IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. SC-3, NO. 4, DECEMBER 1968
A New
Wide-Band
BARRIE
Absfracf—Precision
less
than
monolithic
that
has
stage
de-coupled
a nanosecond
have
amplifiers
recently
planar
process.
The design
a broad
range
of applications
gain
accurately
determined
stage-gain-bandwidth
product
tors,
transfer
and a very
linear
having
is based
and
by
the
GILBERT,
risetimes
of
using
the
on a simple
tectilque
is characterized
ratio
essentially
Amplifier
fabricated
been
of two
by
a
currents,
a
equaf to that
of the transis-
free
temperature
characteristic,
from
MPLIFIERS
of
100
and
sub-nanosecond
of
voltage
the
far
two
amplitude
most
due
to
the
distortion
and
rarely
excess
in modcircuit
con-
amplifier,
which
characteristic,
distortion
[1],
[2]
techniques
applicable
phase
over
at
shift
these
around
A
further
characteristic
swings
throughout
the
capacitances
This
at
have
is
tion-isolated)
inductive
improve
the
corporating
is available.
For
electronically
part
amplifiers
bandwidth,
output
capability.
Suggestions
to
diode,
this
point
has
several
produce
The
have
later
each
all
stages
been made
or even
limitations,
a stage
behave
that
the
having
objective
too,
diodes
from
time
in the
This
and
voltage
of the
the
in-
of
the
dis-
an
to the
f~-
total
In
by
the
impedance
at
I=
but
I,
technique,
The
of certain
in
a new
analysis
also
configurations
is
to the
author’s
knowl-
still
here
we
base
to
was
16, 1968.
in
Fig.
3; this
the
(b),
used
is the
lC.
marriage
prolific
The
widely
linear
will
consider
1, were
first,
(a),
as a multi-
“current
The
source”
married
has proven
offspring
Fig.
as a multiplier
will
this
1 (a)
because
couple
unexpect-
be described
in
more
detail.
It
the transconductance
is proportional
to collector
because
TECHNIQUE
shown
now
second,
~E. However,
this
OF THE
to the emitter
it is far frOm being
transconductance
a precise
tail
ele-
is both
nonlinear
use the
junction-
dependent.
paper,
we will
frequently
(14].
where
forward
The
plying
conduction
= reverse saturation
v=
q=
m..
T=
Mantlscript
received
June 28, 1968: revised
September
This paper was presented
at the 1968 ISSCC.
The author
is with
Tektronix,
Inc., Beaverton,
Ore.
exploited
papers.
diode expressions
introduced
reported
can be
permits.
every
and
and temperature
[7]
is
useful
over a wide range
[9].
circuits,
The
and later
ment,
cir-
gain.
work
from
[5]-
motivation
these
other
of each stage
precision
that,
pair,’)
in Fig.
fruitful,
in this
load
linear
to meeting
several
new configuration.
a
in practically
are shown
edly
common
produce
[10] – [ 13].
current
applicable
to time
gain
FOUNDATIONS
‘(differential
First,
(the
like
to
can be used
parasitic
at
stage
is something
very
found
collector
is a useful
the
to
of
line
works
contribute
[3]
circuit
much
stage
eliminate
com-
some
very
nonlinearities
[7].
is not
however,
in
altogether
main
this,
and
(junc-
delay
voltage
needed—at
of beta, even at gains close to beta in mag-
This
plier
bandwidth,
these
constant
the gain
with
edge, no other configuration
is the
parasitic
a monolithic
had
[8],
Two
way
where
in addition
a property
married
employed
is to make
[4] ), but
compensating
mitigate
emitter
in
Interestingly,
limited
to include
often
satisfactory
that,
developed
speeds,
that
discrete
no
a lumped
described
using
the upper
between
only
multiplier
nitude.
absence
transfer
slope between
four-quadrant
independent
virtual
perfect
terminals.
example,
the
the
Conversion
at nanosecond
present
monolithic
are
but
approach
amplifier
microcircuits.
amplifiers
circuits
elements
of
the
in
with
be made
controlled
that
cell”)
the
is
are
the
limiting
severe
elements
Another
distributed
in
In
inductive
capacitances
cuit
part
bandwidth,
amplifiers
frequencies
consequently,
especially
circuits.
ponents,
tributed
and,
a
such
signal
chain,
problem
of
the
limits.
and output
(a ‘[gain
a theoretically
can then
properties.
form
temperature-insensitive
constant
having
overload
II.
voltage
gain
and
goals, the principle
shows
and temperature-
feedback
are
excess
prevalent
causes
Negative
stages
in
importance
the current-
impedance
instability.
bandwidths
the
problem:
emitter
than
low
bandwidths,
emitter-degenera~d
a familiar
gain
wide
circuit
de-coupled
current
swings,
loop.
to
provide
It became apparent
are of increasing
By
is
dependent
with
gain,
electronics.
figuration
more
a cascadable
could
the input
INTRODUCTION
having
MHz,
stable
suffers
develop
to
that
and current
1.
ern
Technique
MEMBER, IEEE
and lower
dependence.
A
353
voltage
current,
>>1,
current
across the junction
charge on the electron
near unity
[14]
a constant
absolute
quantity
(1)
to
temperature.
mlcT/q
Fig.
1 (a),
is about
and
26 mV
solving
at
for
300”K.
the
Ap-
variable
IEEE JOURNAL
IC2
XIB
Ourwr
CURRENT
L
Is_
OF SOLID-STATE
1~
Q3
(i-Q)I.
d,
~1,
1.
Two
QL
(b)
common
(b)
w
1,
~
(a)
Fig.
(1-X)1*
M
b!%
Q2
&
J
CIRCUITS, DECEMBER 1968
circuits.
(a)
The “current
amplifier.”
The “differential
source.”
Fig.
3.
I
The new circuit
form.
The text shows that the
and x are equal, for any IB and at all temperatures.
reference
to the figure
it will
be apparent
factors
a
that
1.,
(6)
(),=1
or
I.
= FJ,
= H,.
(7)
*I
50 I
mVl -
~
By
Ql,
a: =C
(a)
This is a linear, temperature-insensitive
relationship.
scaling the area of Q2 emitter
relative
to that of
%=
X=1
o
a stage gain can be realized;
matches
(b)
Amplifiers
this
independent
qv
the emitter
voltage
in
area of Q2 emitter
saturation
as an equivalent
log
or
h,
(3)
?
area of Q1 emitter
reverse
currents
area ratio
offset
A =
exp
of
III.
q v,
—.
mlGT
I,
with
mismatch
the improved
will
(4)
V)
merely
shifts
changing
be raised
circuits,
the
same
to be described—
and, hence, jt is the
bandwidth.
of the differential
its
the transfer
shape. This
again
Fig. 3, which
from
the diode
of Fig.
1 (b),
in which
a pair
However,
configuration
this case there are two drive
which
currents
where
O < x <
of the bias current
ignoring
finite
the emitter
1, and will
(the signal
in
input),
voltages
be termed
the “modulation
1~.
the effects of junction
around
y
2)1.
resistances,
area dif-
and summing
the Q1–Q4 loop, we have
logy
collapses to
a = x.
Thus,
like
.Efowever,
(8)
beta and ohmic
in con-
where the effects are
volt-
that,
= xI.
index”
which
junctions
are of the form
Im
mlcT
—log*–
q.
of
can be considered
the base-drive
are current-driven.
1.2 = (1 –
Characteristic
transfer
curves are shown in Fig. 2(a)
for two temperatures.
The transconductance
is clearly
a nonlinear,
temperature-dependent
quantity.
Notice
of area
swing,
amplifier
isrnore
(5)
l+exp*(VO–
mismatch
WO, without
like
IMPROVED CIRGUIT PRINCIPL~
derived
Temporarily
=
stages
and have
circuits
determining
ages are
ferences,
that an emitter-area
curve by an amount
THE
voltage,
Thus, in terms of V,O,we have
ICI r= aIx
cascaded
tested,
as the
We can now examine
diodes of Q1 and Q2. This
expressed
v, = y
nection
little
factor
as a differential
and 181, and 1.2 are the
very
properties
is very
(2)
‘
of IE, where
81
misline-
and have a fixed gain.
qv
‘Xp mkT
A=p=
now emitter-area
do nok jeopardize
directly
they lack the advantages
h+expz
matter
from
been successfully
dominant
we find
usually
have
built
high-speed
there
a=
but
arity.
Fig. 2. Comparison
of transfer
curves for (a) Fig. 1(a) amplifier;
(b) new circuit
shown in Fig. 3. In each case, the curves
for
–50”C
and + 125°C are compared;
in (b), they are virtually
co-incident.
a.
cause a gain-ewor,
the magnitudes
(lo)
of the output
currents
are simply
different.
The circuit
fier, in which
of Fig. 1 (b) can also be used as an amplicase khe signal input is the current IE. With
IC, = xI.
(11)
IC, =
(1
–
X)IE,
GILBERT:
NEw
WIDE-BAND
and the dc stage
gain
AMPLIFIER
355
TECHNIQUE
is
(12)
Typical
transfer
curves
for
this
circuit
are
shown
in
Fig. 2(b).
Notice
the
the
sharp
extreme
overload
range useful.1
tures
from
insensitivity
points,
We will
the
the
whole
theory
due to
and
dynamic
now discuss the effects
simple
ohmic resistance,
to temperature
making
of depar-
area
mismatch,
and beta.
A. Area Mismatches
Equation
(9) assumed all the diodes had equal areas,
hence, the same Is. In practice, however, this will never
be exactly
the case. We will
(a)
define the variable
(13)
Reevaluating
(9) to include
y, we find that
(14)
a=l+z:–l)
which
is no longer linear
that
is, unless
the
pairs
of transistors
with
emitter
are mutually
Since we are concerned
in the transfer
partures
the
inner
y =
and
1,
outer
equal.
achieving
the
ideal
to z, unless
of the
with
function,
from
respect
areas
effects
case must
low
distortion
of even
small
be examined.
deFrom
(14),
(15)
At the extremes
The
incremental
I/Y,
respectively.
departures
curve
To
from
to
y
vary
example,
the
will
be
10 percent
high
atx
The
in
synonymous
error
the
at
2.7
z
mV
=
of
at
then,
small
300”K,
O, rising
For
a total
to
offset
the
gain
10 percent
has
Fig.
of
by
an
4(a).
