An International Journal
Volume 9, No. 2, March 1996
Translinear Circuits in Subthreshold MOS
Andreas G. Andreou and Kwabena A. Boahen'
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Analog Integrated Circuits and Signal Processing, 9, 141-166 (1996)
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Translinear Circuits in Subthreshold MOS
Electrical and Computer Engineering, Johns Hopkins University, Baltimore MD 21218 USA
Computation and Neural Systems, California Institute of Technology, Pasadena CA 91125 USA
Abstract. In this paper we provide an overview of translinear circuit design using MOS transistors operating in
subthreshold region. We contrast the bipolar and MOS subthreshold characteristics and extend the translinear
principle to the subthreshold MOS ohmic region through a draidsource current decomposition. A fronthack-gate
current decomposition is adopted; this facilitates the analysis of translinear loops, including multiple input floating
gate MOS transistors. Circuit examples drawn from working systems designed and fabricated in standard digital
CMOS oriented process are used as vehicles to illustrate key design considerations, systematic analysis procedures,
and limitations imposed by the structure and physics of MOS transistors. Finally, we present the design of an analog
VLSI “translinear system” with over$590,000transistors in subthreshold CMOS. This performs phototransduction,
amplification, edge enhancement and local gain control at the pixel level.
1. Introduction
The Translinear principle [ 11 exploits the exponential current-voltage non-linearity in semiconthctor devices and offers a powerful circuit analysis and synthesis [2] framework. Originally formulated for bipolar transistors El], this principle enables the design of
analog circuits that perform complex computations in
the current-domain including products, quotients, and
power terms with fixed exponents [ 11, [2]. Translinear circuits perform these computations without using
differential voltage signals and are amenable to devicelevel circuit design methodology.
Most of the work on translinear circuits todate, use
bipolar transistorsand the emphasis is on high precision
and high speed. One fascinating aspect of translinear
circuits is their insensitivity to isothermal temperature
variations, though the currents in its constitutive elements (the transistors) are exponent$ally dependent on
temperature. The effect of small lock1variations in fabrication parameters can also be shown to be temperature independent. An excellent up-to-date overview of
translinear current-mode analog circuits using bipolar
transistors can be found in [3].
The increased commercial interest in analog CMOS
LSI and VLSI has renewed interest in the translinear principle for MOS circuit design. A generalized
form of the translinear principle was recently proposed
for MOS operating above threshold [4]; this extension
however does not follow the original definition of a
translinear circuit [11. This extension is simply a design
principle that exploits conservation of energy (KVL)
around circuit loops which have specific topological
properties. A novel class of translinear circuits that
employs multiple input gates, with floating gate MOS
transistors in subthreshold has been recently proposed
and experimentally demonstrated [20].
Another exciting research area that emerged the
last few years, is the synthesis of analog VLSI for
sensory information processing systems [7], [S] employing MOS transistors operating in subthreshold region [5], [6], [7]. We have been exploring translinear
circuits in subthreshold MOS for use in analog neuromorphic LSI and VLSI systems [113, [121, [9], [101.
In this biologically motivated computationalparadigm,
high processing throughput is attained through a tradeoff between massive parallelism and lower speed in
the circuits and therefore subthreshold CMOS operation is possible. Such architectures often necessitate the computation of linear and non-linear functions,
and if a current-mode [ 111, [ 121 design methodology
is adopted, the translinear principle offers an effective way for synthesizing circuits [ 131, [ 141 and systems [15], [16], [17], [18].
A. G.Andreou and K. A. Boahen
In this paper, we discuss experimental circuit designs
based on the translinear properties of subthreshold
MOS transistors in the saturation and ohmic regions.
Our objective is to present a comprehensive overview
on this subject, beginning with the basic devices and
circuits, and following it through to the system level.
The discussion of subthreshold MOS models, and their
characteristicsand limitationscan be found in other excellent references (for example [5], [6]). However, a
basic review of subthreshold MOS and bipolar operation, is provided since the large signal properties of
the devices are key to the subject matter. Most of the
circuit examples given, have been used in analog LSI
and VLSI systems that have been fabricated and tested
functional. The value of these circuits can only be
fully appreciated in the context of the systems that employ them; references to the original journal articles
are given.
The paper is divided into six sections. Section 2 contrasts the translinear properties of bipolar transistors
with those of MOS transistors in subthreshold. Basic
circuit techniques that employ MOS transistors in subthreshold saturation and ohmic regimes are introduced
in section 3. In the same section, we discuss both
translinear loops (TL,) composed of generalized diodes
and current sources, and translinear networks (TN) that
include voltage sources as well. Section 4, focuses on
an analog VLSI translinear system, a contrast-sensitive,
silicon retina [171. A discussion of MOS device limitations and deviations from the first order large signal
models that ultimately affect circuit and system performance is presented in section 5. Section 6 concludes
the paper.
2. Translinear Devices
We begin the discussion of translinear circuits in subthreshold MOS technology with the basic devices.
A translinear element is a physical device whose
transconductance and current through the device are
linearly related, that is, the current is exponentially dependent to the controlling voltage. A two terminal
p-n junction (diode), with its exponential I-V characteristics, is a translinear element and used often as an
example in circuits [3]. Voltage gated, ion channelsconductances- are also translinear devices.
Three-terminal devices are termed “translinear” if
the relationship between the current and the controlling
voltage is exponential and the two terminals across
which the controlling. voltage is amlied exhibit true
diode-like behavior, i.e., increasing the voltage on one
terminal is exactly equivalent to decreasing the voltage
on the other terminal by the same amount. In this
case, a loop of such devices consists of voltage drops
across pairs of control terminals and we exploit the
linear transconductance-current relationship. Bipolar
transistors have both properties whereas MOSFETs do
The large-signal device model equations for both
the bipolar transistor and MOSFET in subthreshold
are discussed in Appendix A where the approximations made during their derivations are clearly stated
and the symbols are defined. In the active-forward region of operation, the function of a bipolar transistor
as a transconductance amplijïer is captured by the following equation:
IC = Is e
where V, = ( k T / q ) and is^ is defined in Appendix A.
The magnitude of the transconductance from the
base is identical to the magnitude of the transconductance from the emitter:
- a IC
We now contrast the operation of a bipolar transistor
as a translinear element with that of an MOS transistor
operating in subthreshold. Much like a bipolar transistor, the MOSFET in subthreshold has exponential
voltage current characteristics (see Figure 1). There are
however, two fundamental differences between MOSFET and bipolar devices that have implications in the
design of translinear circuits.
l. Unlike a bipolar transistor, the current in a MOSFET is controlled by the surface potential, which
is capacitively-coupledto the gate (front-gate) and
bulk (back-gate) terminals.
2. The MOSFET, is symmetric with respect to the
source and drain terminals while a bipolar is not.
In summary, the MOS transistor is a four terminal
device with symmetric drain and source terminals, as
result of lossless channel conduction, and an isolated
control potential capacitively set by one or more control
gates. As we will see in subsequent sections, the latter property of the MOS transistor is a mixed blessing
when the design of translinear circuits is considered.
Translinear Circuits in Subthreshold MOS
Equation 4 can be re-written as a function of dimensionless current quantities iG and i g . Each of these
currents would correspond to the device current if the
surface potential @S could assume the voltage at the
gate or bulk terminal. In essence these currents correspond to ideal diode junctions between the source and
surface potential weighted by the appropriate capacitive divider ratio. Therefore, the equation for the drain
current can be written as:
Fig. I. Measured current ZDS and Zc versus controlling voltage
VGS and VBE respectively. The MOS transistor has dimensions of
(16 x 16pm2) and is fabricated in a 1.2pm n-well CMOS process
and is biased at a drain-source voltage, V~s=1.5Volts. The current
is measured at two different substrate voltage bias conditions. The
bipolar transistor is a vertical device with an emitter area of (16 x
16pm2)fabricated in a 2pm n-well CMOS process and biased with
Vc~z1.5Volts. T = 301.5 K.
