IEEE Solid-States Circuits Magazine - Spring 2022 - 22

you how much peaking your circuit
has. A SPICE ac analysis will cough
up closed-loop response about as
fast as you can type in the request.
But, like all good tools, the Nichols
chart gives you an intuitive feel for
how shifting parameters will change
the result.
Notice that both phase and gain
margin could be read directly
off of Figure 2(b) if the grid
were detailed enough. But
that's only two points on
the trajectory. The ()jH ~
curve could sneak up Mt.
Nichols from any angle.
So, although Bode plots
are usually good enough,
they are clearly not
telling you the whole
story. The Nichols chart
gives you a nice visual landscape
for understanding feedback performance
no matter how unorthodox
your loop is.
By the way, you can easily make
the 2D version of the Nichols chart
by doing an X-Y plot of your normal
Bode data. That is, plot the
open-loop gain (in dB) versus the
open-loop phase (in degrees). Voila.
However, you may soon find that
this is not really necessary. A rough,
hand-drawn sketch is often enough
to trigger the " aha moment. " Such is
the power of Nichols!
The Nichols
chart gives you
a nice visual
landscape for
understanding
feedback performance
no matter
how unorthodox
your loop is.
But Is It Stable?
Okay, here's the big one: absolute
stability. Is it stable or is it not? The
short answer is this:
for most circuits, if the
line on the Nichols chart
passes the mountain on
the right (closer to zero
phase), like the orange line,
it is stable. If it passes on
the left side, it is unstable.
That's it. No big fanfare is
required here because this
is something you knew al -
ready. If you think about it,
" passing the mountain on the right "
is the same thing you would require
on a Bode plot: passing 0 dB with
less than 180° of phase shift. Right?
The aforementioned short answer
is a wild and horribly irresponsible
oversimplification-you're welcome.
But we can't just leave it there because
you don't really need a Nichols chart
for " most circuits. " The whole point
of dragging you into all of this was
to help you cope with unusual situations
like those in Figure 1. So, we're
going to have to dig a little deeper.
But it's worth it because the full story
is nothing short of amazing.
First, be aware that we have
looked at only half of the Nichols
chart thus far. Like all complex functions
that describe real quantities,
G(s) has conjugate symmetry. The
other half, which maps the values of
G(s) on the negative jw axis, is the
mirror image, so we don't normally
need to plot it. In this case though,
it's helpful to see it, so have a look
at Figure 3(b). Note that the inside
of the contour is shaded, and the −1
points at −180 and +180 are outside
the shaded region.
Remember conformal mapping?
No? Me either. Conformal mapping
is the almost-mystical property of
some complex functions that preserve
geometrical attributes across
bizarre transformations. In engineering,
it is used like a change-ofcoordinate
system to transform an
intractable problem into an almost
trivial solution. It can also be used for
general relativity and the modeling
-360 -270 -180 -90 090 180 270 360
Open-Loop Phase (°)
60
40
20
at Infinity
σ = Real Axis
(a)
-20
-40
(b)
FIGURE 3: The closed contour mapping of H(s) using the example of Figure 2. (a) The S-plane and (b) open-loop gain-phase plane of the
Nichols chart.
22
SPRING 2022
IEEE SOLID-STATE CIRCUITS MAGAZINE
jω = Imaginary Axis
Negative
Frequencies
Positive
Frequencies
at Infinity
Open-Loop Gain (dB)
Negative Frequencies
Positive Frequencies

IEEE Solid-States Circuits Magazine - Spring 2022

Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Spring 2022