To
offset
y =
area
2
(for
twice
confirm
can
this
be restored
(26
mV
of the
that
voltages,
18 mV
example,
that
area
one
ratios
photograph
by
times
of
others)
are
It
measurements
ured
2)
into
for two
of similar
Resistances
(bulk)
might
seem
that
of amplifier
serious
due
distortion
to the
inevitable
would
arise
in
mismatches,
tances
fects
All
the
1 T@e
results
were obtained
using
a single-sided
input
configuration
(see Section
III-G)
driven
from the oscilloscope
sweep
output
of 100-volt
amphtude
via a large resistor.
The input
voltage variations
over the dynamic
range were thus negligible, and
a true current-drive
condition
obtained.
to
the
of
also
these
tance. Also,
with emitter
geometries).
voltages
the
and
current
to an extent
emitter
cliode
of
paths,
are
swings.
base
and
resistive
range.
exhibit
described
due
am-
far greater
to
emitter
by
the
These
due
crowding
on the
components
to
device
can
(1)
ohmic
diffusions,
interfaces.
dependent
dependent
a peak-
differential
voltage
contact
and
been meas-
having
prevalent
of
Slope
typical,
of the dynamic
signal
components
resistances
and
at 500 kHz
gain and bandwidth
B. Ohmic
be
are
of —60 dB have
out that
for full-scale
addition
reassuring.
percent
of 80 percent
distortion
an
*2
signals
be pointed
metalization
kind
equal
It should
will
surprisingly
products
amplitude
there
the
are
less than
combined
also
introducing
log
of
intermodulation
In
loop.
this
but
variations
transfer
range.
to
1.
y and
the
dynamic
corresponding
of
caused
linearity
voltage
slope
O and
thus
plifiers
with
that
loop
are
1.
emitters
is shown
the
over
low
distortion
four
shows
Q1–Q4
x =
points
approximation,
1 cause
1.1,
range,
khese
a first
=
y =
around
the
dynamic
at
linearly
when
=
of the
slopes
resisef-
geometry.
be referred
to
ohmic emitter ~esiswe will assume that this resistance scales
area (a safe assumption
for many “stripe”
circuit
With
as an equivalent
these
resistances
inserted
into
the
IEEE JOURNAL
356
circuit
of Fig.
mii!l’
3(1
10g
–-{}
!l
–
Rx
=
When
=
R,{lll,(l
– 2a)}
(16)
resistance
of each
2z) – IE(l
–
area ratio of inner to outer pairs of emitters.
right-hand
the
a.
we find
z)a
equivalent
ohmic emitter
inner transistor
A=
x
modified,
This
side of this
occurs when
A =
equation
vanishes,
G,. =ID/IB,
is,
the
areas are scaled in proportion
to the drive currents.
A # Go, distortion
will
arise.
Equation
(16)
has
no
general
explicit
magnitude
solution
of the
a small-signal
At
the
for
point
be readily
(x
of the transfer
estimate
If
of the
obtained
by
Also, putting
curve
0.5),
the
mean
i
mkT
—+
ARjJ,
mkT
—
q
+ RJE
“,,
Therefore,
Thus,
curve
the
(17)
lB
mean
a = O, a = 1,
normalized
(area
ratio
slope
at
a
must
slope
too
big)
is
the
center than
the
lish
the
betas
to
The
small-geometry
very
the
same
emitter
area.
of Beta
monolithic
I?or
gain.
between
will
be assumed
that
the
range
emitter
beta
of
transistors,
adj scent
to
match
of
is
very
be perfect.
is not
interest.
A
further
a function
In
currents
the
transistors
the
of
In
presence
of Q1 and
over
the
finite
modified
= xI,
Ij,
=
–
(1
–
(1 –
X)lB
(1
@,I(l
–
–
–
(1
z)lB
–
–
a)alE}
(1
(1
–
to
a =
which
again
reduces
result,
since
it implies
is, the
gain
to the emitter
that
Zi)(l
–
–
beta,
the
significance
to
high
x.
This
no matter
circuit
is
is
a)lE
how
still
= 1
an
(19)
the
beta
Go. Of course,
where
with
currents
Q4 to estab-
operation.
reduction
1 percent
Since
from
and
emitter
first
order
discussion
is that
stage
can
realized
made.
above
magnitude
It
will
any
of
be
be appreciated
the
the
actual
with
that
standard
1(b) ) where
of the form
G,. is now
ratio
this
circuit
is
forms
gain
is
given
of
is used
of the
the
nominal
current
to
emitter
impedances
base
circuit,
or the
ratio
gain
(for
in
example,
an
emitter-
A in the
circuit
of Fig.
like
3 in which
l(b)).
Measurements
on an amplifier
betas
of Q2 and
equal
to the
of
cally
5.
in Fig.
The
input
will
but
R,.t
operation
with
The
=
results
shown
transistors
are connected
are
the
substantially
up to lE/IB
Resistance
~cE
92, with
as diodes
no
graphi-
and
thus
=
be a well-behaved
zero.
Since
increases,
with
right
linearity.
a gain
Saturation
not
approaches
IE/IB
Fig.
95 showed
V~~ ~ 0.8 volts. However, the interbias is reduced to V~B—IcR,at, and the tran-
with
nal collector
sistor
Q3 were
ratio
degradation
sharply
z Another
method
of analyzing
this kind
of distortion
in [91, where
expressions
for the form
and amplitude
nonlinearities
are derived.
in
base
(20)
operate
astonishing
low
of
of the
beta
linear
gain
order
accuracy.
possible
the
so that
a)I.
a) I.}aI.
for
the
can be readily
D. Collector
{xIB
Q1 and
is
where @ = large-signal common-base dc current gain.
Substituting
these values into (9) and reducing
the
logarithmic
terms to products
and quotients
as before,
we have
{(1
the
into
and
(18)
–
supply
current
a)alE
–
to
flows
voltages
is of the
degenerated
Ii,
excess
drive
(including
Fig.
by an expression
current
good
assumption
of
Q4 are
the
correct
close
not
C. Ejjects
sufficient
any
of 100 are typical,
collector
gains
of
be
Q3;
compensation
at the extremes, and vice versa. Fig. 4(b) shows transfer
curves for 1~ = 6 mA, 1~ = 6 mA * 3 mA using four
transistors
still
of Q2 and
higher
have
GAIN
Fig. 5. Gain errors due to beta. Curve (a) shows that accurate
gain up to beta is achieved with the new circuit. All prevalent
transistor
amplifiers have gain errors as shown in curve (b),
calculated for p = 100.
when A > G,.
will
NOMINAL
normal-
is
z = O, x = 1 in (16) yields
respectively.
unity.
=
=fl
dx
still
a good
can
that
analysis,z
da
transfer
a, but
distortion
quiescent
ized slope
CIRCUITS, DECEMBER 1968
a)
(1 –
.
where
(9) suitably
3 and
OF SOLID-STATE
the
must
use
when
this
with
temperature
maximum
usable
Circuits
involving
temperature.
a buried-layer
diode
V~~ decreases
a sufficiently
collector.
large-geometry
voltage
current
falk
high-current
device
GILBERT : NEW WIDE-BAND
E. Thermal
The
The need for a pair
so far have assumed that
at the same temperature,
case. For example,
perature
than
package
the
outer
pair
and dissipating
the variable
which
eliminated
all transistors
need not be the
the inner pair may be at a higher tem-
shown by inserting
that
357
TECHNIQUE
Ej7ects
analyses
operate
AMPLIFIER
if they
are in a separate
considerable
the appropriate
power.
It
can be
temperatures
into
(9)
a becomes
put to ground
the
junction
should
by connecting
and supplying
of Q1 and
also be added
input
voltage
Q4 emitters.
to the input
during
Vi.
currents
the
a constant
overload
Z) lB in-
current
of IB to
Clamping
diodes
to control
conditions
= 26 log&
can be
(1 –
point
< O). This input point is a useful
close to ground potential
since,
(21)
the
(Z >
1 or z
current-summing
node
mV at 300 “K
(23)
thus
where
~ = temperature
Another
vealed.
potential
and introduce
magnitude
4(b).
significant
a step change
structures
having
of input
the circuit
to reacquire
the output
re1.2
0.5 at z =
and
and
described.
chstortion
than
for
During
that
these
gain
z <0.8.
0.5)
form
and a small
arises. However,
beta immu-
offset
term
for typical
(a #
betas,
the
are negligible,
variants
designed.
this
of the
Space
here,
principle,
further
in
CONFIGURATIONS
basic
permits
but
given
forms.
versatility
depend
on
the
the
These
circuit
just
discussed
only
a few
of these
generalized
Appendix,
examples
of the
principle.
Input
Diodes
A. Inverted
thermal
effects
are typically
for the circuit
cell described
Many
been
ALTERNATIVE
statement
will
will
have
to
point
serve
of
the
be
the
way
to
to illustrate
the
be inverted
and
distortion
dissipation.
of the step amplitude
The
of each
transistor-to-transistor
show
1 percent
0.2<
no longer possesses the unique
of the original
described
is taken
duration
and, of course, the power
Measurements
Following
a transient
and
distor-
dissipation
equilibrium.
suffer
waveshape,
circuit-to-sink
less than
will
the
for
errors in gain and dc balance
coupling
thermal
and some time
thermal
signal
magnitude,
impedances,
being
in form
applications,
current,
36mV
shown in Fig.
close thermal
in critical
in the quad changes,
whose
put n =
similar
does not arise, but transient
can be a problem
time,
will
circuit
nity
IV.
In monolithic
device
distortion,
This
is thus
to the effects of bulk resistances
this problem
tion
of nonlinearity
excess of 60°K
j<
(22)
of Q2 and Q3 in “K”
source
A temperature
IVi.
of Ql and Q4 in “K
temperature
the
of complementary
altogether
later,
The
input
driven
from
diodes
Q1 and
current
sinks
Q4 can
(instead
of current
sources),
as shown in Fig. 6. Neglecting
the effects of beta,
mismatches,
and bulk resistances, we can writes
area
has less
zI~aI~
this.
= (1 – a)l~(l
– Z)lB
(24)
or
F. DC
Stability
Equation
introduce
a=
(14)
showed
that
a shift
in the
output
(Z = 0.5),
but this
shift
emitter-area
at the
mismatches
quiescent
is not temperature
point
dependent.
The circuit
inal
form.