It should be pointed out that the voltage difference
that controls the current in a MOSFET to yield the
translinear behaviour, is the potential difference between the channel surface potential @, and the potential at the source VS and or drain VDso that the current
between the drain and source for an NMOS is given
Since the MOS transistor has two “gates” the relationship between @,(V,, V,) and the bulk or gate
terminal voltages V, and VG can be obtained using the simple capacitive divider model depicted in
Figure 2. The introduction of the parameter K EZ
C,3,/(Cix C:ep) is convenient for modeling the effect of the two gates. Note that K is a function of the
surface potential @S as Ciepis a function of the applied
gate and substrate voltages.
In saturation i.e. when VDS L. 4Vt, and when the
current controlling voltages are referenced to source,
Equation 3 simplifies to:
In subsequent sections, we will see how the latter
formulation facilitates the analysis of MOS translinear
circuits and an extension of it will be used to analyze
FGMOS translinear loops. Since the dimensioneless
current quantities are related to the surface potential
the will be called @-currentsor psi-currents.
The transconductance from the gate is given by:
gm EZ -
V,, vS=c
-- K IDS
and from the local substrate (backgate) terminal:
- (1 - K ) I D S
The conductance g, at the source is given by:
The transconductances depicted in Equations 6,7 and 8
are linear functions of the current -to a first order- and
hence each MOS transistor in saturation has the equivalent of three different translinear elements. Note,
that the source transconductance is equal to the gate
transconductance by shorting the local substrate and
the source of the transistor (VB = V S )in which case
Equation 4 for the current becomes:
In subsequent sections, we will see how shorting of
the substrate to the bulk, partially circumventsthe nonidealities in the translinear properties of MOS transistors and enables the design of near ideal loops.
The translinear properties of the bipolar and MOS
transistors in subthreshold are evident in Figure 1. In
A. G. Andreou and K. A. Boahen
in the elements connected in the Clockwise
(CW) direction is equal to the corresponding product for elements connected in the
Counter Clockwise (CCW) direction.
As an example let us consider the circuit of Figure 4
consisting of four ideal diodes in the loop X-Z-Y-WX. Following the translinear principle, we can write:
Note that the translinear principle is derived by beginning with Kirchoff’s voltage law or the principle of
conservation of energy, so that:
Fig. 2. (a) Symbol for an NMOS transistor. The current between the
drain and source is controlled by the difference between the surface
and the potential at the source terminal. The surface
is set by the potential at the gate and bulk terminals
through the capacitive divider between C,, and c&pshown in (b).
An NPN bipolar transistor is shown in (C). The current between the
collector and emitter is controlled by the voltage difference between
the base and emitter nodes.
a logarithmic current scale, the transfer characteristics
show linearity with respect to the controlling voltages.
Plots of the normalized transconductance ( g m / I ) are
shown in Figure 3 and demonstrate how the bipolar
device is an ideal translinear element while the MOS
transistor in subthreshold only approximates it over a
limited range.
3. Translinear Circuits
In this section we discuss translinear circuits that employ translinear elements, both MOS operating in subthreshold and bipolar transistors. We follow the convention proposed by Barrie Gilbert in [3] and make
a distinction between a Translinear Loop (TL) and a
Translinear Network (TN).
3.1. TranslinearLoops
In “strictly” TLs the translinear principle [l] can be
stated as follows:
In a closed loop containing an equal number of oppositely connected translinear elements, the product of the current densities
VD(2i-1) -
VD(2i) = 0
Equation 10follows from Equation 11if the voltages
are summed around loops of translinear devices.
In a circuit graph composed of two terminal elements
such as ideal diodes (see Figure 4), there is a direct relationship between the voltage difference among each
pair of nodes transversed by the translinear device, and
the current in the arc that joins the nodes. This is a
consequence of having the voltage nodes that control
the current be the same as the current-output nodes of
the device. In practical systems, the ideal diodes in
Figure 4, would correspond to base-emitter junctions
of bipolar transistors with shorted collector-base terminals.
Analogous behaviaur can be obtained using translinear three terminal devices such as bipolar transistors,
MOS transistors in subthreshold, or any other device
that yields diode-like characteristics. However, in three
terminal devices, the diode-control nodes in the circuit
need not correspond to the current path. In bipolar
transistors the diode control nodes are available and
thus they can be used explicitely in constraint equations such as Equation l l. This is not true for MOS
transistors! As we have seen already, one of the diode
control nodes, (namely the node corresponding to the
surface potential
is not directly accessible. The situation becomes even more complex in MOS transistors
with a floating gate (FGMOS) (see Figure 5) coupled
to one or more controlling gates (see [ 191, [20] and
references therein). At first sight, the floating gate, appears to make the situation worse, but actually it opens
the possibility for a new class of translinear circuits
Translinear Circuits in Subthreshold MOS
Fig. 4. A Translinear loop using ideal p-n junctions (diodes).
Fig. 3. Normalized transconductance curves. The transconductance
is computed through numerical differentiationof the data in Figure 1,
and subsequent smoothing. (Top) For the MOS transistor; (Bottom)
for the bipolar transistor. The dramatic decrease of the transconductance in the MOS transistor at low gate-source voltages is attributed
to the leakage current.
proposed and experimentally demonstrated recently by
Minch [20].
Essentially the physical structure of FGMOS transistors offer an extra degree of freedom which can be
exploited systematically through another set of constraint equations of the form:
where V F G i , QFGi are the floating gate voltage and
charge on the (ìth) transistor and V G j is the voltage of
the ( j th ) control gate. The total capacitance seen in
the floating gate is C T i and A i j is a design parameter
that depends on the ratio of the control gate to floating
gate capacitance,i.e. A i j Cfgj/ C T i . The details of a
systematic analysis procedure for FGMOS translinear
circuits can be found in [20].
3.l.l. Analysis of Translinear Circuits with MOS Transistors in Saturation The current mirror is a trivial
example of a translinear circuit; it has a single loop
with two translinear elements, one CCW and the other
Fig. 5. Capacitive model and symbol for a floating gate MOS (FGMOS) transistor. The device depicted in this figure has three control
gates G 1, G2 and G3.
Two currrent mirrors implemented with complementary devices and connected back-to-back yield the circuit shown in Figure 6. This loop includes four threeterminal devices and corresponds to the ideal diode
example of Figure 4. The circuit can be readily recognized as a BiCMOS implementation of an AB stage
in a digital oriented CMOS process where only one
type (NPN) of bipolar transistors is available [23]. A
composite strÚcture made of an MOS in subthreshold
and an NPN bipolar yields a pseudo-PNP device with
good driving capabilities. Translinear loops using both
PNP and NPN bipolar transistors were first studied by
Fabre [2 11.
Applying the translinear principle to the loop XZ-Y-W-X of Figure 6 yields the following constraint
A. G. Andreou and K. A. Boahen
junctions and obtain
The above relationship can also be derived by summing the voltages around the loop (conservation of energy)
Fig. 6. The translinear loop of Figure 4, implementedusing composite bipolar and subthreshold MOS transistors. The loop is employed
in a current conveyor configuration where the bidrectional output
current Zout equals to the bidirectional current l i n .
+ V2 - V3 - V4 = o
Replacing the gate-source voltages for M l , M2, M3, M4
with their respective drain-source currents using Eq. (9)
(assuming all devices are in saturation, have VSB = O,
have negligible drain conductance, have identical K ,
have identical IO and geometry S ) , we obtain:
Fig. 7. A translinear circuit that performs one-quadrant normalized
multiplication. I I , 12 and 13 are the inputs and 14 is the output.
equation for the currents in the circuit:
This classical four junction loop can be combined
with two current mirrors to implement a current conveyor [22] where Iout = l i n and Vz = v i n .
Our second example is the MOS transistor onequadrant multiply-divide circuit shown in Figure 7. A
large number of these CMOS multipliers have been
employed in the implementation of a correlation-based
motion-sensitive silicon retina [24].
