IEEE Solid-State Circuits Magazine - Winter 2016 - 8

A C ircu it for All Season s

Behzad Razavi

Translational Circuits

T

Early History
The use of time-variant systems (e.g.,
mixers) for the translation of transfer
functions dates back to the late 1940s
and early 1950s. In their 1948 paper
[1], Busignies and Dishal propose the
arrangement shown in Figure 1(a),
where a mechanical wheel holding a
number of capacitors rotates at a rate
of f turns per second, allowing each
capacitor to charge to the input and,
half a cycle later, discharge at the output. The input-output transfer function thus appears as in Figure 1(b),
revealing a "comb" filter response
with band-pass peaks located at the
harmonics of f.
In 1953, Le Page et al. [2] presented the implementation depicted
in Figure 2, which employs switches
A and B to alternately connect each
capacitor to the input and to the output. Also in 1953, Smith [3] provided
a comprehensive and intuitive treatment of these circuits. Shown in Figure 3(a) are Smith's original drawings,
suggesting that a first-order low-pass
RC response can be translated to a
center frequency of fR if the capacitor
Digital Object Identifier 10.1109/MSSC.2015.2495821
Date of publication: 21 January 2016

8

W I N T E R 2 0 16

Rotating
Brushes

vo/Ei

Translational circuits are periodically
driven, time-variant systems and encompass topologies such as "commutated networks" and " N -path filters." These systems cause frequency
"translation" (shift) of impedances and
transfer functions, a useful property
that has led to myriad new and interesting concepts in RF design. In this
article, we study the operation principles of these circuits.

RI
Ei

R2

Capacitors

vo
0

f

2f
Frequency

(a)

3f

(b)

Figure 1: (a) A 1948 electromechanical system and (b) its transfer function.

Cp

ep(k )
+

A

2 πfrt

B
2 πfr
(t - mTi)

C1
+
et(k )
+

C0
eo(k )
R
+

+
eo(t )

e(t )

Figure 2: A 1953 electromechanical system with translational behavior.

is replaced by an array of N commutated capacitors that connect to the
output node one at a time at a rate

IEEE SOLID-STATE CIRCUITS MAGAZINE

of fR . (This action is realized today
by N switches driven by nonoverlapping phases of a clock.)



Table of Contents for the Digital Edition of IEEE Solid-State Circuits Magazine - Winter 2016

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