(25)
thus has a polarity
It
reusable
are
l-z.
has the advantage
at the
reversal
that
over the orig-
the input
currents
of Q1 and Q4, a feature
collectors
By adjusting
the drive-current
balance, the output can
be zeroed, and remains this way over the temperature
range. This is in contrast with the behavior of prevalent
that is put to use in the gain cell described later.
This configuration
lacks the beta im~unity
differential
amplifiers,
where the offset is corrected
voltage, leaving a drift of 85 log ~ pV/O_K.
add to the input
G. Unbalanced
Drive
by a
original
that
the
Currents
the input
base-current
larger
The use of complementary
drive currents is the key
to linear large-signal
operation
of this type of circuit.
However,
a constant-current
the inputs
will
not impair
offset
on one
linearity,
(or both)
but there
will
of
be a
beta
fraction
considering
modulation
change in gain, and dc unbalance.
An
amplitude
ratio
between
the two
inputs
the input
restored
the
ratio.
by
transistors.
If
the
an adjustment
ratio
is known,
in the
emitter
linearity
area
can
that
This,
is receiving
drives
of the
the
together
emitter
with
the smaller
transistor
that
current,
of the
of Q2 and
Q3
the fact
fraction
conducts
causes
of
the
significant
these
index
currents,
it is found
a is reduced
that
the
out-
to
(26)
f3(l-z)+~
where
o=
be
of one of
the base current
currents.
a=
is more
serious—for
example, 1~1 varies from O to 1 mA as l~z
varies from 1.5 mA to zero. This is equivalent
to an area
mismatch,
as (14) will reveal, in which case ~ represents
because
dependence.
By
put
form,
s The
Appendix
from an inspection
P
(27)
(I+ O+M))’
shows how
of the circuit.
these
quantities
can
be
equated
358
IEEE JOURNAL
OF SOLID-STATE
CIRCUITS, DECEMBER 1968
+ZIB
Fig.
6.
Inverted
form
of circuit.
Performance
is similar,
for phase change in output
(a = 1 – z),
except
Fig.
S.
Product-quotient
functions
imputs,
various
‘4
f%I B
This
circuit
earlier,
aE
IB
‘2
circuit.
By
suitable
can be generated.
also possesses the beta immunity
and can be inverted
C. Product-Quotient
J
Fig.
8 shows
output
equal
puts
.
an interesting
to the
is the
emitter
of
variant
products
The
““’man”
significant
connection
Neglecting
able
to
to produce
an
of several
in-
or quotients
speeds.
feedback
Q1.
referred
6 form.
Configuration
at nanosecond
circuit
to the Fig.
second
(12 – IOJJOU,
feature
of Q4’s
order
of this
output
effects,
to the
we have
(29)
= (12 – Io.t)IJ,
or
+IB hE
Fig. 7. Multiple-input
version of Fig. 3. This circuit is useful
standardize the absolute magnitude of analog signals.
to
The magnitude
present
Linearity
is not
(tot hecollector
impaired,
circuit)
but
the
actual
stage
gain
The circuit
ber
(28)
G; = 8G,,
and the output
swing is reduced.
For example, with GO = 2 and D = 50, G; is 1.8, 10 percent below the nominal gain, and the limiting
values of
a are 0.96 and 0.04.
The “inverted”
circuit is identical
in behavior to the
earlier
configuration
matches,
be driven
rangement
collectors
Q4
plied
of area misIt may also
and bases of Q1 and Q4 are joined
and
base
to the
of
Q3
grounded.
emitter
of
diodes
outside
Ql,
The
and
must
are needed
of this
input
lie
in the
match errors.
By putting
the
center
we
the
The
number
We
It
is not
pair
of
may
useful
when
and
but
it
p-n-p
inputs
an absolute
is desired
An
to
the
input
currents.
accepted
the
have
amplitude.
lateral
be
transistors
Fig.
to
produce
n
are
known
to
value
standardize
example
of
of this
might
as the
which
and
7 shows
be
may
signals
be
in
in
This
level
as z’,
z’,
the emitters
Q4, and in Q1 and
of
Q6,
kind
that
“GAIN
now
describe
have
arisen
from
into
cascaded
in
of
Fig.
a
at
the
6,
n
the outputs
Fig.
what
9, and
except
can
be devised
to
perform
is legion.
that
CELL”
is probably
this
amplifiers
is similar
the
work,
to
input
the
currents
the
most
certainly
useful
as
far
is concerned.
‘{inverted”
that
It
circuit
reappear
Q4 are added in phase with
of Q2 and Q3. The gain is thus
collectors
of
Q1
and
is
(31)
widely,
a known
connection
and
any odd numsuch
a certain
vary
to
to
how
outputs.
the
drivers
output
circuits
to
as incorporation
to limit
complementary
inputs
ratio
necessary
Q3 and
will
circuit
Configurations
terms
(30)
O
range.
to accept
power
diodes in series with
V. THE
input
not
the maxi-
get
is shown
B. Multiple-Input
extra
transistors,
ap-
range
to constrain
Iz, although
may assume.
be expanded
to generate
emitter
is then
current,
for ~o.~, determines
Z3/YZ, etc. Of course, there is a practical
limit to the
circuit
complexity
determined
by the stacking
of mis-
functions
and taken
source equal to 1~, and the
current
to 1~. Clamping
voltage
the effects
and temperature.
by a single-sided
signal, by a similar
rearto ?’that suggested for Fig. 3. This time, the
to a positive
of
as regards
ohmic resistances,
may
of inputs
.
11
of the emitter
in the expression
mum value the output
isreducedto
_ IJ.
T
I 0“:
with
shifters.
This
inner
still
form
stage
achieve
is attractive
can
operate
a net
stage
for
several
with
a gain
gain
reasons.
less than
greater
than
Firstly,
the
unity,
yet
unity;
we
GILBERT : NEW
WIDE-BAND
AMPLIFIER
X(L+ IE)
359
TECHNIQUE
(1-X)(1,+ 1,)
j3=40
IB=O
IE = lmA
Ix=o
IB=lmA
(b)
in limiting
(a)
Fig.
Fig. 9.
would
therefore
given
gain.
to
This
current
injected
total
find
the
ing, because the output
directly.
this
cell
output
is well
faster
for
to
cascad-
suited
Finally,
string
all
Exact
Equation
the bias-voltage
be used;
can
only
0.5–1 volt
viously
shown
resistances
gain
omits
that
other
parameters,
and area mismatches,
over the
full
dynamic
per
It
was pre-
such
as ohmic
did not affect the mean
range,
distortion.
Thus, the accuracy
beta are of most concern.
In a cascaded amplifier,
of beta.
but
only
limitations
a modulation
index,
the input
the output
of all stages, except
x, and
values
supply
of Ii
current
and z’ for
stage, the stage gain G;”,
and
are
IL
= cd.
+ I,
AIC = a(I,
G:”
The practical
onstrated
swept
an emitter
the effective
swing AIC.
The results
= a(l
+
+
significance
of a gain
beta of 40, and
(b)
at a higher
must
be less than
tortion
a gain
(by one V~E) than
~B, so that
of less than
can be simply
where VCB is
appears
the
across
Q4 Typically,
is dem-
of Fig. 10, which
show the
cell using
1~ = 200@
(a)
with
transistors
with
a
a beta of 200. For
and In was swept from
expanded
shown
=
bias
two
must
for no distortion
voltage
the
.1
(33)
“
supplied
collector-base
it proves
very
distortion
even under
11 shows the
step response
cell
at very
operated
to each
W.
difficult
large-signal
step response
analyses
and
0.8 voks,
to induce
apprecFig.
integrated
gain
with
the
vertical
RESPONSE
of this
have
for the transistor
not possible
type
been
not surprisingly,
that
by the ft of the
fairly
Various
however,
and
the response
time
devices.
large
does
to determine
of circuit.
made,
are scaled to the currents,
and for
and
of a single
model
the exact
be eliminated,
Q1
conditions.
TRANSIENT
small-signal
is dominated
V~H =
currents
and it is, therefore,
geometries
of
per division.
not exist,
these indicate,
stage
junctions
high-power
high
to 1 percent
1~
dis-
be used.
with
In practice,
iable
thermal
that
VCB
+ v,.
v.,
the outer pair,
to eliminate
Vc~ = 1.6 voIts,
we must use a stage gain of 1.67.
An accurate
of these expressions
scale
is the same. Because the inner pair operates
voltage
(32)
OGO)
transistors
these measurements,
distortion
of gain cell. Vertical
to 1 percent/div.
IX = 20 mA.
shown that for the gain cell no thermal distortion
arises
if the total power dissipated in the outer and inner pairs
scale expanded
OIJ
by the photographs
gain
Thermal
I.
the inputs
to the following
11.
by
the first, are simply the collector currents from the previous stage. Thus, starting with an input bias current, 1~,
1~, we can calculate
Fig.
It
introduced
determined
ac-
for
of transistors
the effects
the
IB2
4-50
+40
+30
+20
4-10
=0
to the
circuit
Gain Expression
(31)
effects of beta
are demonstrated.
of the
node contributes
stage is needed.
A. More
Z~, the
of gain
a
each stage has to supply only the base current for that
stage, which is not signal dependent. Hence a low-power
diode or resistor
sweeping
curacy
the next
advantage:
emitter
each
swing.
circuit
of one stage can drive
leads to a further
at
By
The “gain cell.”
expect
Secondly,
10.
Further,
if the
the effects of r~ can
stage-gains
(greater
O to over 1 mA. The input was modulated
to a depth
of 80 percent at 1 kHz. At more practical
operating currents, the effects of bulk resistances must be taken into
than two or three), an adequate approximation
for the
stage response of the Fig. 3 circuit is obtained by treating
it as a network with a single pole at ft/G,o.
account,
The gain-cell
response is more complex, and can be
by a pole-zero
approximated
by a pole at f ~/2, flanked
preferably
B. Thermal
In
many
scope vertical
mentioned
by scaling
the emitter
areas.
Distortion
exacting
applications,
amplifiers,
earlier
can
for
the transient
be very
example,
thermal
disturbing.
oscillo-
pair. For Go = 1, (G:’ = 2), the pole pair are co-incident
and cancel, leaving a single-pole response with a gain–
distortions
bandwidth
It
moves toward
can
be
product
equal
the origin,
to f,.
For
GO <
1, the
zero
while the second pole moves out,
300
IEEE
+
2..