Applying the translinear principle to the loop GNDA-B-C-GND, we find a total of four equivalent diode
from which Eq. (14) readily follows. Note that the
assumption of identical K holds true to a first order
because VSB = O and the gate of all transistors are
within a few hundred millivolts from each other.
Yet another way of viewing the function of this circuit is that of a log-antizog block. Transistors M1 and
M2 do the log-hg, M 4 does the antilog-ing and M3 is
a level shifter.
Another single quadrant multiplier is shown in Figure 8. This circuit was proposed and its function experimentally demonstrated in [29]. The operation of
the circuit can be understood by noting that a single
transistor (M4) can perform a single quadrant multiplication because the voltages on the gate and bulk control
the current in a multiplicative fashion (see Equation 4).
Since in subthreshold the transistors saturate at only a
few & of drain source voltage, the bulk terminal of the
device can be connected to the drain without turning
on the bulk-source junction.
An expression for the output current I4 can be obtained by applying the translinear principle around
the four loops (Vdd-A-Vdd), (Vdd-B-Vdd),(Vdd-CVdd),(Vdd-D-Vdd) to obtain the following equations
Translinear Circuits in Subthreshold MOS
Fig. 8. A four transistor translinear circuit that performs a onequadrant normalized multiplication and exploits the back-gate in an
MOS transistor. Device pairs M l , M 3 and M 2 , M4 share local
substrate terminals (in this case n-wells). 1 1 , 12 and 13 are the inputs
and 14 is the output.
for the psi-currents introduced in Equation 5:
The actual currents in the four MOSFETs
M1 ,M2,M3,M4, can be written as a function of the psicurrents:
z1 = IDSI
= SI no iK1
G1 B1
where the devices have been assumed to have the
same S and Zno. If now the assumption is made that
K I = ~2 = ~3 = ~ 4 Equations
15 and 16 yield the.
following expression for the output current 14 in terms
of the input currents 11,12 and 13 :
In the original implementation [29] it was suggested
that to improve accuracy, the voltage on the local substrate (n-well) of devices M3 and M1 should be set at
a value close to that of node B , the local substrate of
of devices M2 and M4. This is indeed necessary, as
the bulk voltage determines K of all transistors, which
Fig. 9. Circuit that converts a bidirectional current on a single wire
into two unidirectional currents on separate wires. This is a currentmode absolute value circuit. The sign of the bidirectional input
current is assumed to be positive when it adds positive charge to
node C . The bidirectional current Z B D is the input, the unidirectional
currents 11 and 12 are the outputs, and Z B sets the operating point of
the circuit.
was assumed to be the same for all transistors. Another implicit assumption here is that the gate voltage
is approximately the same for all transistors.
Our next example addresses the problem of converting a bidirectional current to two unidirectional
currents which is the equivalent to a current-mode
half-wave rectification. A translinear circuit that computes this nonlinear function is shown in Fig. 9. The
bidirectional current ZBD is steered through transistor M3 when ZBD > O and through transistors M4,5
when ZBD < O. Concentrating on transistors M1,2,3,4,
we identify a loop (VDD-A-B-C-VDD) and apply the
translinear principle to yield:
form a simple current mirror.
These equations may be solved for 11 and 12 in terms
of ZB and ZBD:
Which shows that 11 E ~ Z B DI and 12 2c O when
ZBD >> ZB and vice versa. The absolute value is ob-
A. G. Andreou and K. A. Boahen
AI* =
Ii n
similarly for A I* I3 - 14, I B E I3 14.The differential output current A I* is a scaled version of the differential input current A I . The voltage between node
B and Vdd should be such that the current source IB
stays in saturation.
The mirror composed of transistors M5 and M6 converts the unidirectional differential signal AI to the
bidirectional signal Iout so that:
= -IB lin
Fig.10. A translinear circuit that computesthe normalized difference
of two current signals. II and 12 are the inputs and the bidirectional
current Zout is the output normalized to 11 12.
tained by connecting the two output wires together in
which case:
This circuit has been employed in a CMOS integration of an autoadaptive linear recursive network for the
separation of sources [ 141.
The next translinear circuit performs a current-ratio
computation. This functional block, is part of the readout amplifier in an analog VLSI system that integrates
monolithically a one dimensional array of photodiodes
and selective polarization film to form a polarization
contrast retina [25].
The simple translinear circuit in Figure 10 is excellent for rescaling differential current signals and thus
computing the contrast. I1 and 12 represent currents
fi-om two selected photodiodes. The heart of the computation circuit will be recognized as a Gilbert gaincell [3] implemented in subthreshold MOS.
The analysis of this circuit is typical for translinear
circuits that involve differential current signals. Application of the translinear principle around the loop
A-B-C-D-A yields:
3.1.2. Analysis of Circuits with MOS Transistors in
the Ohmic Regime In this subsection, we extend the
translinear principle to subthreshold MOS transistors
operating in the ohmic region. In Appendix A, (see
Fig. 25), we show how the source-drain current of a
MOS transistor can be decomposed into a source component I Q ~and a drain component I Q ~and
, that these
components superimpose linearly to yield the actual
current ISD E Ies - I Q ~ .
In the ohmic region, these components are comparable. Decomposition and linear superposition may be
used to exploit the intrinsic translinearity of the gatesource and gate-drain “junctions.” This is the basis for
extending the translinear principle to the ohmic region.
On the otherhand, in the saturation region, we can exploit the translinearity of the gate-source “junction” directly because the drain component is essentially zero
and decomposition is of no consequence.
The translinear circuits based on subthreshold ohmic
operation are only possible because of the symmetry
between drain and source operation of an MOS transistor. One could argue that decomposition is also possible with bipolar devices. However, while the difference
of two exponentials is the exact form for MOS devices,
it is only an approximation for bipolars, due to the fact
that the forward and reverse current gains of the device never reach unity [46], [47] (see Equations 45 in
Appendix A). This distinction is fundamental and important difference between MOS and bipolar transistors
arising from lossless transport in a MOS channel versus lossy transport in bipolars due to recombination in
the base. It is possible to use CMOS compatible lateral
bipolar transistors as symmetric devices [26] but at the
expense of a large base current that increases the power
dissipation in the system.
Translinear Circuits in Subthreshold MOS
Fig. II. A translinear circuit that employs subthreshold MOS transistors in saturation and ohmic regime and computes the product of
two input currents 11 and 12 normalized to 11 12.
To demonstrate the application of the translinear
principle to circuits that include MOS transistors in
the ohmic regime, consider the one-quadrant currentcorrelator circuit in Fig. 11. Transistor M2 operates in
the ohmic region. Proper circuit operation requires that
the output voltage is high enough to keep M3 in saturation. This circuit was first introduced by Delbrück [27]
and later incorporated in a larger circuit that implements the non-linear Hebbian learning rule in an autoadaptive network [28], [30] and in a micropower autocorrelation system [313.
An expression relating the output current, 13, to the
input currents, I1 and 14,can be derived by treating the
source-gate and the drain-gate “junctions”of the ohmic
device as separate translinear elements and applying
the translinear principle. For the two loops formed
by nodes GND-A-GND and GND-A-B-C-GND in
Fig. 11, we obtain
Fig. 12. A current-mode circuit -translinear network- that implements a normalized cubic non-linearity. 11 is the input, 12 is the
output and the voltage source VR normalizes the result.
and (20), the output current is given by:
Il 14
+ 14
3.2. Translinear Networks
In the previous section we have discussed “strictly”
translinear loops (TL).Translinear networks [3] differ
from translinear loops in that they contain independent
voltage sources and the following equation can be employed in their analysis:
source-drain current of the MOS transistor can be decomposed into a source component IQ,and a drain
component IQd-COntrOkdby their respective “junction” voltages VQ, and Ved. These opposing components superimpose linearly to give the actual current
passed by M2, i.e.,
where E is the independant voltage source and Ç
is a constant coefficient that lumps device design and
fabrication parameters. The above extension to the
translinear principle was proposed by Hart [33].
