Gain margin and phase margin
•
A( s)
A
(
s
)

f
Closed-loop gain of amplifiers with feedback is
1  A( s) 
replacing s=j2πf gives the closed loop gain as a function of
frequency A ( f )  A( f )
1  A( f ) 
For a given frequency f1, if βA(f1)=-1, the close loop gain becomes
infinite. This corresponds to a pole on the jw axis at s= j2πf1. The
transient response then contains a constant-amplitude sinusoid.
In considering the stability of a feedback amplifier, we examine the
bode plot for the loop gain βA(f) to find the frequency fgm for which
the phase shift is -180 degrees. If the magnitude of the loop gain is
less than unity at fgm, the amplifier is stable. On the other hand, if
the loop gain magnitude is greater than unity, the amplifier is
unstable.
For a stable amplifier, the amount that the loop gain magnitude is
below 0db is called the gain margin. A gain margin of zero implies
that a pole lies on jw axis. In general, a larger gain margin results in
less ringing and faster decay of transient response.
Another measure of stability from the loop gain bode plot is the
phase margin, which is determined at the frequency fpm for which the
loop gain is unity (0db). The phase difference between the actual
phase and -180 degrees is the phase margin.
f
•
•
•
•
Stability of amplifier
•
•
Since we want to design feedback amplifiers to avoid transient response
ringing and frequency response peaks, a generally accepted rule of thumb is
to design for a minimum gain margin of 10db and a minimum phase margin
of 45 degrees.
Typical magnitude and phase plot for stability consideration:
Pole compensation I
• Negative feedback is useful to reduce distortion, stabilize gain and
increase bandwidth. But to achieve these benefits, the loop gain A0 
must be much larger than unity.
• On one hand, we can design an amplifier with a large open-loop
gain. This calls for several stages of amplification, and multi-stage
amplifiers invariably introduce multiple poles.
• On the other hand, a large value of feedback coefficient may lead to
instability in a multiple-pole amplifier.
• Thus, we must deliberately modify the pole locations (or equivalently
frequency and transient response) of the amplifier before feedback
can be used effectively. The is called compensation.
Pole compensation II
• Instability occurs if the magnitude of loop gain A0  is greater than
0db at the frequency for which the phase is -180.
• Each pole potentially contributes a phase shift between 0 to -90 at
any given frequency.
• For a single amplifier, instability is not a problem, since the extreme
phase is -90.
• For a two pole amplifier, the extreme phase is -180, which occurs
until frequency approaches infinity. However, it is possible for the
phase to become very close to -180 at the frequency for which the
loop gain is 0db, resulting in very small phase margin, hence long
transient ringing and large frequency response peaking.
• For three or more poles amplifier, a phase shift of -180 is possible
before the loop gain magnitude has dropped below 0db. Thus, an
amplifier having three or more poles can become unstable.
• There are several approaches to compensate the pole location. One
popular method, called dominant-pole compensation, is to add
another pole at a very low frequency, such that the loop-gain drops to
unity by the time the phase reaches X (e.g. -135 degrees). In this
way, a phase margin of X+180 (e.g. 45 degrees) could be achieved.
Pole compensation III
• Most of the amplifier design are OpAmp based design.
• A good example for pole compensation at the concept
level is illustrated in the book Page 623, E.g. 9.11.
Before compensation
After compensation
Frequency response
Transient response
Three examples of IC OpAmp I
Three examples of IC OpAmp II
http://en.wikipedia.org/wiki/Operational_amplifier
Three examples of IC OpAmp III