Handbook for Sound Engineers

708 Chapter 20

. (20-11)

The nature of the feedback is best explored by
studying the quantity AB as a function of the frequency
as displayed in a Nyquist diagram. The quantity AB is
called the loop gain of the hypothetical amplifier.
Included in the diagram is a circle of unit radius
centered on the point l, j0.
The perimeter of the unit circle divides the plane into
two regions. For all of the points outside of the circle
, the feedback is negative, whereas for any
point on the curve within the unit circle, ,
the feedback is positive. The hypothetical amplifier for
which Fig. 20-9 was drawn thus has negative feedback
at low frequencies but exhibits positive feedback over a
range of high frequencies. Note that AB is negative,
real, and has its maximum absolute value at Z=0. This
is characteristic of a dc-coupled amplifier having a loop
gain transfer function that has poles but no zeros.
Furthermore, AB resides in the second quadrant until Z
exceeds Z 1 for Z 1 < Z < Z 2 , AB is in the first quadrant
but the feedback is still negative. Whenever the
frequency is such that Z > Z 2 , AB falls within the unit
circle and the feedback becomes positive. This region
must be handled with extreme care.

As will presently be discussed in detail, negative
feedback is a stabilizing influence on amplifier perfor-
mance but positive feedback is a destabilizing influence
and can, in fact, lead to an uncontrolled oscillatory
condition. The final conclusion to be drawn from Fig.
20-9 is that for Z > Z 3 , AB is in the fourth quadrant and
finally approaches zero as the operating frequency
becomes very large. As Z is allowed to vary from zero
to infinity, the angle associated with AB undergoes a
change of 270°, which is characteristic of a transfer
function that, in the absence of zeros, possesses three
poles. If AB for the hypothetical amplifier had possessed
just a single pole, the entire Nyquist diagram would
have been restricted to the second quadrant and the
feedback would have been negative for all frequencies.
On the other hand, if AB had possessed just two poles,

the Nyquist diagram would enter the first quadrant at

high frequencies but would approach zero without ever

crossing the real axis. The feedback would be positive at

high frequencies but not to an excessive degree.

The critical point to be avoided for stable operation

is the point 1, j0 on the positive real axis. If the Nyquist

curve passes through this point under any condition, the

loop gain becomes one with an angle of zero. As a

consequence, |1AB| becomes zero and Ac becomes infi-

nite. Physically this implies that the amplifier will

produce an output even in the absence of an input signal

from the outside world. That is, what was intended to be

an amplifier has become an oscillator. The type of

Nyquist diagram that is to be avoided is one that encir-

cles the critical point such as displayed in Fig. 20-10.

Fig. 20-10 was obtained by a modification of the ampli-

fier described by Fig. 20-9. This modification amounted

to changing AB at zero frequency from its former value

of 6 to a new value of 10 with all other factors

remaining the same.

Unlike Fig. 20-9, Fig. 20-10 is the Nyquist curve for

an unstable amplifier. The curve does not pass through

the critical point but it does encircle the critical point.

Consider for the moment that the amplifier is initially

off. Under such a circumstance, A is zero and, conse-

quently, AB is also zero. Under these conditions the

Nyquist curve is collapsed into a single point at the

origin. Following turn-on, there is a period of time in

which A and AB are growing toward their final values.

In this interval, the Nyquist curve is in effect growing

outward from the origin. At some instant during this

growth period, the Nyquist curve will intersect the crit-

ical point 1, j0 and the amplifier will break into oscilla-

tion. Precaution must be taken, therefore, when dealing

with feedback loops in which the loop gain transfer

function has three or more poles.

It was mentioned earlier that negative feedback can

be a stabilizing influence on amplifier operation. A

Figure 20-9. Nyquist diagram for loop gain.

1 –<1AB

1 –1AB!

1 –1AB

WW

W

WW

W¾

WW

Real

Unit

circle

Imaginary

Figure 20-10. Nyquist curve for an unstable amplifier.

Real

Imaginary