We begin the discussion of TNs using a simple circuit
that has the topology of a current mirror and incorporates a voltage source in the loop (see Figure 12). If the
A. G.Andreou and K. A. Boahen
Fig. 13. A translinear circuit using FGMOS transistors to compute
the ratio of a cubic to square functions. 11 and 12 are the inputs and
13 is the output.
input current is I I , the output current is 12, and VR is
a constant voltage source, application of the translinear principle around the loop (GND-A-B-C-D-GND)
The voltage source is necessary for circuit operation and it normalizes appropriate the output current.
This circuit has been employed in a small system that
implementsthe Herault-Jutten independent component
analyzer [141.
An FGMOS-based circuit that has the same functionality as the circuit in Figure 12, is shown in Figure 13.
We begin the analysis of the circuit by noting that the
current in the channel of an FGMOS, is controlled by
multiple gates that can be thought as extensions to the
front gate of the transistor. As such, Equation 5 can be
re-written for an N-input NMOS transistor as:
where it has been assumed that all N gates of the
i-th transistor have the same strength, i.e. have the
same coupling capacitance to the floating gate. The
charge Q F G on the floating gate is incorporated through
a geometry related multiplicative constant SQ so that
when the charge Q F Gis zero, S Q = l.
An expression for the output current I3 can be obtained by applying the translinear principle around
three loops that include the floating gates and source of
transistor M 3 together with the floating gate nodes and
sources of the other transistors. When the three loops
are traversed, the following equations for psi-currents
are obtained:
We have adopted a notation where for example, i3a
denotes the psi-current in device 3 controlled by the
voltage on its gate a. Using Equation 24, the current at
the source of M1 , M 2 , M 3 , can be expressed as functions
of psi-currents so that:
where the devices are assumed to have the same S
and I n o ; the back-gate contribution to the current in
each device is eliminated as all transistors have the
source shorted to the substrate. Now, by making the
assumption that K I = ~2 = ~3 = ~ 4 and
no charge
on the floating gates, Equations 25 and 26 yield the
following expression for the output current I3 in terms
of the input currents I I and 12:
The assumption of equal K is reasonable so long
the voltage on the floating gate is such that all devices
stay in subthreshold. An alternative way of obtaining
a functional description of the circuit can be found in
the paper by Minch et. al [20].
A TN that incorporates bipolar transistors, an MOS
in subthreshold, and an independent voltage source
Vxu is shown in Figure 14. We will use Equation 22
to derive the relationship between I l , 1 2 , I z w and Vxy.
The MOS transistor is assumed to be ideal with K = l.
A discussion of networks with non-ideal devices will
be done in the next section.
A ratio relationship between I Z w and I1 - 12 can
be derived by employing Equation 22 applied to the
two loops (X-Z-Y-X) and (X-Y-W-X) to yield the
following equations:
Translinear Circuits in Subthreshold MOS
Fig. 14. A BiCMOS translinear network that exploits the MOS
subthreshold ohmic characteristics.
Using the adopted conventions for current decomposition in NMOS transistors (see Figure 25), I Qand
together with Equation 28 we obtain the following expression that relates the currents in the circuit.
This is a lumped parameter model where G1 and G2
correspond to resistances per unit length. The voltages
on nodes P and Q referenced to ground, represent the
state of the network and can be read out using a differential amplifier with the negative input grounded.
The equivalent circuit using idealized non-linear
conductances is shown in Figure 15(right). The difference in currents through the diodes D1 and D2 are
linearly related to the current through the dimsor MOS
transistor.' This relationship can be derived from Equation 55 describing subthreshold conduction, and the
ideal diode characteristics where ID= Is exp[VD/ Vt].
An expression can be derived for the current IPQin
terms of the currents I p and I Q , the reference voltage
V, and the bias voltage V,, when diodes are replaced
by transistors:
The current In0 and S is the zero intercept current
and geometry factor respectively for the diffusor transistor M h . Is is the reverse saturation current for the
diode that is assumed to be ideal. The currents in these
circuits are identical if
C1 =
It is immediately apparent that the current ratio
Izw/(I1 - 12) can be controlled both by a fixed parameter (S)that is designed prior to the fabricaition of the
circuit and a variable quantity (e-vxy/K)that can be
programmed (post-fabrication) during circuithystem
operation. This property will be utilized in the design of linear MOS transistor-only spatial averaging
3.2.l. Translinear Spatial Averaging Networks Often, models of neural computation necessitate the realization of spatial averaging networks [7]. To demonstrate the analogies between linear and translinear networks as well as their subtle and important differences,
we begin with networks that employ linear conductances, voltages and currents and contrast them with
translinear current-mode [ 161 networks.
A voltage-mode circuit model for a loaded network
is shown in Figure 15(left) for which:
(F) "1
Increasing VC or reducing V, has the same effect as
increasing G1 or reducing G2. The state of this network
is represented by the charge at the nodes P and Q .
Since the anode of a diode is the reference level (zero
negative charge), the currents I p and IQrepresent the
result. Unfortunately, the anode of a diode or a diode
connected transistor is not a good current source.
When diodes are not explicitly available in the process, diode connected PMOS or NMOS transistors can
be used as shown in Figure 16. When the loads are
PMOS, the current current I ~ isQgiven in terms of
voltages normalized to ( k T / q ) :
When NMOS transistors are used as loads, there is
the additional benefit, that of exploitingthe current conveying properties of a single transistor [ 161, to obtain
the current outputs I p and IQ, on nodes that are low
A. G. Andreou and K. A. Boahen
Fig. 15. Building blocks for linear loaded networks. Using segments that employ ideal (left) linear and (right) non-linear elements.
where s h and Sv are geometry parameters for transistors
M h and M,, respectively.
The one dimensional MOS transistor-only network
corresponding to the Helmholtz equation shown in Figure 17 can model the averaging that occurs at the horizontal cells layer of the outer retina. This is equation is
the basis of the well known silicon retina architecture
proposed by Mahowald and Mead [34], [7].
Summing the currents at node j we get:
Using the results from the previous section for the
currents Iij and Ijk given by Equation 32 substituted in
Equation 33 yield:
Fig. 16. Current-mode building blocks for linear loaded networks
using (top) PMOS transistor implementation, (bottom)NMOS single
transistor current-conveyor implementation.
conductance (the drain terminal are now excellent outputs for the currents). Using Equations 8.45 in [1619
the current I P Q is given as:
Normalizing internode distances to unity the above
equation can be written on the continuum as:
I*@) =I(x)
d 2I (x)
+h dx2
This equation yields the solution to the following
optimization problem: Find the smooth function I (x)
that best fits the data I * @ ) with the minimum energy
in its first derivative. Input is the currents I * @ ) and
output the currents I (x).
The parameter h =
exp& vc - K ~ V is~ the
cost associated with the derivadve energy-relative to
the squared-error of the fit.
The diffusive network in Fig. 17 was recently described in terms of “pseudo-conductances” [35]. We
Translinear Circuits in Subthreshold MOS
Fig. 17. A one-dimensional MOS translinear network to perform local aggregation -spatial averaging-. The back-gate terminals of all devices
are connected to the substrate.
have used the chargelcurrent-based formulation first
proposed in [ 151 to explain its behaviour. This &mentmode approach relies an intuitive understanding of the
device physics and yielded the insight which enabled
us to extend the translinear principle to subthreshold
MOS transistors in the ohmic region as well as the decomposition of the current into dimensioneless components correspondingto ideal junction. We now have
a comprehensive current-mode approach for ailalyzing subthreshold MOS circuits. The essence of this
approach is the representation of variables and parameters by charge, current, and diffusivity. Voltages and
conductances are not used explicitly.
