A JSSC Classic Paper:
The Gilbert Cell (continued from January
2003 issue)
In the last column about the genesis of
the first translinear multiplier "core," an attempt was made
to capture the series of events that led up to its discovery. Here,
that account is aided by presenting the actual circuit forms, and is
extended by a discussion of the rea-sons for its importance when it
was first reported at the 1968 ISSCC and in two linked articles in the
December 1968 JSSC, "A new wide-band amplifier technique,"
and in the fifth most cited article in the JSSC, "A precise four-quadrant
multiplier with subnano-second response." Also, an explanation
of its continued value during the intervening thirty-five years to the
present is given.
However, before doing that, it is useful to consider some key ideas
about current-mode signal processing. This term was coined-possibly
not for the first time-and popularized by Gilbert to make a clear distinction
between the fundamental nature of these interesting new cells, which,
it quickly became apparent, belonged to a genuinely separate design
class. Numerous current-mode cells-fixed and variable-gain amplifiers;
multipliers and dividers; squaring, square-rooting, N-dimensional vector
normalization, vector addition, and subtraction cells; absolute-value,
maximum, and minimum cells; and so on-were discovered within the first
year. Many appeared in the two widely referenced JSSC articles that
prompted the IEEE to suggest writing this account.
Two key criteria define a current-mode circuit. First, all of the state
variables must be in the form of currents. Formally, this refers to
the system variables that appear in the equations that define the boundary
function; informally, we can think of them as the primary signals. Second,
the circuit's essential function must be independent of any deliberately
introduced time constants. Their inclusion requires mode transformations,
specifically, the introduction of state variables in the form of voltages.
This is implied in the first criterion, but underscores the fact that
filters and oscillators, for example, do not belong in the class of
current-mode circuits.
Because this term nowadays is used so freely, we may add three more
practical criteria. Third, the circuit concept must be inherently complete
with regard to its current-mode aspects; that is, it should not depend
on the use of any unspecified supporting agents, or special functional
derivatives of the state variables that may require recourse to conventional
circuit techniques involving voltage-mode state-variables. Fourth, when
the proposed current-mode circuit addresses the realization of a function
that formerly was implemented in a conventional manner, it ought to
offer clear, compelling, and defensible advantages over prior art, rather
than just replicating the function in a different, albeit interesting,
way.
Finally, the intrinsic merits of a novel current-mode circuit should
be readily and widely understandable by the design community at large,
thus holding out the prospect that it will be utilized rapidly by other
designers and become widely adopted in actual products. While these
criteria may seem excessively strict, the newly proposed current-mode
circuit must earn its reputation through widespread acceptance and use,
or risk being soon for-gotten, along with hundreds of other curiosities
that fill the pages of transactions and journals.
The Elements Come Together
BJT current mirrors and differential pairs were in wide use in 1965.
As a linear-circuit element, the mirror was a truly new form, only possible
in a monolithic realization, and was the first and simplest current-mode
cell. By contrast, the differential pair is a mode-transforming element.
An input voltage, VIN, applied between the bases unbalances
the collector currents, making an output IOUT. But this cell
has a serious flaw: its IOUT vs VIN relationship
is very nonlinear, having the form tanh(qVIN /2kT), with
a quasi-linear region of a few millivolts. On the other hand, its transconductance
is an almost exactly linear function of the tail current, a property
later identified by the name translinear. This was useful in 1966 when
analog multiplication was still of broad general value and of special
interest to the author for use as the core of a 500-MHz variable-gain
amplifier.
 |
| A B |
|
Figure 1. Two Current mirrors merged to realize
a current-mode multiplier.
|
As noted in the previous article, the question arose:
Is there some way of merging these two basic cells to realize a super-cell,
having the linearity of the fixed-gain current mirror but the variable-gain
properties of the differential pair, by exploiting the excellent linearity
of its transconductance value with respect to the gain-controlling (tail)
current? This was not only possible, but direct and effortless, as shown
in Figure 1. This one-step metamorphosis lead immediately to a current-mode
analog multiplication element that was linear with respect to both of
its inputs. This cell, the first four-transistor current mode cell,
was the cause of all the excitement behind that early paper.
Since the X-input can represent a bipolarity signal, but the Y-input
only can have one polarity, it is called a two-quadrant multiplier.
The extension to four-quadrant operation (the more generally expected
function of an analog multiplier) was straightforward, and also was
presented in the paper, as was an alternative form in which the emitters
of Q1 and Q4 were driven by the X-input currents and connected to the
bases of Q2 and Q3 (Figure 2a). That form has been very widely used
through the present time. Often, these six-transistor cells were called
"transconductance multipliers." The term is clearly incorrect,
since all inputs and outputs are in current form; the transconductance
aspects of its operation are all internal to the cell, and completely
incidental to its operation. However, by removing the outer diode-connected
transistors, and driving the inner bases directly with voltages, the
four-quadrant form is still capable of analog multiplication. That form
does invoke a transconductance mode, although only very small base voltage
swings can be allowed before the linearity degrades severely. In turn,
that results in a very poor signal-to-noise ratio.
However, when these multipliers were deliberately over-driven at their
bases (Figure 2b) they could be used in a current-steering, or switching,
mode to realize the core of a low-noise commutating mixer. This is a
very important mode of use, and the form has become known as the "Gilbert
Mixer," in both its BJT and more recent MOS embodiments. It should
be noted that, while processing these early multiplier patents, this
more basic form of the core was found embedded in a patent concerning
a synchronous demodulator by H.E. Jones, issued in 1963. In spite of
strenuous efforts to correct this incorrect assignment at conferences,
in Short Courses, and at informal gatherings, the name persists.
  |
| A B |
|
Figure 2. A Current-mode linear four-quadrant
multiplier (a) and a fully balanced mixer (b).
|
Since its appearance, Gilbert's 1968 paper has become the fifth most
frequently cited paper from the JSSC and the earliest JSSC paper to
be cited over 100 times. The original paper is available on the Solid-State
Circuits Digital Archive 2002 DVD at www.sscs.org/Archive
(IEEE product # JD3755B); or link to the article in IEEE Xplore®
through the JSSC Classics icon at http://sscs.org/jssc.htm.
See "So Many Articles, So Little Time."
Barrie Gilbert
Analog Devices
barrie.gilbert@analog.com