JOURNAL
OF SOLID-STATE
*
(a)
(b)
Fig,
12,
Measured
transient
responses
of Fig.
3 circuit.
+
(a)
causing
an
overshot
arises,
fast
rise
the
second
followed
pole
by
have
been
on
made,
ohms-per-square
Parameter
~c:
CCB
For
response
a slower
time
and
integrated
using
base
ohms-per-square
summarized
the
dominates
Measurements
9
response.
and
both
diffusion)
process.
The
13.
Measured
transient
GO >
1, the
reverse
consists
of
initial
constant.
an
For
GO >>1,
f ,/GO.
is at approximately
forms
the
of
Fig.
standard
and
device
a
3 and
process
shallower
Fig.
(200
4S0
characteristics
are
below.
200-Ohm
Process
450-Ohm
Process
600 lMHz
2 pF
0.7pF
1200 MHz
2 pF
0.7 pF
b’
40
rb
15 ohms
2.s
&
(b)
Fig.
situation
CIRCUITS, DECEMBER 1968
200
27 ohms
response
of
the
gain
cell.
Response time was measured in a test system having
an overall risetime
of less than 75 ps. The waveforms
were plotted on an X–Y plotter,
which provided
better
resolution
than photography
for comparing
several collated responses.
The results confirm the predictions
of the small-signal
analyses. In Fig. 12, the responses of the Fig. 3 circuits
are shown,
using
1200 MHz
devices. The form
(a) the 600-MHz
devices,
and
(b) the
of the response closely
ap-
proximates
a single time constant proportional
to gain.
Fig. 13 shows the more complex “gain cell” response for
Test
l’c~
Vcs
Vc~
Conditions
= 2 volts, Ic
= 10 volts
= O volts
(a) the 600-MHz
=
10 mA
VCE = 2 volts, Ic = 5 mA
IBB = 10 mA, Ic = O
It should
neutralizing
devices and (b) the 1200-MHz
devices.
be mentioned
that the test jig incorporated
capacitances and balun transforms
at critical
sites to eliminate
preshoot and other aberrations
in the
responses, and those components markedly
influenced the
GILBERT : NEW WIDE-BAND
apparent
risetimes
at
AMPLIFIER
the
lower
performance
obtained
from
fier
VII-D)
supports
(Section
VII.
361
TECHNIQUE
gains.
the totally
this
The
0.3
improved
integrated
I
ampli-
conclusion.
CASCADED AMPLIFIERS
02
A repetitive
problem in de amplifier
design is that of
cascading
stages without
“level
shifting,”
that is, the
buildup
of supply–voltage
requirements
as each stage
J
is added.
J, I
Several
problem.
techniques
They include
are used to overcome
this
II
the use of
0.1
1) complementary
p-n-p–n-p-n
supply rails,
2) zener-diode
or resistive
3)
very
For
the most
swings
are small,
circuits
under
presented
It
the
diode
overload
recovery,
Typical
that
will
Fig. 14.
dinearity,
results
are
noise
for these
supplied
emitter
each stage of an amplifier
can be treated
Go is the low-frequency
(vii
–
1)’”.
currents,
as a
Consequently,
range (zero to
cur-
function
(34)
o
Also, if the total current-gain,
the N stages we can write
G, is shared equally
optimum
number
of
over
– 1)”2
N@j
function
of interest.
stages
required
(35)
is plotted
Firstly,
to
in Fig.
there
is an
maximize
the
bandwidth.
Secondly, the maxima
are well defined for
low gains but vague at high gains. Finally, the bandwidth
of a large number of stages is relatively
insensitive
to
overall
In
gain.
discrete
less than
reasons.
the
designs,
it is frequently
optimum
The possibility
fier removes
of the difference
number
necessary
of stages for
of integrating
to use
economic
the entire
ampli-
(not
shown)
at the extremes
A limited
been made.
gain.
The
overall
terms
be
the
gain
in the denominators
this
current
prevent
problem
of
been
sources for the emitters.
the transistors
of the dynamic
have
saturating
range.
investigation
of this form of amplifier
has
Some resuhk
for a four-stage
high-gain
X1OOO) design are shown in Fig. 16. They
and overload
(Fig.
16(a) ), and the transient
(Fig.
16(b) ). In
the
recovery
and 1200-MHz
C. Cascaded
Gain Cells
demon-
to a 3-PS ramp
response and noise level
second photograph,
of 600-MHz
transistors
the
responses
are compared.
A series of three gain cells is shown in cascade in Fig.
17. The power is supplied as currents
(to the emitter
nodes) which set both the gain and the output swing
capability.
A low-power
bias string, shown here as pairs
of diodes,
balanced
sets the operating
nature
signal currents
undecoupled
this limitation.
Cascaded
the
of overcoming
devised, using dependent
amplifier.
B. A Practical
must
with
the gain can be swept over a very wide
about /P), but the gain is a sensitive
Methods
(about
This “bandwidth-shrinkage”
14. It shows three things
do. Currents
and these, together
determine
strate the linearity
E
transistors
at each interface,
of N stages is
Diodes
F = (vii
to maximize
(36)
F, of N cascaded stages is thus
F = $
15
number of stages required
bandwidth.
as the diode-connected
amplifier
(36).
bandwidth,
Curves showing
-1
be
rent gain of the stage.
The overall
io
appropriate.
circuit
where
N
Stages
transistor
single pole at j/Go,
forms
to the
level shifting.
response.
earlier
,5
1
some cases, stages
without
of Cascaded
0,
far
voltage
suited
cascaded
In
is by
where
in a cascaded
be given where
was shown
using
only
of interest
drift,
A. Bandwidth
method
indefinitely
and transient
will
circuits,
and hence is eminently
Several
factors
parameters
is possible
approach.
stability;
level,
third
use this
can be cascaded
gain
the
discussion.
that
Other
but
two
for each stage.
designs,
attractive,
between
level-shifting
low bias voltages
monolithic
devices
of the
voltage
amplifier
in the base circuit,
line to supply
The overall
of each stage. The
results
in very
and permits
small
a common
all the bases in a multistage
gairi, for sensible values
of beta, is
Amplifier
The first cascaded form we will discuss is made by
taking the Fig. 3 circuit and connecting the collectors of
one stage to the bases of the next, as shown in Fig. 15.
The amplifying
transistors
operate with VO. = O, just
~,
=
IE1
+
~E2
+
1.3
+
IE4 ,
(37)
IE1
With this method of cascading, the correct static bias
conditions to handle the steadily increasing signal ampli-
362
IEEE JOURNAL
OF SOLID-STATE
CIRCUITS, DECEMBER 1968
INPUT
II
II
II
OUTPUT
%)
I-----K
+
.
Fig.
15.
A cascaded amplifier,
HF
STII
input
output
Xlo
Xloo } 0/=
(a)
(b)
Fig. 16. Performance of amplifier similar to Fig. 15. In (a), a 2.5-KA input current
(top trace) is accurately reproduced at x 1000 (second trace). When overloaded
ten or a hundred times, (lower traces) recovery is rapid. The top portion of the
waveform is within the linear range. Time scale is 1 ~s/div. Waveforms
(b) show
transient response at x 1000,. to a 1-PA input step using 600-MHz (top) and 1200MHz (bottom) transistors. Time scale is 20 ns/div.
xrE,
d~E,+.
lNPUT
OUTPUT
(L-X)
1.,
,1
Fig.
17.
Cascaded gain cells.
... ~E4)
363
tude
along the chain
signal
are automatically
and bias currents
increase
is in
contrast
with
must
be taken
to provide
A corollary
same point
conventional
of this
amplifiers
adequate
is that
on the overall
present,
because
at the same rate.
care
range at each stage.
all stages overload
transfer
$ < 0 or z > 1. Furthermore,
This
where
at the
characteristic,
at overload,
when
no transistors
saturate, and, because of the modulation
reduction
discussed above, the input to each successive stage never
quite
drops to zero, Consequently,
teristics
the recovery
charac-
are good.
D. An Integrated
Fig.
Gain-Cell
Amplifier
18 is a photomicrograph
of a five-stage
amplifier
on a 50 X 60 mil die, using the 1200-MHz
transistors.
It
comprises an input stage of the type shown in Fig. 3
followed
by four
current-source
which
cascaded
gain-cell
transistor,
stages each with
and a patchwork
a
of resistors
can be used to set the base bias voltages
and emit-
ter currents.
The
mask
achieving
layout
optimum
to maximize
is of
considerable
performance.
a crossover
region
cause of the low impedance
lous attention
is essential
X25
this
equal
x5
Zo”t
(a)
the
inner
of the circuit,
and
outer
distortion
of two
gain of 16. (The
- 50”C
+ 25 “C
+125°C
plifier
must
input
The
system
below,
transistors
have
due to bulk
resis-
be used, producing
The
(d)
risetime
used for
input
from this
significant
measurements
some corrections
m
Kg
w
50-ohm
ditions,
pulser
driving
By
Xlo
(f)
Fig. 19. Performance
of the integrated
amplifier.
(a) Input/
output delay at x20, 1 ns/div.
(b) Transient-response
variations for gains of x5 to x M, 1 ns/div. (d For gains x50 to
x300, 5 ns/div. (d) Shows transient response for x25 at —50”C,
25”C, and 125”C. (e) Demonstrates
swept-gain linearity,
X 1
to x30. (f) Shows recovery from x 10 overload.
lines terminated
no power gain
below 40. It is, however,
input
and output
suitable
gain from
acceptable
am-
of Fig.
19.
was
were necessary
risetimes.
to the amplifier
50-ohm
these
was supplied
by a push–
Q1 and Q4 via
tors, to ensure current-drive
conditions.
puts were connected to the sampling
xl
an
at a gain
of the integrated
and in the waveforms
0.4 ns, and, accordingly,
pull
1.”$
meticu-
stage operates
of the performance
is given
to the measured
IB
Be-
resistances
of unity.)
However,
quite large variations
value can be tolerated
without
introducing
A summary
(c)
base circuit.
distortion.