Bult and Geelen proposed an identical network for
linear current division above-threshold and used it in a
digitally-controlledattenuator [36]; they also analyzed
its subthreshold behavior. However, they stipulate that
all gate voltages must be identical and control the division by manipulating the geometrical factor W / L of the
devices. We have shown here, and previously in [ 151,
that this constraintmay be relaxed in subthreshold without disrupting linear operation. This is a real bonus because it allows us to modify the divider ratio or space
constant of the network after the chip is fabricated by
varying (V, - Vr). Tartagni et al. have demonstrated
a current-modecentroid network [37] using subthreshold MOS devices whose operation is described by the
current division principle.
3.3. A general resultfor MOS translinear loops
Three of the circuits discussed in the previous subsection, namely the translinear multiplier of Figure 8, the
MOS implementation of the Gilbert gain stage in Figure 10, and the current correlator (Fig. 11) have been
experimentally shown to exhibit near “exact” translin-
Fig. 18. Translinear loop composed of five MOS transistors in subthreshold. All devices are in saturation except device M5 which is in
the ohmic regime and therefore can be decomposed as two devices
in saturation ,back to back sharing same gate and substrate.
ear behaviour even though they are build from MOS
transistors and they do not have their source connected
to the local substrate.
A recent result by Eric Vittoz [32] can be employed
to partially explain this rather surprising behaviour.
He considers translinear loops constructed from MOS
transistors in subthreshold saturation with common
substrate connection (similar to the one shown in Figure 18). If the pairing of transistors in the CW and
CCW direction, is such that they have their gates connected to gates and sources connected to sources and
they are alternated (much like even-numbered and oddnumbered devices in Figure 18) Vittoz shows that the
translinear loop does not suffer from the MOS transistor non-ideal translinear behaviour. He notes also
that loops containing transistors in the ohmic regime
can also be included in this formulation as they can
be decomposed as two parallel connected saturated de-
A. G. Andreou and K. A. Boahen
vices sharing common gate and comrnon substrate (see
Figure 25.)
However, to account for the near “exact” operation
of the multiplier in Figure 8, Vittoz’s argument must be
extended to include loops that go through the back gate
of the MOS transistors as illustrated in Figure 18. The
global substrate restriction can thus be removed and
replaced by a local substrate connection, and the result
still holds true. In a standard CMOS process, this will
of course be possible only for one type of devices.
Now, we will re-examinethe operation of the circuits
in Figure 10, Figure 8, and Figure 1l .
Consider the largest loop (A-B-C-D-A) in Figure 10. Devices M1 and M2 have common gate and
common bulk and so do devices M3 and M+ When
adjacent devices are paired in different ways we observe that M3 and M2 share the same source and bulk
which is the case also for M1 and M4.
In the the largest loop (A-B-C-D-A) of translinear
multiplier circuit of Figure 8 in we can verify that transistors M1 and M2 as well as M3 and M 4 share common
gate and source. The alternative pairing, finds M 4 and
M2 sharing same bulk and source which is also the case
for M3 and M l .
When devices in the loop are operating in the ohmic
regime, such as M2 in the circuit of Figure 11, we can
verify that the loop (GND-B-C-GND) incorporates
two adjacent sets of devices M3 and M2 share same
bulk and source/drain while M3 and M 4 share bulk and
gate; the bulk in this circuit is the same for all devices.
Translinear circuit dynamics
The dynamics of translinear circuits and systems have
not been discussed in this paper. However, it was
pointed out in [45], that in networks with non-linear
conductances without complementary non-linear reactances, the state equations that describe the dynamics
of the system are non-linear. Given an architecture and
a particular network, a method was outlined to test for
stability [45].
4. A TranslinearSystem: A ContrastSensitiveSilicon Retina
Image acquisition under naturally occuring, uncontrolled lighting conditions is required by autonomous
robotis, in prosthetic devices for the blind, and autonomous motor-vehicle navigation. Today this task
is accomplished in two separate steps. First the light
intensity is recorded through a standard imager such
as a CCD camera. The intensity field is subsequently
processed outside the camera to discard any absoluteluminance information and form a representation where
only relative illumination, i.e. contrast, is retained. Additional processing such as edge extraction and or low
bit-rate encoding may follow. However, even though
the precision necessary for these tasks rarely exceeds
8 bits, the signal itself has a very large dynamic range,
many orders of magnitude, which makes the problem
difficult. This issue becomes acute when the illumination varies within a single frame something not uncommon in natural scenes (see Figure 19). The detrimental
effects of non-uniform illumination in the performance
of a face recognition system have been investigated experimentally by Buhman, Lades and Eeckman [38].
We will now present one solution to the problem
of robust image acquisition and preprocessing under
variable illumination conditions: a neurornorphic analog VLSI silicon retina. This is a contrast-sensitive
edge-enhancing imager that includes a rudimentary,
yet effective, local gain control mechanism at the pixel
level. The architecture is inspired by the processing
performed in the outer plexiform layer of the vertebrate retina [ 151, [171. The resulting image captured
with such a system is shown in Figure 20.
The biologically motivated solution is attractive
from a computational perspectiye because contrast, an
invariant representation of the visual world, has been
obtained with a front-end that is robust, small, and extremely low power (a few mW). There is also another
benefit; the output representation has limited range
and therefore subsequent processing/communication
stages are not burdened with handling and processing
signals of wide dynamic range. A performance comparison between the contrast sensitive silicon retina
front end [15] and a conventional camera, in a face
recognition experiment is reported in [38].
4.1. Biological Organization
The analog silicon system in the core of the array
is modeled after neurocircuitry in the distal part of
the vertebrate retina-called the outer-plexiform layer.
Figure 21 illustrates interactions between cells in this
layer [39]. The well-known center/surround receptive
field emerges from this simple structure, consisting of
just two types of neurons. Unlike the ganglion cells
Translinear Circuits in Subthreshold MOS
Fig. 19. (Bottom) “Mark” & captured by a conventional camera. (Top) Intensity profile at image line 110 (white line). The light souIce is
positioned to the right side of the image and it introduces a large gradient in illumination within a single frame. This is clearly shown in the
intensity histogram. The dynamic range of the scene exceeds the dynamic range of the camera. Aperture control on the camera provides a
rudimentary global gain control mechanism. Information in this imagé is lost at this very first step because there is no gain control (adaptation)
at the pixel level.
Fig. 20. (Bottom) “Mark” as captured by the translinear silicon retina. (Top) Histogram for the output of the system. The light source is again
positioned to the right side of the image and it introduces a large gradient in illumination within a single frame. The image captured by the
silicon retina discards absolute illumination and preserves only local contrast information through local gain control at the pixel level. Unlike
the image in Figure 19, the presence of a large illumination gradient does not degrade image acquisition here.
in the inner retina and the majority of neurons in the
nervous system, the neurons that we model here have
graded responses (they do not spike); thus this system
is well-suited to analog VLSI.
The photoreceptors are activated by light; they pro-
duce activity in the horizontal cells through excitatory
chemical synapses. The horizontal cells, in turn, suppress the activity of the receptors through inhibitory
chemical synapses.. The receptors and horizontal cells
are electrically coupled to their neighbors by electri-
A. G. Andreou and K. A. Boahen
Fig. 21. One-dimensional model of neurons and synapses in the
outer-plexiform layer. Based on the red-cone system in the turtle
cal synapses. These allow ionic currents to flow from
one cell to another, and are characterized by a certain
conductance per unit area.
In the biological system, contrast sensitivity -the
normalized output that is proportional to a local measure of contrast- is obtained by shunting inhibitiori.
The horizontal cells compute the local average intensity and modulate a conductancein the cone membrane
proportionately. Since the current supplied by the cone
outer-segment is divided by this conductance to produce the membrane voltage, the cone’s response, ,will
be proportional to the ratio between its photoinpùt and
the local average, i. e. to contrast. This is a very simplified abstraction of the complex ion-channel dynamics
involved. The advantage of performing this complex
operation at the focal plane is that the dynamic range
is extended (local automatic gain control).
