IEEE Solid-States Circuits Magazine - Fall 2021 - 15

SHOP TALK: WHAT YOU DIDN'T LEARN IN SCHOOL
Chris Mangelsdorf
Duct Tape for the Frequency Domain
If This Ol' Hack Don't Fix It, It Ain't Broke!
D
Dominant pole compensation has a
problem. It has always had a problem.
But don't panic, this is not
a recall notice. And it's not about
amplifiers. An ancient solution to
the dominant pole problem is one
of the most useful, general-purpose
techniques that you can have in your
tool kit. To show off its utility, we'll
use it to subdue the ringing in power
supply wiring, but you can apply it
to patch up all kinds of stability, resonance,
and phase problems in the frequency
domain.
The Defect
Compensating feedback systems with
a single, dominant pole is robust,
convenient, and older than all of us.
For a wide range of feedback factors,
the phase margin of the unloaded
closed loop is a comfortable 90º.
So, what's the problem? Well, 90º of
margin doesn't permit the addition
of another pole within the useful
frequency range. If you've got a general-purpose
operational amplifier
(op-amp), for example, you can't load
it with a capacitor, at least not one of
any substance. So, unlike the feedback
factor, you have to approach
your load with caution. This just
doesn't fit with the plug-and-play
image of the commodity op-amp.
Wouldn't it be nice if these useful little
bricks could drive capacitors with
impunity, too?
Digital Object Identifier 10.1109/MSSC.2021.3111427
Date of current version: 17 November 2021
Shrooms to the Rescue
Some smart person or persons in
distant antiquity-perhaps as far
back as the ancient Mayans-envisioned
a creative solution to the tyranny
of the dominant pole. (It may
be that hallucinated is a more accurate
verb here. Early Mesoamerican
civilizations were quite advanced
in psychopharmacology.) Instead of
a dominant pole, a dominant halfpole
could be used! What the heck
is a half-pole? I'm afraid you need to
ingest the mushrooms to fully comprehend
such a thing, but the point
is this: rolling off an op-amp with
only -10 dB/decade of attenuation
would create an amplifier phase shift
of only -45º, leaving a total
unloaded phase margin of
135º. (Abstinent readers
without mushrooms are
referred to the blue curve
in Figure 1.) Now you
could add a full -90º of
phase to the loop with
a really ugly capacitive
load and still have 45º
of phase margin left.
Nirvana! Or whatever perfection was
to the ancient Mayans.
This -10 dB/decade roll-off is known
bulk materials exhibit an alternating
pole-zero behavior in distributed
fashion and do the square-root-of-s
trick naturally. But it turns out that
for most applications, a precise 45º
of phase is not needed, so just a few
singularities are good enough. In fact,
a single pole and a single zero can
sometimes do the job.
An ancient
solution to the
dominant pole
the most useful,
general-purpose
techniques that
you can have in
your tool kit.
Objection, Your Honor!
At this point, counsel for the defense
demands to know why, if the squareroot-of-s
is such a great technique,
it isn't used everywhere, especially
in general-purpose op-amps. The
answer is simple: bandwidth and
gain. Negative feedback systems
become progressively
more ideal as the
loop gain increases.
The closed-loop transfer
problem is one of
function becomes more
precise, the gain-related
portion of the input offset
is reduced, the linearity
improves, some sources of
noise are reduced, and so on.
All the good stuff that comes
as slow roll-off or square-root-of-s
compensation (where s is our complex
friend from the Laplace transform),
and it is achieved by alternating poles
and zeros to approximate half a pole.
The closer the poles and zeros are in
frequency, the closer the resulting
Bode plot approximates the preternatural
-10 dB/decade. In fact, some
with feedback just gets better as we
raise the loop gain. Loop gain is the
carpet under which we sweep all
our circuit's " dirt. " We recognize this
immediately when it comes to dc performance,
but it holds true at higher
frequencies, as well. So, the long,
drawn-out decline of square-root-ofs
compensation doesn't do you any
favors in this department.
Figure 1 illustrates both the con -
ventional dominant pole and the
IEEE SOLID-STATE CIRCUITS MAGAZINE
FALL 2021
15
IEEE Solid-States Circuits Magazine - Fall 2021

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