(b)
(c)
are paired
also serve
the metallization
minimum
a stage gain
overall
must
to achieve low distortion.
circuit,
area. For
tances
1[.
for the
level
to balancing
in
in order
to output, thus maintainAlso, to reduce collector-
capacitance, the gain-cell transistors
common-collector
isolations,
which
to provide
In
importance
example,
the j~ of each stage, the transistors
be increased in size from input
ing a constant current density.
substrate
into two
For
l-k~
Also, both
oscilloscope
resisoutvia
at both ends. Under these conwas realized
for current
gains
entirely
practical
to match
both
to 50 ohms.
choice
of current
ratios,
a wide
range
of
zero to X 1000 could be obtained,
although
high-speed
performance
was possible
only
over a limited range. Some idea of the effect of gain on
transient
response is provided
by Fig. 19(b) and (c).
Considerable
care had to be exercised to eliminate
ringing and preshoot, particularly
in respect to ground-plane
IEEE JOURNAL
364
currents.
A bifilar
the input
drive.
that
resulted
sults
from
published
ringing.
transformer
putting
earlier
[8]
gether.
packaged
Series
stages to bring
showed
a large
of
cells were breadboarded
resistors
the ringing
were
Here
is a summary
amplifier.
—
o- —
required
to-
between
down to even this
level,
and
-3
-6 : I
G@)
~
X25
-9-
of the performance
of the
fully
X50
40
20
Corrected
Rlsetime
(ns)
Current
Gain
5
10
12
8
3
—
—
7:5
8.4
9.2
8.5
9.2
9.0
9.3
2
—
31
KM
Fig. 20.
—
>500
>500
500
420
370
170
The
effect
of temperature
low
temperatures,
current
(Fig.
latter
response
the risetime
due to increases
were the accuracy
The
on transient
and, as expected,
to include
precision
amplifiers,
four-quadrant
19(e) ) and overload
shows
the
response
8(I percent of the dynamic
increase in the input.
recovery
for
range,
was
swept
(Fig.
an output
and that
19 (f)).
equal
for
x 5 to
x50,
of the dynamic
a signal
Other
equal
The
key
of logarithmic
factor,
swing:
swing:
These
details,
permit
in
faster
‘~;~l~c;l~t
are very
‘f
This
technology
gain-cell
amplifiers
paper
has
technique
for
amplifiers
tion.
It
suited
demonstrates
that
amplifiers
can be made
circuit
with
cascadable
practical.
dc
useful
500-MHz
without
This
im-
certainly
believed
to
linear
planar
both
sides is taken,
a common
of no consequence.
The
leaving
the
product-quotient
term
for example
~
1.,1.,ID31D4
currents
=
(38)
“
of the diodes in the loop can be
as the ratio
power
gain
and
of
could
a
indicates
all diodes connected in the
loop,
., and the minus sim
those in the opposite direction.
Since 18 is pro-
portional
the ratio
to area, the y ratio can also be calculated
as
of the areas of the emitter diodes in the loop.
Also, every I, varies with temperature
in the same way,
drops out of the analysis
at this
and thus temperature
point, too, explaining
the excellent freedom from temperature
effects observed in these amplifiers.
This reduces the key
equation
fabrica-
is now
20 dB
to
=
‘Y
II
(40)
~(–).
Stated at length, in a closed loop containing
an even
number of perfect exponential
diode voltages (not neces-
operating
O to
to the form.
-H 1(+)
integrated
designed
components,
concept
(39)
wide-band
alone,
A commercial
be
transistor
de-coupled
response,
additional
building-block
is
very
transistors
mode.
a controllable
to
of
to monolithic
using
current-gain
giving
a
almost
to be fabricated.
what
design
especially
in a strictly
Subsequent
will
SUMMARY
described
the
is, thus,
form
where the plus sign indicates
same direction
around the
to + 125°C
encouraging.
device
VIII.
new
is of the
of an even number
set equal to zero, having
which
for a
320 mW (64 mA at 5 volts)
Within ~0.35 dB over _ 350C
results
circuits
ImIJ,s,Is,
{
provements
these
summation
&60 mA into two 25-ohm loads
3 volts peak–peak, differentially
Gain stability:
Zero drift:
fill
be taken
logarithmic
arguments may be compounded into a single
product-quotient
term, and finally
the antilogarithm
of
extracted
current
for
terms
rnkT/q,
The saturation
Output voltage
Dissipation:
will
gain
gain of X257 are
Output
equation
to 50 percent
performance
stages, power
etc., and would
Principle
given in (9), an algebraic
equal to unity,
and for
range.
to
a x 10
The CW response is shown in Fig. 20, for current
from
input
multipliers,
APPENDIX
checked
of gain in response to a linearly
high-impedance
amplifier.
a long neglected need. Some of these topics
up in subsequent papers.
decreases at
in ft. Also
i( Km
400
CW response of integrated
The Generalized
also checked,
I
2cU
w ( MHz) -w
Gain–
Bandwidth
Product
(GHz)
Bandwidth
(MHz to –3 dB)
Overshoot
(percent)
0.58
0.63
0.69
0.81
0.92
2.0
4,0
8.5
12.0
15
20
25
50
100
200
300
~
in which
these slowed the response.
integrated
G.
+3 o
amount
were for an amplifier
gain
damping
to balance
to note the improvement
CIRCUITS, DECEMBER 196$
all the stages on one die. Re-
These waveforms
individually
required
was
It is interesting
OF SOLID-STATE
+6 J-
be
fully
be expanded
sarily
the
two-terminal
currents
devices)
are such that
in diodes whose voltage
arranged
in canceling
the product
polarities
spect to a node in the loop is exactly
of the
are positive
proportional
pairs,
currents
with
re-
to the
365
IEEE JOURNALOF SOLID-STATECIRCUITS,VOL. SC-3, NO. 4, DECEMBER1968
product
of
negative
with
portionality
tion
latter
the
currents
respect
in
being the ratio
currents
diodes
to that
whose
of the product
of the former
voltages
node, the constant
are
of pro-
of the satura-
set of diodes to that
of the
set.
ACKNOWLEDGMENT
During
the course of this work, the discussions held
with L. Larson of the Advanced Instruments
Group and
G. Wilson of the Integrated
Circuits Group proved stimulating and helpful, The experimental
work of E, Traa is
also gratefully
acknowledged.
REFERENCES
and S. K. Salmon, “Intermodulation
in com[11 A. E. Hilling
mon-emitter
transistor amplifiers,” Electron. Engrg., pp. 360364, JUIY 1963.
distortion
in broadband ampli[21 L. C. Thomas, “Eliminating
fiers: Bell S’VS. Tech. J., p. 315, March 1968.
amplifiers,”
in Amplijier Han&
[31 J. S. Brown, “Broadband
book
New York:
McGraw-Hill,
1966, sec. 25, pp. 25-57.
broadband
transistor
amplifiers~
[41 L. F. Roeshot, “U.H.F.
EDN Mug., January-March
1963.
diode-transistor
[51 W. R. Davis and H. C. Lin, “Compound
Proc. IEEE (Letstructure for temperature
compensation,”
ters), vol. 54, pp. 1201-1202, September 1966.
“Gain stabilization
of transistor
voltage ampli[61 A. Bilotti,
fiers,” Electron. Letters, vol. 3, no. 12, pp. 535-537, 1967.
“Tunable
resonant circuits
suitable
[71 A. M. VanOverbeek,
for integration,”
1965 ISSCC! Digest oj ‘Tech. Papers, pp.
92-93.
multiplier
principle,”
[81 B. Gilbert, “A dc-500 MHz amplifier
1968 IL$SCC Digest OJ Tech. Papers, pp. 114-115.
‘(A
precise
four-quadrant
multiplier
with
sub[91 —
nan&econd reeponse,” this issue, pp. 365-373.
synchronous
detector utilizing
101 H. E. Jones, “Dual output
tranewtorized
differential
amplifiers’
U. S. Patent 3241078,
June 18, 1963.
“Applications
of a monolithic
analog multiplier,”
111 A. Bilotti,
.1121 1968 ILSY3CC Digest of Tech. Papers, pp. 116-117.
W. R. Davis and J. E, Solomonj “A high-performance
monolithic
IF amplifier
incorporating
electronic
gain-control,”
1968 I&SCC Digest of Tech. Papers, pp. 118-119.
[131 G. W. Haines, J. A. Mataya, and S. B. Marshall, “IF amplifier using C-compensated
transistors,”
1968 LSSCC Digest
of Tech. Papers, pp. 120–121.
of surface recombination
and channel
[141 C. T. Sah, “Effect
on P-N junction
and transistor characteristics,”
IRE
Trans.
Electron
Devices,
vol. ED-9, pp. 94-108, January 1968.
A Precise Four-Quadrant
Subnanosecond
BARRIE
Multiplier
Response
GILBERT,
Abstract—This paper describes a technique for the design of
two-signal four-quadrant multipliers, Iiiear on both inputs and useful from dc to an upper frequency very close to the -f~of the transistors comprising the circuit. The precision of the product is shown to
be limited primarily by the matchmg of the transistors, particularly
with reference to emitter-junction areas. Expressions are derived
for the nonlinearities due to various causes.
MEMBER, IEEE
3) A residual
zero
4) A scaling
and/or
fectly
FOUR-quadrant
satisfy
the
5) An equivalent
multiplier
would
per-
sign. Ideally,
no limitation
on the rate of variation
All practical
multipliers
suffer from
following
(1)
XY
of X and Y, and produce
algebraic
an output
there
would
havbe
of either input.
one or more of the
shortcomings.
1) A nonlinear
puts.
2) A limited
varies
with
temperature
dc offset on one or both of the inputs;
This
medium-speed
multipliers,
technique
has gained
makes use of the relationship
method
XY
correct
that
favor
[1].
expression
Z = constant,
ing the
is
voltages.
“quarter-square”
A
for any values
constant
supply
when the other
“null-suppression”),
6) A dc offset on the output.
the
lDEAL
response to one input
(imperfect
In the field of high-accuracy
I. INTRODUCTION
~
with
dependence
and
of the in-
tional
+
Y)’
having
characteristics,
–
(x
–
bipolar
together
Y)’]
(2)
square-law
with
volt-
several
opera-
amplifiers.
Much work has been put into harnessing
exponential
voltage-current
characteristics
diode
for
single diodes
multiplier
applications,
(or transistors)
the excellent
of the junc-
either
in conjunction
by
with
using
opera-
tional amplifiers
[2], or, more recently, pairs of transistors connected as a differential
amplifier. [3]– [6], In the
rate of response.
majority
the
Manuscript
received June 28, 1968; revised September
This paper was presented at the 1968 ISSCC.