The biological system, is mapped onto silicon using
circuits of minimal complexity that exploit native properties of subthreshold MOS transistors. High computational throughput at low levels of energy dissipation
is achieved by employing low precision analog processing in a massively parallel analog architecture that
exploits the translinearproperties of subthreshold MOS
in saturation and ohmic regime.
4.2. Silicon System Architecture
The core of the silicon retina is an array of pixels with a
six-neighbourconnectivity (see Figure 22). The wiring
is included in the layout of the cell (see Figure 24) so
that they may be tiled in a hexagonal tesselation to form
the focal plane processor. This is a mesh processor
architecture where two layers of processors, C and H ,
communicate both intra and inter layer through local
paths. This parallel processing scheme features locality
of reference and thus minimizes communication costs.
Fig. 22. Floorplan and system organization. It comprises of two
functional components, the core, and the support circuitry. Focal
plane processing is performed in the core area.
Fig. 23. One-dimensionalimplementationof outer-plexiformretinal
processing. There are two diffusive networks implemented by transistors M4 and M5, which model electrical synapses. These are coupled together by controlled current-sources(devices M1 and M2) that
model chemical synapses. Nodes H in the upper layer correspond
to horizontal cells while those in the lower layer ( C ) correspond to
cones. The bipolar phototransistor QI models the outer segment
of the cone and M3 models a leak in the horizontal cell membrane.
Note that the actual system has a six neighbor connectivity.
Support circuitry in the periphery extracts the data
from the core and interfaces with the display. The
chip incorporates a video pre-amplifier and some digital logic for scanning the processed images out of the
array. This circuitry is discussed in detail in the paper
by Mead and Delbrück [41]. Standard NTSC video
is produced off-chip using an FPGA controller and a
video amplifier.
The basic analog MOS circuitry for a one dimensional pixel with two neighbor connectivity is shown
in Figure 23. We begin with the non-linear aspects
of system operation, its contrast sensitivity. The nonlinear operation that leads to a local gain-control mechanism in the silicon system is acheived through a mechanism that is qualitatively similar to the biological
counterpart, but quantitatively different (see discus-
Translinear Circuits in Subthreshold MOS
Fig. 24. (Left) Photomicrograph of the chip. The surface is covered by second metal except where there are openings for the phototransistors
(the dark square areas). Note the hexagonal connectivity of the pixels. (Right) Layout df the basic cell.
sion in [ 151). Refering to Figure 23, the output current
I c ( x m , y,) at each pixel, can be given (approximately)
in terms of the input photocurrent I (x, y,) and a local average of this photocurrent in a pixel neighborhood
( M , N ) . This region may extend beyond the nearest
neighbor. The fixed current I , supplied by transistor
M3 normalizes the result and KP is a parameter.
At any particular intensity level, the outer-plexiform
behaves like a linear system that realizes a powerful
second-order regularization algorithm [40] for edge detection. This can be seen by performing an analysis of
the circuit about a fixed operating point. To simplify
the equations we first assume that g = ( I h ) g , where
( I h ) is the local average. Now we treat the diffusors
(devices M4) between nodes C and C’ as if they had a
fixed diffusitivity jj. The diffusitivity of the devices M5
between nodes H and H’ in the horizontal network is
denoted by h. Then the simplified equations describing
the full two-dimensional circuit on a square grid are:
Y,> =
+ h x { I h ( X m , y,>
- ~ h ( x i y, j ) }
Using the second-difference approximation for the
laplacian, we obtain the continuous versions of these
M x , y) = I(& y)
+ jjV21,(x,Y )
with the internode distance normalized to unity. Solving for Zh (x, y), we find
This is the biharmonic equation used in computer vision to find an optimally smooth interpolating function
I h (x, y) for the noisy, spatially sampled data I (xi, y j ) ;
it yields the function with minimum energy in its second derivative [40]. The coefficient h = jjh is called
the regularizing parameter; it determines the trade-off
between smoothing and fitting the data.
4.3. Layout Considerations
The two-layer architecture for the silicon retina can
be accomodated in a cell area of 80h x 94h using a
single poly two metal technology. In the implementation reported in [15] and here, a double poly, double
metal technology is used and the cell area is 66h x 73h.
First metal and polysilicon wires are used for interconnects; second metal is used to cover the entire array,
shielding the substrate fi-om undesirable photogenerated carriers. Transistors are implemented using both
polysilicon layers.
The system has been fabricated with 230 x 210 pixels on a 9.5 x 9.3 mm die in a 1.2pm n-well double
metal, double poly, digital oriented CMOS technology. The chip incorporates 590,000 transistors in the
48,000 pixels and support circuitry, with the core operating in subthreshold/transition region consuming less
than 1OOmW.
A. G. Andreou and K. A. Boahen
Fig. 25. Large signal model for an NMOS transistor (left). Adopted conventions for current decomposition in PMOS and NMOS devices (right).
4.4. Discussion
The silicon retina, presented in this paper is essentially
an analog floating-point processor. As a first step, the
system computes the range (the voltages on the H nodes
correspond to the value of the exponent in floatingpoint data representation). This is the operating point
of the system and is a function of the spatial coordinates
and this is how local automatic gain control is achieved.
At an operating point, sophisticated spatial filtering is
performed to smooth the sampled data and enhance the
edges. Having separated the problem of precision and
dynamic range, the signal processing within the range
can be done with low precision analog hardware.
The benefit of a robust architecture on the ultimate
system performance is evident in the design of the silicon retina. The regularization properties inherent in
the architecturemitigate the intrinsic random variations
in the device characteristics, leading to robust performance. This methodology allows analog VLSI implementations using the poorly matched, small geometry,
nano-power devices available in garden-variety digital
VLSI CMOS technology. Thus we see how a neuromorphic architecture can account for the properties of
the computational substrate, and yield robust operation in the presence of noise (structural variability) in
the individual devices, The translinear propeties of the
MOS transistor in subthreshold ohmic and saturation
are key to an area efficient implementation which is
commensurate with integration at the focal plane.
13 +................i...................1...................t...................I................... ................L1
5 '1
( V-"2 )
Fig. 26. Experimentally determined variation of parameter
function of the substrate voltage V S B .
as a
tr bslinear circuit design. This was also shown to be
to MOS transistors in subthreshold as well.
The absence of a base current makes it in a sense an
"ideal" element.
However, one must be aware of certain characteristics of the MOS transistor that have a detrimental
effect on circuit behaviour. Some of them are already
depicted in the large signal model equations. Device
characteristics that are not modeled in the simplified
transistor model are discussed with the help of experimental data from measurements on transistors operating in subthreshold.
5. Subthreshold MOS Device Limitations
5.1. Transconductance Limitations
As we have seen earlier, linear relationships between
conductance/transconductances and current, and the
equivalence of controlling the current from the collector and emitter terminals, are the basis of bipolar
Unlike bipolar transistors, the gate and the source of
an MOS transistor are not equally effective in controlling the current. Increasing their voltages by the
same amount, does not change the current by the same
Translinear Circuits in Subthreshold MOS
amount because the transconductances are different
(compare Eqs. (6) and (8)). Whereas changing the
source voltage changes the the barrier by an equal
amount, only a certain fraction ( K ) of changes in the
gate voltage affects the surface potential and hence the
barrier height. This behaviour is depicted in Figure 1
and manifests itself as a slope for the current characteristics that has lower value compared to that of a bipolar
transistor. An even lower value (1 - K)for the slope is
seen from the back gate terminal.
The experimental data from Figure 1 suggest that K
can be pushed closer to unity by biasing the device with
a large surface potential (i.e. large VSBand VGB).The
dependence of K on the substrate voltage is measured
and plotted in Figure 26. As the substrate is reversed
bias, the depletion capacitance C&,p, in Equation 58
becomes smaller and hence the gate has larger influence
on the channel conductance. Under these conditions,
the subthreshold MOS device becomes closer to an
ideal translinear element whose transfer characteristics
come closer to those of bipolars.
