IEEE Solid-States Circuits Magazine - Spring 2022 - 24

Plot Twist
Well, things did get weird with our
original example of Figure 1. Looking
at Figure 4, the Nichols plot for this
example, you begin to see how you
might get confused about inside and
outside. (And it does get weirder than
this.) The biggest problem is that −180°
is indistinguishable from +180°, and
−360° is really 0°. It can be very confusing,
and phase wrapping just makes
things worse. The version of the full
Nichols plot for this problem that I like
is de picted in Figure 5, where only the
" high-gain " version of the loop (the blue
line) is shown.
The S-plane source of the mapping
is not illustrated in Figure 5, but
it looks like Figure 3(a), including
the shaded region, which is meant
to enclose all the RHP singularities.
Aside from the occasional dent to
avoid singularities lying exactly on
the
j axis-~
, this D-shaped region is
always the same.
There are several thought-exper-360
-270 -180 -90 090 180 270 360
Open-Loop Phase (°)
60
at Origin
iment tests that you can use to distinguish
inside from outside, but
one of the easiest is this: right turns
in the S-plane translate into left
turns on the Nichols chart and vice
versa. So, at
j~ infinity=+
, where
the s-plane contour turns right to
start around the semicircle, the
Nichols chart contour turns left in
the lower-left corner. Also, for our
particular example, where S goes
from j
~ 0= - to j~ 0= +
-60
, there is no
turn, so a jump from −180 to +180
gives a continuous contour.
Such thought experiments con-120
at
Infinity
-180
-240
FIGURE 5: The closed contour mapping of H(s) using the example in Figure 1 (blue line only).
firm that the shaded area of Figure 5
does indeed correspond to the RHP
of the S-plane. And because the
peaks of Mt. Nichols are outside
this region, we can be sure that the
closed loop will be stable. Looking
back at Figure 4, however, we see
that the low-gain version of the loop,
the red line, goes around the other
side of Mt. Nichols and thus will not
be stable. Skeptics are referred to the
step-response transients in Figure 6.
All of this is not really doing
3
2.5
2
1.5
1
0.5
0.1
0.2
0.3
Time (s)
FIGURE 6: A closed-loop, unit-step response for the two systems in Figures 1 and 4. The blue
and red lines are high- and low-gain versions, respectively. Do not mess with Mt. Nichols!
24
SPRING 2022
IEEE SOLID-STATE CIRCUITS MAGAZINE
0.4
0.5
the subject justice but hopefully is
enough to get you started. If you want
to get serious about this, there are
many good explanations composed
by people who do know what they're
doing [3], and you should seek their
expertise. But before leaving the
subject, there is one more shortcut
that will help: simulators will often
answer the question of absolute stability
for you. You need to be wary
because an unstable system can balance
in a " metastable " state for quite
a long time, but transient simulations
will eventually blow up if the system
isn't stable. MATLAB even has a function,
called isstable(), which answers
the question directly. (Do not confuse
this function with istable().
See " Practical Tips for Trying This
Yourself " ). So the question of inside
Amplitude
Open-Loop Gain (dB)
Negative
Frequencies
Positive
Frequencies

IEEE Solid-States Circuits Magazine - Spring 2022

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