The author is with Tektronix,
Inc., Beaverton,
Ore.
elements
age–current
tion
on one or both
employs
= *[(X
16, 1968.
diode
of cases, the strong
voltage
proved
temperature
a problem,
commercially
available multipliers
oven to reduce this dependence.
dependence
and at least
are equipped
with
of
two
an
IEEE JOURNAL
366
Another
plier,
problem
analyzed
respect
to
of the “differential-amplifier”
in
the
[7],
is the
nonlinear
base-voltage
input.
y:E
multi-
response
To
achieve
with
d,
Q3
Q2
useful
+
linearity,
the dynamic
range on this input must be restricted to a very small fraction
of the full capability,
leading
to poor
perature
noise
dependence,
The
problems
associated
can be largely
overcome,
current-voltage
rendering
linear,
fects.
This
sions
entirely
realization,
will
for
type
the
base
diodes
inputs,
controlled,
free from
Q1
with
the
directly
from
this
Hg.
pairs
BASIC
in
a
Q2-Q3
cross-connected,
pair
of transistors,
is the addition
the circuit.
of this
x=2x–1
expres-
(6)
y=zy–1
without
is comprised
Q5-Q6,
having
on the bases by
connected
of diodes
The X signal input
y are dimensionless
and
Q1-Q4,
the
transistors
as diodes.
linearizes
resistance),
that
xI~
is yIE and (1 —y) lE, where z
in the
that
range
2) perfect
The
impair
shown
[7] that
the ratio
of the emit-
In
[7]
tudes
more
effects).
We
can thus write
it was
to
transistors
1.,
= (1 – x)yIE
matched
characteristics
emitter
(no ohmic
betas.
from
this
ideal
case
now be analyzed.
DUE
found
TO AREA MISMATCHES
that
balance
the
“offset
emitter
in a differential
conveniently
transistor
mismatch
= XYI.
1) perfectly
departures
will
DISTORTION
(or areas)
I.,
(7)
zero to
required
second order
to which
the linearity
ter currents in the Q2-Q3 and Q5-Q6 pairs is the same
as that in the Q1-Q4 pair and independent
of the magniof 1~ and IH (neglecting
had
amplifier
ratio
of the
emitter
“cell”
a pair
output
saturation
currents
For
in that
the
four-
paper,
(8)
was defined. It was then shown that for y # 1 (imperfect
matching),
the output currents were no longer simply in
is
Thus,
the normalized
I
——=
z – ;:’
Zy +
Icz
+
Ice
Ic3
–
output
Z is
(1 – Z)(I
– y)
–
a=l+z;
This
= 1 –2y’–2x+4xy.
It
is seen that
bipolar
currents,
but had the form
can be expressed
(9)
–l)”~”
in a form
linearity
due to area mismatches
For y =
1, this simplifies
that
shows the non-
as a separate
term
DA
(5)
the circuit
to 0.5. If
signals
z as the input
(4)
1c5.
X)g — (1 — y)z
–(l–
are equal
the
1,s,1s4
the same ratio
I OUb =
of
be expressed
junctions.
discussed
(3)
voltage
of
amplifier-could
as a ratio
of the two
voltage’’—the
currents
7 = I,ylIs,
The differential’
is
analysis, and makes no
It did, however, assume
exponential
and 3) infinite
extent
III.
unit y.
It was previously
— 1 to +1, the output
This is an exact large-signal
assumptions
about temperature.
diodes,
is the pair of currents
indexes
multiplier.
z = XY.
1. It
driven
pair
and (1 —%)1~. The Y signal
and
four-quadrant
substitute
on
CIRCUIT
in Fig.
of transistors,
collectors
a further
It
THE
basic scheme is shown
of two
basic
where X and Y are in the range
II.
their
The
deter-
proof here.
The
1.
ef-
heavily
paper
j)IE
(1-
theoreti-
some mathematical
Q6
Q5
thus
nonlinearities
draws
_
@
as
temperature
mainly
of the
in [7];
be quoted
tem-
of multiplier
using
and the analysis
laid
worsened
by
current
magnitude
the groundwork
this
is concerned
of the
practical
with
and substantially
paper
mination
and
c“,
poor zero stability.
however,
convertors
the circuit
cally
performance
including
CIRCUITS, DECEMBER 1968
OF SOLID-STATE
is balanced
we apply
bias
when
currents
x and g
such
X and Y can be used as the inputs,
that
DA w z(1 – x)(1-–
and
This
is a parabolic
fi~ of 0.25(1
1 The output
may also be taken
as a single-sided
the collectors
of (J2 and Q6, in which case it is Z =
signal
} (1 +
from
XY).
to
function
-y).
of x having
(11)
a peak value
– 7), which leads to the useful rule of thumb
l)~(percent)
% V,(mV)
at 300°K
(12)
GILBERT:
PRECISE FOUR-QUADRANT
MULTIPLIER
367
V>+i
Z=+l
(PROPOWiONRL DISTORTION)
2+
Z.-1
x.-l
X=+1
TRANSFER
X.+1
x=-i
>\sToRTlor4
CURVES
(a)
Y=i-i
Z-+1
Zao
Wwrtm
t-cl
(CONSTAI.IT
?JK.TMRIIX-1)
v= +1
v= 0
Zt
Z.-i
~=.i
X=+1
TRANSFER
X.+1.
x=-i.
-D\ STO CKr, ON
m sues
(b)
Fig.
Distortion
2.
introduced
by area mismatches
1/y2.
(a) y, = y,. (b) y,=
where Vo = (k T/q) l?g y, the total loop offset voltage.
However, notice that DA is not a function of temperature.
In the case of the four-quadrant
multiplier,
there are
two
such
circuits
working
in
conjunc$lon,
1) Q2 and Q3 have the same mismatch
that is yl = y2;
2) Q2 and Q3 have the opposite
71= Is,
Is,
z = XY+
(13)
--
distortion
perfectly
(with
(YI
=
will
to the x-input)
if Q1-Q2-Q3-Q4
respect
of v. For
1)
example
but
Q1-Q5-Q6-Q4
(Y2 # 1), there will be no distortion
ing maximal when y = O.
The output
do not
when y = 1, becom-
can be expressed as
Z = XY
+ 2yD4,
with
respect
products
for this
(14)
– 2(1 – y)Dm
shows that
the common
dZ/dy
= O) is shifted
Notice
also that
be seen that
the y input
the
the cases where
– 2)(1 – ~,).
the linearity
is not affected
purposes
=
is yl
(15)
point
to X =
respect
to
we can consider
perfectly,
%. Thus
the null
is unaffected
by
curves and distortion
in Fig. 2(a),
of intersection
(1 – Y)/(1
slope, and varies
which
P
also
(where
+ Y), Z = 0.
is always
of the same
in proportion
to it.
(17)
– 22(1 – X)(1 –’y)
Y input is balanced, there is a residue on the output of
peak amplitude
0.5 (1 –Y). The point P is thus at X = O,
as shown in Fig. 2(b).
The general co-ordinates
of P are
by area mismatches.
Q1 and Q4 match
(16)
1).
For the second case
Z = –0.5(1–Y),
of Z with
of demonstration
–
which
corresponds
to a constant
parabolic
distortion
component
added to the signal. In this case, when the
and
z(l
–7)(2zJ
to the X input
the nonlinearity
z = XY
DA2 x
– 3)(1
case is shown
sign as the output
D A, R Z(I – 4(1 – I’J
For
2*(1
this mismatch situation.
The general form of the transfer
where
It will
of
When the y input is balanced,
Y = O, y =
Z = O for all values of X. Stated differently,
suppression
match
mismatch
that
In the first case,
and
be a function
polarity
l/-y2, or yl Z –yz.
-.
1s21s4
The total
as Q5 and Q6,
Q5 and Q6, but the same magnitude,
so we must
define two area ratios
now
(exaggerated).
but
P(x,
z)
=
(
—.
1 — VY,7,
l+
G’’y2+G
72
—
~717z
)
.’
(18)
IEEE JOURNAL
368
x-INPUT
(SCOPE SWEEP)
OF SOLID-STATE
CIRCUITS, DECEMBER 1968
+5V
~
JJ}Rzzl
.
143K
~
-Isv
(::
Q2
LK
- 5V
Q3
Ot.z%v
(v,)
180K
~oK
Q1
57.2K
#
-15V
(a)
—
/
w
Q5
ZEROTO
250/JA 1
(39
I
+
“j--~
(c)
C/lo
. p“
(e)
Fig.
3.
Experimental
circuit
for
investigation
of nonlinear
effects.
Venjication
To verify
the above theory,
cases discussed,
a circuit
and demonstrate
was built
as shown
Use was made of the equivalence
and offset voltage. The equivalent
and Q6 could be varied
giving
the two
in Fig.
3.
of area mismatches
areas of Q2, Q3, Q5,
by the bias voltages
and Vz,
VI
Fig
4.
Demonstration
are
shown,
that
can
The devices were operated
250 PA, Vo = 2.5 volts)
at low powers
so that
tures were close to 300 ‘K.
(IB = IH =
the junction
tempera-
The use of low operating
the distortion
due to ohmic
curresis-
tances, discussed later.
To demonstrate
the nonlinearity
more clearly, a linear
ramp was used as the X input, and a simple R–C differentiator
produced
cremental
a waveform
slope of the transfer
corresponding
function.
to the in-
This
technique
provides
a very convenient
sensitive
measurement
of
distortion,
and became a valuable
tool during the investigation
of improved
multiplier
designs,
without
which it would
point-by-point
linearities.
Fig. 4(a)
adjusted
derivatives.
have been necessary
DVM
measurements
from
the
nonlinearity
deviation
the
DA
area
the ideal line),
linearity
transistors.
slope is within
(which
mismatches.
excellent
well-matched
of the dynamic
term
from
0.3 percent
with
constant
cent over 75 percent
+0
range.
– 1 per-
In terms
is a measure
this
amounts
of
of the
to less than
at any point.
to’ resort to tedious
to reveal the non-
point
shows the transfer
distortion,
Seven static
values
curves
with
and Fig.
of Y, from
VI
the slope
+1,
low.
starting
30-percent
The deviation
from
percent;
4(b)
With
0.68)
in Fig.