Another way to circumvent the potential divider
problem is to move the local substrate voltage together
with the source voltage-which is exactly equivalent
to increasing VGB by the same amount and does not
change VSB.In practice, this may be achieved simply
by shorting the source to the local substrate. Clearly,
this is only possible with devices in separate wells. For
this reason, it is preferable to bias the device at large
values of surface potential as described in the previous
paragraph, since that works for both types of transistors
and is more area efficient (there is no need for separate
wells and a triple well process).
Another departure from the simplified model equations presented earlier is evident when we consider the
dependence of the drain,current on the voltage at the
source of the transistors. According to the model (equation 55) this dependence should follow the exponential
law and the parameter in the exponent is the inverse
of the thermal voltage. An experiment to verify this
was conducted and the results are shown in Figure 27
where the slope of the curve has the value of only 35.4
V-’ . This suggests that the source conductance cannot
be adequately described though Equation 8 and that
parameter r] m Os9 must be introduced in
of the source and drain voltages in the original device
Equation 55. Both NMOS and PMOS transistors in
different fabrication processes show similar behaviour
that is correlated to the zero bias leakage current of the
10 1 O
VSB (Volts)
Fig. 27. Measured drain current IDS as a function of the source
voltage VSBfor a (16 x 16pm)NMOS transistor fabricated in a 2pm
n-well CMOS process. The solid line is an exponential function fit to
the data. The experimentswere performed at a temperature such that
the thermal voltage V, = 0.0259 Volts and hence 1/ V, = 38.5V-’.
1o -0
10 -I0
10 -”
Fig. 28. Drain saturation characteristics of an NMOS FGMOS transistor. The dimensions of the device are W = 8pm and L = 4pm
and the first-poly second-poly coupling area is 2pm x 2pm. Note
that the current is plotted in a logarithmic scale.
A. G. Andreou and K. A. Boahen
1.1 ou
Above Threshold
-1 n
--1 oop
Fig. 29. (Left) Density plots of currents in a 32 x 32 array of 4 p m x 4 p m NMOS transistors. Each transistor is represented by a square pixel.
Current level is coded by the shade of gray, where the minimum and maximum values are represented by black and white, respectively. The
current at a nominal current level of lOOnA is obtained by setting VGSto be the same for all transistors in the array. The devices are biased
in saturation. (Right) Measured drain current ZD versus gate-source voltage VGSfor 32 small geometry transistors (4 x 4p.m) fabricated in a
2 p m n-well CMOS process; drain-source voltage of V~s=1.5Volts. The fuzziness in the current, (mismatch between ¿levices), is constant in
subthreshold (on a log(Z) scale) and decreases as the device ent!qs the transition and above threshold regime.
\ ,
5.2. Output Conductance Limitations
The non-zero output conductance, or what is often
called the Early effect also degrades the performance
of the circuits. All well known circuit techniques that
reduce the output conductance, such as cascoding and
regulated cascoding are beneficial to translinear circuits. A second effect contributes to the drain conductance of FGMOS transistors. As pointed out in [19],
the parasitic coupling of the drain and source to the
floating gate through capacitances Cfgd and Cfgs(see
Figure 5) yields an exponential dependance of the output current on the drain voltage. This is evident in
the experimental data shown in Figure 28 for NMOS
devices; PMOS transistors exhibit similar behaviour.
The experimental data in Figure 28 are fit to an output
conductance model and the overlap capacitances Cfgd
and Cfgsestimated as 0.15fFlpm [19]; this number
could be as large as O.SfF/pm.
5.3. Device Matching Limitations
Another important area of concern in designing
translinear computational circuits, is the poor matching characteristics of MOS transistors in subthreshold
(much worse than bipolars). This is more acute in
analog VLSI systems applications, where small geometry transistors must be used, typically 4pm x 4pm or
6pm x 6pm, (in a one micron process) to achieve high
densities. Low power design condiserations suggest
that it is preferable to operate the devices in the region
where the transconductance per unit current is maxi-
mum [16], i.e., in the subthreshold and transition regions. Small device geometries and high transconductance per unit current makes the drain current strongly
dependent on spatial variations of process-dependent
parameters, particularly l o . Characterization of the
fabrication process and the qhtching properties of the
basic devices is thus of paramount importance because
it provides the necessary information for designing low
power systems. The experimental data in Figure 29
show that there are three different effects that can be
responsible for the poor matching characteristics of
MOS transistors in subthreshold; these were discussed
in [42]. After discounting the two deterministic effects,
we are left with the random variations.
Random mismatch in the subthreshold region can
be characterized in terms of the simple model parameters Io and K . The parameter K is very stable, with
a normalized standard deviation of o ( K ) / ( K ) M 0.3%,
where (.) denotes mean value. The small variations
in K suggest that doping and gate-oxide thickness are
extremely uniform. The fuzziness in the data points
in Figure 29(Right) is therefore not due to changes in
slope. This implies that characteristics are displaced
from one transistor to the next, implicating the flatband voltage which depends on the contact potential,
and this fixed interface charge density Qg (implanted
ions and trapped electrons). The variations in the current can of course be related to this fixed-charge distribution through the transconductance and the gate-oxide
Translinear Circuits in Subthreshold MOS
Fig. 30. Dependence of normalized standard deviation of IDS on
transistor size; the lines are best fits to the data. All devices have
square geometries with area A = L2. Notice that the normalized
standard deviation of IDS saturates at large transistor geometries.
Figure 30 shows the dependence of the normalized
standard deviation of the drain current, O ( I ) / (I ) , on
transistor size. Each data point represents measurements from approximately 1000 transistors. The normalized standard deviation of the current is inversely
proportional to the square root of the device area, A,
and is given by:
where 00 is the mismatch per unit length for a given
device type and process.
5.4. Noise Limitations
Shot noise in the MOS transistor operating in subthreshold has a one-sided power spectrum given in [49]p
Si,shot (LC)) = 4qIDS
For a device in saturation the noise is exactly half.
For sub-threshold currents between 1 nA and
100 nA, flicker noise for mid-to-low frequencies must
be included. A model for flicker noise is given by [49].
si,f l i c k ( w ) = C&WL
M a process-dependent constant with a typical value
of 4.0 x 1026C2/rn2.
Fig. 31. Power spectral density for a PMOS transistor in saturation,
where W = 1148 pm and L = 4 pum. The model is given by solid
lines, the data are marked by x's. The three curves correspond to
nominal current values of (a) 1 nA, (b) 10 nA, and (C)100 nA for an
equivalent square device. Some amount of excess noise is evident at
low current levels.
Assuming shot and flicker noise are independent, a
complete noise model for a transistor operating in the
subthreshold region is
Figure 31 shows the noise power spectral density
for a PMOS transistor. One free process-dependent
paramciter M is used to model the flicker noise. Note
that, at low enough current levels, flicker noise cannot
be detected within the audio frequency range. This
property is seen for curve (a) of Figure 31 in which
there is little evidence of flicker noise for frequencies
above 50 Hz.
5.5. Bandwidth Limitations
The maximum useful frequency of operation possible
with an MOS transistor, is determined by its transition
frequency f~ defined [49] as (gm/ 2 n C ) where C is
the total input capacitance i.e. the capacitance per unit
area times the area of the device:
so that for subthreshold operation:
A. G. Andreou and K. A. Boahen
The maximum value of drain current IDSmax with
the MOS transistor still in subthreshold region is given
by 151:
a sabbatical leave at Caltech. We thank Carver Mead
for his continuing support and encouragement. Chip
fabrication was provided by MOSIS.
Appendix A: Device Models
From the above equations a maximum transition frequency in subthreshold f T m a x can be approximated to:
where is the effective carrier mobility and L iy$he
device channel length. The transition frequency ‘of a
device is essentially the bandwidth (as determined by
the internal gain and parasitic capacitances of the transistor). For six to ten micron length devices (typical
in analog VLSI today), functional systems in the hundreds of kHz range are possible while for submicron
devices, the limit extends to the MHz range.
