    MethodAutomaticmethod to use
    SamplingPeriodNonethe 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

open allclose all

Basic Examples  (1)Summary of the most common use cases

The gain and phase margins of a system:

In[1]:=1
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https://wolfram.com/xid/08u4mw1rwt9mo32-g9e95k
Out[1]=1

Scope  (3)Survey of the scope of standard use cases

A system with multiple crossover frequencies:

In[1]:=1
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https://wolfram.com/xid/08u4mw1rwt9mo32-j355e4
In[2]:=2
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https://wolfram.com/xid/08u4mw1rwt9mo32-lvltzl
Out[2]=2

The number of gain crossover frequencies:

In[3]:=3
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https://wolfram.com/xid/08u4mw1rwt9mo32-6kqjq3
Out[3]=3

A discrete-time system:

In[1]:=1
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https://wolfram.com/xid/08u4mw1rwt9mo32-vwmxxm
Out[1]=1

Margins of a time-delay system:

In[1]:=1
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https://wolfram.com/xid/08u4mw1rwt9mo32-j0v9ji
Out[1]=1

Generalizations & Extensions  (1)Generalized and extended use cases

GainPhaseMargins[TransferFunctionModel[g,var]] equals GainPhaseMargins[g]:

In[1]:=1
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https://wolfram.com/xid/08u4mw1rwt9mo32-etd1u7
In[2]:=2
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https://wolfram.com/xid/08u4mw1rwt9mo32-70c9js
Out[2]=2

Options  (2)Common values & functionality for each option

FeedbackType  (2)

The system is assumed to be the loop transfer function of a negative-feedback system:

In[1]:=1
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https://wolfram.com/xid/08u4mw1rwt9mo32-4vlsy6
In[2]:=2
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https://wolfram.com/xid/08u4mw1rwt9mo32-sar8oa
Out[2]=2

Specify the system as part of a positive-feedback system:

In[3]:=3
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https://wolfram.com/xid/08u4mw1rwt9mo32-uh5jej
Out[3]=3

Specify the system as a closed-loop system:

In[4]:=4
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https://wolfram.com/xid/08u4mw1rwt9mo32-xy8ucx
Out[4]=4

A positive-feedback system:

In[1]:=1
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https://wolfram.com/xid/08u4mw1rwt9mo32-0idpgi
Out[1]=1

Properties & Relations  (2)Properties of the function, and connections to other functions

The gain and phase margins can be visualized on all the frequency response plots:

In[1]:=1
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https://wolfram.com/xid/08u4mw1rwt9mo32-0xtr5m
In[2]:=2
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https://wolfram.com/xid/08u4mw1rwt9mo32-yiamlb
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/08u4mw1rwt9mo32-equ7un
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/08u4mw1rwt9mo32-45zcbj
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/08u4mw1rwt9mo32-ulqmjm
Out[5]=5

The gain margins in decibels:

In[6]:=6
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https://wolfram.com/xid/08u4mw1rwt9mo32-hojcpe
Out[6]=6

The phase margins in degrees:

In[7]:=7
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https://wolfram.com/xid/08u4mw1rwt9mo32-rhlt4b
Out[7]=7

If the crossover frequencies are in radians/second, they can be converted to hertz as follows:

In[8]:=8
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https://wolfram.com/xid/08u4mw1rwt9mo32-cy7716
Out[8]=8

The setting StabilityMargins->True computes and draws the gain and phase margins:

In[1]:=1
✖
https://wolfram.com/xid/08u4mw1rwt9mo32-67m1gp
Out[1]=1
Wolfram Research (2010), GainPhaseMargins, Wolfram Language function, https://reference.wolfram.com/language/ref/GainPhaseMargins.html.
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Wolfram Research (2010), GainPhaseMargins, Wolfram Language function, https://reference.wolfram.com/language/ref/GainPhaseMargins.html.

Text

Wolfram Research (2010), GainPhaseMargins, Wolfram Language function, https://reference.wolfram.com/language/ref/GainPhaseMargins.html.

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Wolfram Research (2010), GainPhaseMargins, Wolfram Language function, https://reference.wolfram.com/language/ref/GainPhaseMargins.html.

CMS

Wolfram Language. 2010. "GainPhaseMargins." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GainPhaseMargins.html.

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Wolfram Language. 2010. "GainPhaseMargins." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GainPhaseMargins.html.

APA

Wolfram Language. (2010). GainPhaseMargins. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GainPhaseMargins.html

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Wolfram Language. (2010). GainPhaseMargins. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GainPhaseMargins.html

BibTeX

@misc{reference.wolfram_2025_gainphasemargins, author="Wolfram Research", title="{GainPhaseMargins}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GainPhaseMargins.html}", note=[Accessed: 16-April-2025 ]}

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@misc{reference.wolfram_2025_gainphasemargins, author="Wolfram Research", title="{GainPhaseMargins}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GainPhaseMargins.html}", note=[Accessed: 16-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_gainphasemargins, organization={Wolfram Research}, title={GainPhaseMargins}, year={2010}, url={https://reference.wolfram.com/language/ref/GainPhaseMargins.html}, note=[Accessed: 16-April-2025 ]}

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@online{reference.wolfram_2025_gainphasemargins, organization={Wolfram Research}, title={GainPhaseMargins}, year={2010}, url={https://reference.wolfram.com/language/ref/GainPhaseMargins.html}, note=[Accessed: 16-April-2025 ]}