IEEE Solid-States Circuits Magazine - Fall 2021 - 22

DOUBLET VOODOO
If you propose a square-root-of-s solution,
such as Figure S1, to your colleagues, be
prepared for the following reaction: the local
circuit shaman will rise up, trembling
with righteous indignation, his or her voice
quavering with ominous foreboding, and
accuse you-yes, you-of visiting famine,
pestilence, and long settling tails on the
land. All your efforts will be doomed to a
grim future of inaccurate misery! Not quite
the reception you were hoping for, huh?
Unfortunately, there is quite a bit of folklore
and superstition surrounding pole-zero
doublets that is both scary and patently
false. The fearsome reputation of doublets
probably arose from instances of imperfect
pole-zero cancellation, a situation
that can catch an inexperienced designer
off guard. These closely spaced poles and
zeros also have a unique ability to lie hidden
in a Bode plot, only to appear as long
time-constant tails in the fine settling of a
feedback loop. (Perhaps, if you ask me nicely,
this could be the subject of a future column.)
When you add poles to a network, you add
Square-Root-of-s
1.2
0.2
0.4
0.6
0.8
1
-0.2
-5
-4
-3
-2
-1
1
(b)
1.2
1.4
1.6
1.8
2
" natural frequencies " to the behavior. The undriven
circuit response is a weighted combination
of these natural frequencies. A squareroot-of-s
network used at frequencies below
crossover will have time constants longer than
the settling of the main loop. No surprise. The
effect that these time constants have on the
overall settling varies with things like polezero
separation and the loop gain at the given
frequency, but it is usually not a deal breaker
[see Figure S2(a) and (b)]. For one thing,
high-accuracy settling requirements are rare
these days. But what you should do-what you should always do-is check the transient simulation of your network. To paraphrase the famous Paul Brokaw, a
transient simulation answers the questions you forgot to ask. (Hint: a noise spectrum is another simulation that can reveal hidden defects.)
As for the hostile circuit shaman, tell them this: the plots in Figures S1(a) and S2(a) are unloaded. When you add the 20-nF cap load, which was the
0.2
0.4
0.6
0.8
1
-0.2
2
4
6
8
10
Time (µs)
(c)
FIGURE S2: Current step responses for the examples in Figure S1. (a) The unloaded response
showing the slow settling of the square-root-of-s circuit. (b) The percentage settling error of
the unloaded square-root-of-s loop compared to the unloaded original. (c) The loaded transient
response. Do you notice any difference between the two curves?
12
14
16
18
20
purpose of the square-root-of-s network in the first place, it forms an 8-kHz pole with the 1 KΩ output resistance and drags the crossover down
to 200 kHz. Now the offending doublet staddles crossover, making it a classic lead compensation network. Boom! Even the orneriest curmudgeon will
back down when you say, " lead compensation. " Figure S2(c) will back you up.
is more of a mindset, a way of
approaching frequency domain
problems. That mindset is best represented
by the geometric insights
from Figure 6. It provides ideas
about how to tailor your Bode plot
and what tradeoffs, if any, you
might incur. It also gives you a starting
point for where you might place the
22
FALL 2021
poles and zeros, but from there, you're
welcome to improvise. As always, feel
free to share your thoughts (and your
shrooms) with me at shifobrains@
ieee.org.
References
[1] T. H. Lee, The Design of CMOS Radio-Frequency
Integrated Circuits. Cambridge,
U.K.: Cambridge Univ. Press, 2004.
IEEE SOLID-STATE CIRCUITS MAGAZINE
[2]
J. K. Roberge, Operational Amplifiers:
Theory and Practice. New York, Wiley,
1975.
[3] C. Mangelsdor f, " Oscillat ion and
ringing: Finding worst-case capacitive
load [Shop talk: What you didn't
learn in school], " IEEE Solid-State Circuits
Mag., vol. 13, no. 2, pp. 16-20,
Spring 2021. doi: 10.1109/MSSC.2021.
3072801.
1.1
0.8
0.9
1
2
2.1 2.2 2.3
(µs)
(a)
Original
Percentage Difference
Loaded Transient (V)
Unloaded % Difference
Unloaded Transient (V)
(V)

IEEE Solid-States Circuits Magazine - Fall 2021

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