6. Conclusions
In this paper we have provided a comprehensive
overview of the application of the translinear principle to MOS circuits operating in subthreshold. Our research was aimed at exploring different ideas on neuromorphic analog network computations and their VLSI
implementations. The results of our investigation are
encouraging; analog circuits designed with components of limited precision, when assembled in large
networks following a design methodology, based on
translinear circuit techniques, can successfully perform
linear and non-linear computation with energetic eficiency unmatched by any other digital counterparts.
Our 590,000 transistor analog VLSI, contrast sensitive, silicon retina is another step towards the direction
envisioned by Barry Gilbert: [44] “. . .convergence of
IC technology capabilities and neural network requirements makes wafer-scale integration of meganetworks
a very real possibility . . .”.
The research was partially supported by NSF grant
ECS-9313934; Paul Werbos is the program monitor,
and by a contract from the Army Night Vision Laboratory at Fort Belvoir. The final version of this document
was prepared while one of the authors (AGA) was on
A*[email protected]
The Ebers-Moll model [46], [47] for an npn bipolar
transistor is:
IC = ~ F I F IR
I F = I E s (q eV B7E - 1)
IR = I c s ( e 7
FIES = ~ R I C S
where IC and I E are the collector and emitter currents
respectively and
V B E is the base to emitter voltage,
VBCis the base to collector voltage,
IES is the saturation current of emitter junction
with zero collector current,
ICS is the saturation current of collector junction
with zero emitter current,
a~ common-base current gain.
aR common-base current gain in inverted mode,
i.e. with the collector functioning as an emitter
and the emitter functioning as a collector.
By convention, the currents for bipolars are positive
when flowing into its terminals.
Combining Eqs. (49, (46), and (47), the collector
current can be expressed as:
- 1) - -
( e qvBc
y - 1))
For an ideal device with common-base current gain,
C Y F , and common-base current gain in inverted mode,
CXR,very close to unity, the above equation becomes:
IC = I E s (qvBE
e 7 - e 4vBC
r )
However, regular bipolar transistors do not have both
near unity.
CXF and U R
Translinear Circuits in Subthreshold MOS
When the collector to base voltage equals zero or the
collector is reverse biased with respect to the base, the
above equation simplifies to the familiar:
where A E is a design parameter, the area of the emitter
junction. JES and Is are the saturation current density
and current for the emitter respectively. In this case,
IR << I F and the equations above give IC = - a I~E .
Using the relation I E IC IB = O (KCL) we get the
familiar result
+ +
where ß F is the common-emitter current gain.
A.2 MOS Transistor Model
A charge-based formulation [7], [ 161 that preserves
the symmetry between the source/drain terminals of
an MOS transistor is presented. Which terminal of the
device actually serves as the source or the drain is determined by the circuit, the bias conditions-and even
the input signals. This symmetric view of an MOS
transistor enabled us to extend the translinear principle
to operation in the subthreshold ohmic regime [ 161.
The MOS device has a very simple current-charge
relationship because diffusion and drift are both proportional to the concentration gradient. As shown in
Appendix A of [7] and in [ 161, this yields a quadratic
expression for the current that consists of two independent opposing components I Q and
~ Zed-in the absence
of velocity saturation and channel-length modulation
effects. These components are related to charge densities at the source Q: and QZ at the drain of the device.
The device drain-source current can thus be written
les - I
tances Cix and C:ep are the gate oxide and depletion area capacitances of the channel. A key property
of the MOS device that makes this possible is lossless channel conduction. Unlike a bipolar transistor,
the controlling charge on the gate is isolated from the
charge in transport by the almost infinite gate-oxide
resistance. Therefore, there is no recombination between the current-carrying charge in the channel and
the current-modulating charge on the gate.
The familiar ohmic/saturation dichotomy introduced
in voltage-mode design can be reformulated in terms
of the opposing drain and source driven current components. In saturation, [ZedI << II Q I~and I M I Q ~
and therefore the current is independent of the drain
~ Zed and
voltage. In ohmic, Zed N IQs and I = I Q therefore the current depends on the drain voltage as
well as the source and gate voltages. The functional
dependence of the current components on the terminal
voltage is fixed and remains the same throughout the
ohmic and saturation regions.
The charge densities at the source and drain terminals
can be related to the terminal voltages. In general the
charge-voltage relationship is much more complicated
than the current-charge one because both the mobile
charge and the depletion charge are involved in the
electrostatics. The device current in Equation 52 can
thus be written as a function F,of the terminal voltages
with a general functional form for the current-voltage
relationship valid for all the regions of operation given
This functional form was first introduced by [48] for
above threshold operation and is also discussed in [49].
q$r an n-type device, F is a nonpositive, monotonically decreasing function of VGBand a monotonically
increasing function of VSB.
In subthreshold region, the following factorization
of F : is also possible [6], [7].
Q ~
W is the width, L is the length of the channel and
(P) is the effective channd mobility. The capaci-
where Ç and 3-1 are exponential functions. This shows
that the source-driven and drain-driven componentsare
controlled independently by VSB and VDB.However,
- i / c B , acting through the surface potential, also controls
both components in a symmetric and multiplicative
fashion. In this mode of operation the MOS transis-
A. G. Andreou and K. A. Boahen
tor has been called a difisor [ 151 in analogy with the
variable conductance electrical junctions in biological
An expression for the current in an NMOS transistor
operating in subthreshold can thus be written [6], [7]
divider ratio closer to unity. A larger surface potential also reduces C&p. The parameter K takes values
between 0.6 and 0.9.
l. The diffusor is a term adopted in [151to describe the exploitation
of diffusion transport in MOS transistors to spread signals in a
manner analogous to gap junctions between neural cells.
I n 0 Sexp(~nVGB / Vt
x [exp(-VSB/h) - exp(-VDB/Vt)l
and for a PMOS
Sexp(-Kp VGB/Vt)
x [exp(VSB/Vt) - exp(VDB/Vt)]
The terminal voltages VGB, VSB, VDB are referenced
to the substrate. The constant IOdepends on mobility
(P) and other silicon physical properties. S is a geometry factor, the width W to length L ratio the device.
For devices that are biased with‘ VDS 2 4 V,, (saturation) the drain current is reduced to:
= S I n o exp(1 - Kn)vBS/vt) exp(KnVGS/Vt)
This shows explicitly the dependence on VBS and the
role of the bulk as a back-gate that underlies this. This
equation, having only the dependence on VGS, and
VBs, is used for circuit designs where devices operate
in saturation as transconductance amplifier. However,
channel-length modulation (Early effect) -which we
have ignored completely-becomes significant in saturation. So the device equations must be augmented
with terms that model this effect to accurately predict
the output conductance.
The parameter K is defined as
K =
+G e p
The physical significance of K is apparent if the observation is made that that the oxide and depletion capacitances form a capacitive divider between the gate
and bulk terminals that determines the surface potential [7]. Lighter doping reduces C&,p, and pushes the
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A. G.Andreou and K. A. Boahen
Andreas G. Andreou received the M.S.E. and Ph.D.
in electrical engineering and computer science from
Johns Hopkins University, Baltimore, in 1983 and
1986, respectively. From 1987 to 1989 he was a postdoctoral fellow and associateresearch scientist at Johns
Hopkins where he became Assistant Professor in 1989
and Associate Professor in 1993. His research interests
are in the areas of device physics, integrated circuits
and neural computation.
Dr. Andreou is a member of Tau Beta Pi and a member of I.E.E.E.
Kwabena A. Boahen is a Ph.D. student at Caltech
in the Computation and Neural Systems program after
completing a B.S.N.S.E. degreein electrical and computer engineering at Johns Hopkins University, Baltimore, MD. His research at Caltech involves analog
VLSI models of biological computation, with an emphasis on retinal computation and chip-to-chip communication.
Mr. Boahen is a member of Tau Beta Pi and a student
member of the I.E.E.E.
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