4(c).
as X varies
Notice
from
– 1 to
high and finishing
30-percent
the ideal line is now about 8
VI = +10
mV, VZ =
the theoretical
value
this
large would
be
–10
mV
(Y1 = 1.47, 72 =
of Z at X
=
0.0 should
be
–0.197. This is close to the value of –0.185 measured
from the waveforms
of Fig. 4(e). An interesting
feature
of the derivatives
shown in Fig. 4(f) is the one for Y =
O. Its linear form confirms the parabolic
shape of the
distortion
term.
IV.
Fig.
DISTORTION DUE TO OHMIC
5 shows
resistances
in
represent
are the
due
both
However,
the
the
circuit
emitters
all the bulk
the
In
+1,
4(d) ] falls
of course, an offset voltage
cuit.
– 1 to
[Fig.
exceptional.
Vz
and
is at —O.18, as shown
that
ularly
for minimum
demonstrate
achieved
The departure
actual
(19)
also eliminated
These
be
to
With V, = V, = – 10 mV, (Y, = 72 = 1.47) the theoretical point of intersection
is shifted to X = –0.19. The
and
rents
of distortion
due
Scales are arbitrary.
base
fact,
to
if
device
addition
the
of
will
effects
geometry
to
the
These
beta
is such
partic-
emitter
be current
and
linear
transistors.
of the diffusions,
referred
elements
crowding
the
the
all
resistances
resistance,
these
with
of
RESISTANCES
cir-
dependent,
nonlinearities.
that
the
cur-
GILBERT : PRECISE FOUR-QUADRANT
369
MULTIPLIER
,
(JIE
Equation
t
(23)
makes
~E and ~B are small
the reasonable
compared
assume RB = 1 ohm and Ifl
mV, about
10 percent
Using
above
a rule
the
of thumb
For
that
example,
2.5 mA, giving
=
of iiT/q
assumption
to lcT/q.
+E
2.5
=
at 300”K.
approximate
analysis,
we can state
for the peak magnitude
of DR, for the
quad Q1-Q2-Q3-Q4:
D.l
% +o.37(y4E
for ~z, 4~ in millivolts,
Similarly,
B.,
(1-;)[.
Fig.
5.
Circuit
having
ohmic
emitter
distribution
is equalized
sets of devices, the current
in the appropriate
dependence
R
+0.37
The net nonlinearity
resistances.
vary
rent-density
can be neglected.
Using the variables shown in Fig. 5, the loop equation
for the quad Q1-Q2-Q3-Q4
under these conditions
be-
with
connected
B.
with
which
has no explicit
solution
for
a in
terms
other variables.
However,
guessing that the
will be small, we will make the substitutions
(20)
of the
distortion
a=x+D~
– d,) Y
+0.37(4JE
the substitution
introduced
marized
where
DR is the fractional
Equation
(20) simplifies
w
–D~
distortion
due to resistances.
to
($ +
1) The
distortion
with
form
common
Thus,
(that
—
— yI,R.(1
the
– 2X – 2DJ
Solving
for
variable
from z to X,
distortion
– l,R,(l
term,
and
– 22).
changing
(22)
input
(23)
I#IBl(l
–
–
Y)]
(28)
nonlinearities
resistances
can be sum-
respect
quite
voltage
geometries
with
respect
has a
of the
to the Y input.
of intersection
of the transfer
at X = Y = Z = O.
arises when +H = +~.
large
and/or
to the X input
and is a fixed percentage
point
ohmic
is, it is possible
emitter
– X)
y)@E
2y —1. The
emitter
output z.
2) There is no distortion
sistance
DN
by y and
as follows.
4) No distortion
DE
lcT
;
D,)
{(1 –
of Y =
curves is always
log (1 –
and
(and
(percent),
by balanced
3) The
and
both circuits,
of each quad
are also weighted
– @~}Y –
symmetrical
(21)
outputs
terms)
(27)
– +,).
= DR2 – B,l
=
– 2CZ)
y)l$.
come from
The
in percent.
we have
out of phase. Thus
= *o.37[{y4.
– 2Z) – @.RE(l
will
and fi~l
circuit,
{(1 -
the y input.
hence the distortion
comes
= I,R.(1
at 3000K,
for the Q1-Q5-Q6-Q4
(26)
– +,)
low
resistances
to use devices
beta),
provided
terms are balanced.
in the ratio
ID/1~,
can be tolerated
with
high
that
the base and
base re-
By scaling the device
the closeness with
which
~a = ~, is then a matter of device matching.
In practice the resistors labeled RB in Fig.
5 will
be equal.
It
conditions
there will
be a residue
can be shown
that
under
in the multiplier
O, having the S-shaped form described
ing a peak amplitude
(at 300°K)
of
where
fi~
these
% &O.05 IE(RZ2
i-
Rm
–
not
output
for Y =
by (25),
and hav-
RE5
–
(29)
REJ
4* = IBRE
(24)
and
$b. =
these representing
base circuits
the extra
Notice
~L3~B,
that
voltages
due to resistances,
A(X)
= ~X(X2
in the emitter
and
and
–
1)
(25)
which describes the form of the distortion,
and has peak
values of *13.096 at X = *0.577,
and zeros at X * 1
and O.
where ~E is in percent,
parent
that
thk
lZ
has been experimentally
that
in rdliamperes,
RE in ohms.
residue term is independent
linear
emitter
confirmed.
resistances
It
reduce
of d.,
a fact
will
be ap-
the output
swing capability
because in the limit
(when the diode
voltages are small compared to the “ohmic”
voltages)
the circuit becomes completely
canceling
for all values
of X or Y. Also, the case where these resistances are
unbalanced will give rise to an equivalent
offset on one or
both of the inputs.
370
IEEE JOURNAL
by inserting
when
the 50-ohm
(29)
of full
predicts
pearance
of
Y input
percent.
of the distortion
of about
The
final
1)
the
2)
the
cases
can
transistors
measured
the display
and the distortion
Fig.
7(b)
gives
has
the
ap-
was modu-
20 percent.
imperfection
Three
percent
The
in which
when the Y input
V. DISTORTION
beta.
of *1.25
balanced.
50 times
*1.1
to a depth
in just the Q2-Q3 pair,
distortion
in Fig. 7(a),
vertically
values
lated
the
is shown
was expanded
CIRCUITS, DECEMBER 1968
resistors
a peak
scale with
nonlinearity
peak
OF SOLID-STATE
DUE
to
TO BETA
consider
is
that
of
finite
be considered:
have
identical,
constant
transistors
have
differing,
but
transistors
have
identical,
beta;
still
constant,
beta;
3) the
current-dependent
beta.
The first case was dealt with
that
the only
error
by the factor
Fig.
6.
Demonstration
of distortion
a small
due to ohmic
resistances.
(for
1~ less than
driven
(as, for example,
(f)
the output
alpha
sion of the circuit
in [7] where it was shown
is that
current
In the ver-
by a single-sided
the test configuration
offset term
is reduced
fll~).
input
current
shown in Fig, 3),
also arises, and the dc output
for
Y
= O is approximately
2(X, o) =
(30)
(1 – @ &
where ~ is the large-signal common-base current gain.
The second and third cases have not been completely
analyzed,
and it is doubtful
involving
all the betas and their
of any
Fig.
7.
distortion
(b)
(a)
beta
Characteristic
distortion
due
to
value.
mismatched
Vertical scale expanded to 0.33 percent/div.
centjdiv.
(a)
resistances.
(b) 3.3 Per-
terms
tortion
arise.
However,
and
are usually
sufficiently
Vfwijication
theoretically
pA; thus +B = +B = 12.5 mV. The derivative
heating
shown in Fig. 6(b),
shows little
over the full dynamic
degradation
of linearity
range.
The
topic
circuits
was
than IB.
In practice,
(28)
tortion
*1.
The
a nonlinear
measured
approximations,
value
(28)
or +~ become
will
comparable
term
of &4.5
is *3.3
percent
percent.
err on the high
with
kT/q).
at Y =
(Due
to the
side when ~~
See Figs.
typical
distortion
no serious
with
disbetas
DISTORTION
in
[7],
using
circuit
of
that
dissipation
in the inner
be arranged,
can operate
with
monolithic
to
category
can usually
are equal. This
in response
this
it was shown
arises due to the differential
if the power
the
in
where
a step
circuits
input
In
less
the thermal
is very
than 1 percent of the output amplitude,
no more than a few microseconds.
VII.
Because
and
6(e) and (f) demonstrate.
most
that
in
current
dis-
much
less
and persists
for
6(c)
and (d). By omitting
the resistors in the Q1-Q4 emitters, the distortion
is of the opposite polarity,
as Figs.
The
be
for
variations
a small
transistors
distortion
with
if necessary,
By omitting
the resistors in the Q2-Q3 and Q5-Q6
emitters, a net error of +~ = 12.5 mV remains. Equation
predicts
thermal
no distortion
pair
the
small
THERMAL
of devices
and outer
and,
of
dealt
would
of 100.
VI.
Using the test circuit of Fig. 3, to which emitter resistors were added, these nonlinearities
were demonstrated.
In Fig. 6 (a), all resistors were 50 ohms and 1~ = 1~ = 250
expressions
the possibility
over
arise using typical
in the neighborhood
waveform,
is now
to device,
should
to
there
explicit
nonlinearities
device
from
range,
Clearly,
whether
is due to the case where
the ohmic voltages do not match, due probably
to mismatches in rb and beta. This can be demonstrated,
too,
tions
the
of the very
cross
small
connection
due to capacitances
properly
tation
TRANSIENT
balanced
inputs
RESPONSE
voltage
swings
of the
transistors,
are very
small,
are used.
is the jt of the transistors.
The
at the
inputs,
the
aberra-
especially
main
speed
when
limi-
GILBERT:
PRECISE FOUR-QUADRANT
MULTIPLIER
371
w
iK
n
u
13
Q13
Q14
14.
Q15
11
2K
i.25*A
D3
I
t
i
6
v
Lu
lZJ
!
5
l.’ZS~A
12K
(Rl)
b
Q2
1
13
I
Soo
&
Q1
Dl
D2
Q4
2
L
1
4
2
El
Fig.
8.
Circuit
used
to
~
examine
high-frequency
behavior.
Fig.
Transient
and
(b).
10.
Complete
response
Figs.
a 200-MHz
staircase
12
for each input
9(c)
and
input,
output
monolithic
with
VIII.
Fig.
for
show the CW
the
other
of the sampling
10 is the circuit
a minimum
MHz)
(d)
(b)
is producing
thus,
about
all
the
ftIB/IB.
and the bandwidth
pect
X
The
a risetime
input
output.
