IEEE Circuits and Systems Magazine - Q4 2021 - 40
Education
Switched-Capacitor
Circuits
Gabor C. Temes, Un-Ku Moon, and David Allstot
T
he first publication
of a switchedcapacitor
resistor
occurred nearly 150 years
ago in James Clark Maxwell's
pioneering book A
Treatise on Electricity and
Magnetism [1]. He pointed
out that the average current
through a periodically
inverted capacitor C
remains the same when it
is replaced by a resistor of
value
RT /,C2=
where T
is the period of inversions.
He used this equ i v a -
lence to give a method
for measuring capacitance
using a circuit employing
a battery and galvanometer.
The next publication
came nearly a century later, when D. L. Fried proposed
the idea of sampled-data analog filters [2], containing
only switches, capacitors and (if necessary) amplifiers.
His paper showed how to realize the equivalent of a resistor
using two switches and a capacitor. A motivation
for using such circuits may be found from the history of
analog filters. These were developed for telephony, and
used initially resistors, capacitors and inductors. Inductors
were bulky and lossy, and were replaced at the earliest
opportunity by alternative circuits using amplifiers.
The resulting active-RC filters were realized by discrete
elements: capacitors, resistors and integrated-circuit amplifiers.
With the development of integrated-circuit technology,
there was strong motivation to put these filters
on a single-substrate IC. However, the absolute values of
Digital Object Identifier 10.1109/MCAS.2021.3118195
Date of current version: 12 November 2021
40
IEEE CIRCUITS AND SYSTEMS MAGAZINE
resistors and capacitors could only poorly be controlled
by the fabrication process: errors of 20-30% were common.
Since the errors of resistors and capacitors were
not tracking each other, the errors of time constants given
by RC products were unacceptably high. This made their
frequency responses unpredictable. Trimming could be
used to tune such filters, but this was expensive. When
the resistors were replaced by their switched-capacitor
(SC) equivalents, the RC time constants were replaced by
time constants of the form /.TC C12
Since the switching
period T can be accurately controlled, as can the ratio of
on-chip capacitors, the SC filters (SCFs) could be implemented
with high accuracy. The design of such filters
initially was based on those of active-RC ones, but it was
soon recognized that they can be more effectively designed
directly in the sampled-data domain, in terms of
the z variable, similarly to digital filters. Although other
design techniques exist, the most popular one constructs
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IEEE Circuits and Systems Magazine - Q4 2021
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