National Instruments 370753C-01

Chapter 5 Classical Feedback Analysis

margin( ) loops over all the frequency points in the response and

performs the following computation for phase and gain margins at each,

denoting gain margin as Mg and phase margin as Mp:

This loop finds all the frequency intervals within the response which

contain –180° phase crossings and 0 decibel gain crossings. margin( )

then interpolates to find more exact frequency values for the crossings.

A gain margin value is returned for every pair of phases between which

a –180° phase value must occur, and a phase margin is returned for each

pair of gains between which a zero-decibel gain value must occur.

margin( ) also computes the frequency-rate of change for both the phase

and the gain of the response.

–180° and 0 dB crossings are difficult to detect accurately if the points in

the frequency response are too widely spaced.

You can examine the gain and phase margins of your open-loop system

quickly using margin( ), without having to draw the bode gain and phase

response plots first. Input to margin is the frequency response H of the

system. Referring to the system defined in Example 5-3, you can see that

you already have H as the output from bode( ), but you can calculate it

explicitly using freq( ) as shown in Example 5-4.

Example 5-4 Obtaining Gain and Phase Margin Using margin( )

H = freq(sys,{Fmin=0.01,Fmax=10,npts=300});

[Gm,Pm] = margin(H)

Gm (a pdm) =

- domain |

----------+----------

0.595514 | 44.5062

Pm (a pdm) =

domain |

----------+----------

0.0257558 | 12.3814

Mp i() phase i() Δphase

Δω

-------------------Δω gain i 1+()

Δω

Δgain

---------------

–

⎝⎠

⎛⎞

Mg i() gain i() Δgain+()

Δphase

--------------------------------------------- Δω phase i 1+()

Δω

Δphase

-------------------

–

⎝⎠

⎛⎞

–=