At
Y =
is doubled.
variation
of about
step response
two
The
pairs Q2-Q3 or Q5-Q6
3-d13 bandwidth
O, each pair
We would,
in response
to one between
to the
receives
Y input
therefore,
is,
1~/2,
the
are possible
single-sided
ferential
null
components
should
sup-
extra components
operation
operational
and square-rooting
in the range
V’Y
k~) in the range O *5
output from pin 6 is
mA.
can be obtained
It
to give
The X input
point
The y input
a high impedance
volts.
to perform
di-
modes. These varia-
a summing
O *1
into
with
(dc to >100
amplifier
by pin changes only.
In into
suitable
to achieve medium-
or more versatility
current
voltage
multiplier
is a dif-
(approx.
can be shown
is a
at ground
that
400
the
ex-
to a step on the
these
for
by the
so as to be usable
Wide-band
built-in
squaring
potential,
At Y = +1, one of the transistor
9(a)
base in the oscil-
of a complete
or with
accuracy.
is available,
vision,
time
driven
the responses. The
20 dB at “500 MHz.
is designed
operation,
by using
tions
It
of additional
at a higher
Fig.9.
Performance
ofintegrated
version
of Figs.8.
(a) and (b).
Transient
response on Xand
Yinputs,
respectively,
at lns/div.
using dc control
on other input.
(c) and (d) 200-MHz
carrier
on X and Y inputs,
respectively,
staircase
voltage
on other
input.
Peak swing is 90 percent
of full scale in all cases.
input
in Fig.
response
A. COMPLETE MONOLITHIC MULTIPLIER
integration.
accuracy
is shown
(d)
loscope used to examine
pression was better than
(a)
multiplier.
(31)
extremes,
be fairly
in-
the scale factor
being
determined
by the +15-volt
sup-
dependent of the X amplitude.
Measurements
on an early integrated
multiplier
were
made to examine the high-frequency
behavior. The circuit, shown in Fig. 8, uses an “inverted”
pair of input
ply and the ratio of RI to Rz. The diode Ds ensures temperature stable scaling, and also makes the scaling factor proportional
to the positive
supply over a limited
diodes [7], which
emitter-degenerated
Input and output current balance, and the rest of the
circuit
currents,
are determined
by the five-collector
are conveniently
driven from pairs of
stages for the X and Y inputs.
range.
IEEE JOURNAL
372
.
CIRCUITS, DECEMBER 1968
OF SOLID-STATE
+15-Is ~
XY
(of:.)
{
(Oxov)
161514131z
109
I[
IoK
i234567S
(O*1OV)
x
{
s
‘*K
.
+
(a)
+Is
-1s
‘“*
0s%
IOK
(IMtov)
(c)
.
>
+!5+s
.
I
1“”””””
:
1
.
(b)
(d)
nKa4-F..c@aFuf
Y I*tol-
AAAA
AA
R?ODodr
‘w
K
(e)
GAtrd
+
Fig.
11, Circuit
of Fig. 9 connected
as (a) medium-accuracy
(b) wide-band
gain and nolarity
control,
(c) scmarer.
(d)
and (e) fully co;rected
m;ltiplie;.
lateral
five
p-n-p,
Q15.
collectors
eration.
matching
one
of the
tional
configuration,
hasto
be stabilized
*2
to
that
percent
collectors
Q13
op-
matching
can
errors
be achieved.
and
an external
the
balanced
is connected
through
by
currentsto
is vital
indicate
less than
that
of the
diffusions)
Measurements
in
~o-
an opera-
Q14.
This
capacitor
loop
connected
between
pins
13 and
14. The
second
collector
nominal
1.25
mA
balance
the
input;
collectors
currents;
collector
and
4 supply
sets
up
perfect
to
tails
4 give
for
access
connection,
absolute
come
coefficient
of
the coefficient
The
balance
to
linearity-connection
resistors
the
current
3 and
allowing
For
output
X
the
supplies
via
multiplier
a
3
5
Q16
Q20.
Pins
tional
to
the
the
through
Fig. 12. Typical
performance.
(a) As balanced
modulator,
carrier
frequency
5 MHz,
peak
output
swing
is 90 percent
of full
scale.
(b)
Output
expanded
ten times
in vertical
and horizontal axes-vertical
now 1.67 percent
of full scale/div.
(c) Null
suppression
for full-scale
5-MHz
carrier on X (upper
trace)
and
Y (I:wer
trace),
expanded
to 0.1 percent/dl~.
(d) Offset ramp
apphed
to both inputs
produces
the parabohc
output,
1 ys/div.
The
(base
considerably
tice
multiplier,
sauare-rooter,
to
by
this
percent
of hT/q
operational
device
these
bases
voltages
temperature.
close
0.38
the
voltages
The
ideal,
“K,
per
of
to
Q3
and
be
applied.
should
be propor-
aluminum
having
slightly
Q6,
l-ohm
a temperature
greater
than
at 300”K.
amplifier
permitting
increases
several
the versatility
modes
to
of
be imple-
IEEE JOURNALOF SOLID-STATECIRCUITS,VOL. SC-3, NO. 4, DECEMBER1968
mented.
Pin
output
pin
7 is normally
from
grounded,
the multiplier,
6 is also grounded.
and the
the multiplier
possible
inverted
pin 5, is connected
Hence,
to pin 8;
block
and D2,
having
voltage
able
must,
therefore,
negligible
conduction
range at the X-input
to conduct
heavily
Q2 and Q5 under
be fabricated
Fig.
be Schottky-barrier
along
11(a)
circuit.
with
through
the
collector
diodes
These
may
the standard
silicon
circuitry.
the versatility
in Fig. 12 show linearity
at 5 MHz,
of
conditions.
(e) illustrate
The waveforms
suppression
diodes
for most of the working
summing point, but being
before
overload
and the output
in the
Thanks
of the
and null
squaring
[21
[31
transistor
[61
of producing
either
a complete
linearity
input
monolithic
error
feasibility
multiplier
of the order
has been demonstrated.
Applications
with
of 1 percent
Better
are due to the Integrated
[11 G. A. Kern
[51
worst-case
a
[71
on
linearity
is
BILOTTI,
Abstract—A fully balanced analog multiplier using differential
transistor pairs is briefly described. Several circuit functions usually
required in communication systems can be derived from the basic
circuit. In particular, the different modes of operation leading to FM
detection, suppressed carrier modulation, synchronous AM detection, and TV chroma demodulation are discussed. Experimental
data obtained with a monolithic analog multiplier are also presented.
CIRCUIT
functions
[81
Analog
SENIOR MEMBER,
Group
at
in communi-
Multiplier
IEEE
siders an integrated
circuit
the block diagram in Fig. 1,
II.
Fig,
2 shows the
three differential
circuit
required
Circuits
and T. M. Kern,
have
objective
that
fully
balanced
pairs
block.
by
is to
arrangement
elsewhere
achieve
of the
Qi, Qz, and Q8 forming
The essential
been discussed
here
can be represented
BASIC CIRCUIT
transistor
an analog multiplier
I. INTRODUCTION
ANY
voltages.
Z7ectronic Analog
Computers.
New York:
McGraw-Hill,
1956, pp. 281-282.
G. S. Deep and T. R. Viswanathan,
“A silicon
diode analogue
multiplier,”
Radio
and Electron.
Engrg.,
p. 241, October
1967,
H. E, Jones,
“Dual
output
synchronous
detector
utilizing
transistorized
differential
amplifiers,”
U. S. Patent
3 241 078,
June 18.1963.
A. R. Kaye, “A solid-state
television
fader-mixer
amplifier,”
J. SMPTE,,p.
605, July 1965.
W. R. Davis
and J. E. Solomon,
“A high-performance
monolithic
IF amplifier
incorporating
electronic
gain-control,”
1968
ZSSCC Digest o} Tech. Papers, pp. 118-119.
A. Bilotti,
“Applications
of a monolithic
analog
multiplier,”
1968 I&SCC
Digest
o) Tech. Papers, pp. 116-117.
B. Gilbert,
“A new wide-band
amplifier
technique,”
this issue,
pp. 35&365.
E. Traa,
“An
integrated
analog
multiplier
circuit,”
M.SC.
thesis, Oregon
State University,
Corvalis,
June 1968.
of a Monolithic
ALBERTO
offset
have been measured.
Tektronix
for the fabrication
of many
experimental
circuits,
and to G. Wilson and E. Traa [8] for helpful
discussions.
A technique
has been described that overcomes the
inherent temperature
dependence and nonlinearity
of a
and the
transistor
ACKNOWLEDGMENT
[41
multiplier,
of
REFERENCES
SUMMARY
four-quadrant
adjustment
of over 500 MHz
now
configuration.
IX.
by
Bandwidths
Q2,
Q3, Q5, Q6 works with a collector-base
voltage
of O
*100 mV (the base voltage swing). The overload diodes,
D1
373
features
[1],
of this
[2]
and the
an understanding
of the
cation systems can be derived by way of analog
multiplication.
Fig. 1 shows a, functional
block
consisting
of an analog multiplier,
symmetrical
input
limiters,
and an optional
output low-pass filter. When
modes of operation
possible, and through
these modes,
to consider a number of possible system applications
[3].
Assume for a moment low-level
driving at the inputs of
VI and Vq. The current Ib established
in the current
properly
source transistor
is split
pair
voltage in transistor
M
perform
combined
FM
with
detection,
AM detection,
amplitude
tions based on frequency
passive networks,
phase
comparison,
modulation,
translation.
this block
can
synchronous
and other funcThis paper con-
termines
and
the
Q3. The
resistor
B
bias
output
and,
Q1. This
therefore,
collector
is proportional
in proportion
the
current
to the
to the
current
gain
of the
summed
product
applied
division
pairs
in the
of the
two
deQ2
load
ap-
signals VI and V2. The circuit topology is such that
if VI = O, the output currents due to a signal V2 are of
plied
Manuscript
received
June 5, 1968; revised
September
25, 1968.
This paper was presented
at the 1968 ISSCC.
The author
was with Sprague Electric
Company,
North
Adams,
Mass. He is now with the Faculty
of Engineering,
University
of
Buenos Aires, Buenos Aires, Argentina.
equal
giving
magnitude
and opposite instantaneous
polarity,
a zero net output. The same is true for an applied