Gain and Phase Margins

` `
As described in Nyquist Example #3, gain margin
and phase margin are two measures of relative stability. They measure how
"close" a system is to crossing the boundary between stability and
instability, in one direction or the other. This example will show how
those margins are found on the Bode plots of a given system. As usual,
the Bode plots are made of the open-loop system G(s)H(s); gain and phase
margin measure how close the roots of the closed-loop characteristic
equation 1+G(s)H(s) are to the jw axis.

` ` Gain margin is the amount of
change in the value of the gain of the transfer function, from its present
value, to that value that will make the magnitude Bode plot pass through
the 0 db at the same frequency where the phase is -180 degrees. The phase
margin is the amount of pure phase shift (no change in magnitude) that
will make the phase shift of G(jw)H(jw) equal to -180 degrees at the same
frequency where the magnitude is 0 db (1 in absolute value).

` `
The equations from Nyquist Example #3 related to
gain and phase crossover frequencies and defining the gain and phase
margins are repeated here. See that example for further description of
the definitions.

` `
We will look at a 5th-order system and study the effect that changing the
gain of the transfer function has on the values of gain and phase margin.
The system is Type 1, has 5 poles and 2 zeros. None of the poles or zeros
is in the right-half of the s-plane. Only positive values of K will be
considered (the closed-loop system is unstable for all K < 0). From this
knowledge of the open-loop transfer function, you should know that the
magnitude plot has a slope of -20 db/decade at low frequencies and a slope
of -60 db/decade at high frequencies. The phase curve has a value of -90
degrees at low frequencies and a value of -270 degrees at high
frequencies. The open-loop transfer function and the Bode plots are

Bode plots with K = 2

` `
Now for the gain and phase margins. The gain margin is defined only at
that frequency where the phase shift of G(jw)H(jw) is -180 degrees (the
phase crossover frequency w_{phi}). On the Bode plot, the gain margin in
decibels is the negative of the magnitude (in db) at the phase crossover
frequency. On the next figure, the dashed horizontal lines are at 0 db
and -180 degrees. The phase crossover frequency is the frequency where
the phase curve crosses the dashed line at -180 degrees. For this system,
the phase crossover frequency is 2.5 r/s. At that frequency, the
magnitude curve is at -26.8 db, so the gain margin (in db) is +26.8 db.
This is indicated by the vertical dashed line from the magnitude curve up
to 0 db at w_{phi}. The gain margin of 26.8 db corresponds to a gain margin
of 21.9 in absolute value, which is the reciprocal of the absolute value
of the magnitude at w_{phi}. That is, if the gain is multiplied by 21.9
from its present value of K=2, the closed-loop system will have poles on
the jw axis.

Gain and Phase Margins with
K=2

The phase margin is defined only at that frequency where the magnitude
of G(jw)H(jw) is 0 db (1 in absolute value). This frequency is the gain
crossover frequency wx. On the Bode plot, the phase margin is the
distance from the phase curve to -180 degree value, measured at w_{x}. The
phase margin is positive if the phase curve is above the -180 value;
otherwise it is negative. For this system, the gain crossover frequency is
0.175 r/s. At that frequency, the phase shift of G(jw)H(jw) is -109.2
degrees, so the phase margin is +70.8 degrees. This is indicated by the
vertical dashed line from the phase curve down to the -180 degree line at w_{x}. This phase margin is the amount of negative phase shift which could
be added to G(jw)H(jw) at the gain crossover frequency which would put
closed-loop poles on the jw axis.

**For this system, there are no open-loop poles in the right-half
plane, and there is only one gain crossover frequency and one phase
crossover frequency. For this case, the following statements are
equivalent:
**

- the closed-loop system is stable;
- the phase margin is positive;
- the gain margin (db) is positive (GM > 1 in absolute value); and
- the gain crossover frequency is smaller than the phase crossover frequency.

Thus, with K = 2, this system is closed-loop stable. The values of phase margin and gain margin are relatively large, so it would take quite a bit of perturbation to the gain or phase to make the system unstable.

` `
If the gain is increased so that K = 25, the phase curve is unchanged, but
the magnitude curve moves up by 21.9 db at all frequencies
[20*log10(25/2)].

Gain and Phase Margins with K=25

The phase crossover frequency is unchanged, but the gain margin has been reduced from 26.8 db to 4.9 db, a reduction of 21.9 db (which corresponds to the gain increase from 2 to 25 when converted to db). The gain margin is just barely visible on the Bode plot. The gain margin in absolute value is 1.75. That is, the gain can be increased from 25 by a factor of 1.75 without the closed-loop system becoming unstable.

The gain crossover frequency has increased in value since the magnitude
curve has been shifted upward. The new value is w_{x} = 1 .77 r/s, and the
phase margin is 43.4 degrees. The phase margin is still fairly good, but
the gain margin is small. This illustrates the fact that neither gain
margin nor phase margin by themselves is an adequate measure of true
robustness of closed-loop stability to perturbations to the system. Since
the phase margin is positive, the gain margin in db is positive, and w_{x} <
w_{phi}, the closed-loop system with K=25 is stable.

` `
When the gain is increased further so that K = 400, the phase curve is
still unchanged, but the magnitude curve moves up by another 24.1 db at
all frequencies [20*log10(400/25)].

Gain and
Phase Margins with K=400

Even though the phase crossover frequency is unchanged, the gain margin has been further reduced, from 4.9 db to -19.2 db, a reduction of 24.1 db (which corresponds to the gain increase from 25 to 400 when converted to db). The gain margin in absolute value is 0.109. Therefore, the gain must be reduced from 400 by a factor of 0.109 in order to make the closed-loop system stable.

The gain crossover frequency has increased in value again since the magnitude curve has been
further shifted upward. The new value is w_{x} = 5.5 r/s, and the phase margin is -30.7 degrees.
The phase margin is now negative. This coincides with the fact that the gain margin is
negative (less than 1 in absolute value), and the gain crossover frequency is greater than
the phase crossover frequency. The magnitude curve is above 0 db at the phase crossover
frequency, and the phase curve is below -180 degrees at the gain crossover frequency. Each of
these indicates that the closed-loop system is unstable when K = 400.

` `
This example shows the effect that a change in open-loop gain ahs on the
closed-loop stability of the system. By changing the value of K, we can
increase or decrease the values of gain and phase margin. This makes the
closed-loop system "more stable" or "less stable", and can even make the
closed-loop system unstable.

**Where there are no unstable open-loop poles, and only one value for w _{x} and one for
w_{phi}, the conditions of positive phase margin, positive
gain margin, and w_{x} < w_{phi} will either be all present (closed-loop
stable) or all absent (closed-loop unstable).**

Table of Gain and Phase Margins

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*Lastest revision on
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