This paper analyses a filter known as the Tau Cell using symbolic methods and shows that the operation of this filter is independent of the magnitude of the input DC offset. This means that the circuit places no restrictions on whether the input DC offset is a sub-threshold current or not. The circuit behaviour predicted from symbolic analysis was observed in similar circuits on a chip fabricated using MOSIS AMI 1.6μm technology. This paper highlights the utility of symbolic analysis and shows that it is a powerful tool for circuit analysis and design

Hamilton, Tara Julia

Jin, Craig

Postula, Adam

Schaik, Andre Van

Thanigaivelan, Balavelan

University of Queensland eSpace

 Symbolic Analysis of the Tau Cell Log-Domain 
Filter using Affine MOSFET Models 
Balavelan Thanigaivelan#1, Adam Postula#, Craig Jin*, André van Schaik*, Tara Julia Hamilton^ 
##School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane Australia 
1
velan@itee.uq.edu.au 
*School of Electrical and Information Engineering, The University of Sydney, Sydney Australia 
^School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney Australia 
 
Abstract— This paper analyses a filter known as the Tau Cell 
using symbolic methods and shows that the operation of this 
filter is independent of the magnitude of the input DC offset. This 
means that the circuit places no restrictions on whether the input 
DC offset is a sub-threshold current or not. The circuit 
behaviour predicted from symbolic analysis was observed in 
similar circuits on a chip fabricated using MOSIS AMI 1.6µm 
technology. This paper highlights the utility of symbolic analysis 
and shows that it is a powerful tool for circuit analysis and 
design. 
 
Keywords—Symbolic Circuit Analysis, Log Domain Filters, 
Small-Signal Modelling 
I. INTRODUCTION 
Symbolic analysis is used to aid the design and analysis of 
analog integrated circuits because it provides a more intuitive 
understanding of circuit behaviour than numerical analysis. 
The symbolic simulator plays an important role, especially in 
the design of novel circuits, by providing direct insight into 
the qualitative influence of each circuit parameter on the 
desired circuit behaviour. In this paper we use a symbolic 
circuit analysis toolbox, developed in Matlab to analyse a log-
domain filter known as the Tau Cell [1], shown in Fig. 1. 
Log-domain filters take advantage of the exponential 
relationship between current and voltage when the transistors 
that make up the circuit are in the weak inversion (sub-
threshold) region. The requirement that the transistors in a 
log-domain filter are in weak inversion places limitations on 
these circuits. Circuits in weak inversion can suffer from 
difficulties with matching and hence, the precision of the 
circuit can be compromised. 
Restriction of the DC offset to a low current level can also 
restrict the allowable signal swing in the circuit and hence the 
gain of the circuit. This is of particular concern when dealing 
with second order filters with high Q values. 
The Tau Cell [1] is a log-domain building block for 
creating a number of different filters. This paper shows, using 
symbolic circuit analysis, that this building block is virtually 
unaffected by the magnitude of the DC offset and hence, that 
it can be used to create filters which enjoy many of the 
benefits of log-domain devices without some of the drawbacks. 
The main advantage will be that the DC offset can be adapted 
to accommodate the amplitude of the input signal over several 
orders of magnitude without degrading the performance of the 
filter. 
The circuit behaviour predicted from our symbolic analysis 
of the Tau Cell is verified in numerical simulation and 
observed in the behaviour of similar circuits on a fabricated 
integrated circuit. 
II. BACKGROUND 
A. Symbolic Circuit Analysis 
Symbolic analysis is a method of analysis in which some or 
all of the circuit parameters are retained as symbols. Therefore 
it provides an opportunity for interactive design of novel 
circuits and allows further exploration into the behaviour of 
existing circuits. The symbolic circuit analysis tool 
complements numerical analysis tools by providing additional 
intuitive understanding of the circuit behaviour. The 
usefulness of symbolic analysis in varied fields of circuit 
analysis, design and testing is widely demonstrated [2]. The 
main problem with the existing symbolic analysis tools is that 
the size of the symbolic expressions is large, even for small 
circuits, making them difficult to read and hard to interpret. 
Moreover, the expressions obtained from a low accuracy 
analysis do not correlate well with the expressions obtained 
from high accuracy analysis. In other words, it is hard to keep 
track of the errors between a low accuracy and high accuracy 
analysis. 
 
Fig. 1. The Tau Cell 
978-1-4244-7456-4/10/$26.00 © 2010 IEEE 1095
 
 
The symbolic analysis toolbox we developed and used to 
analyse the Tau Cell is based on affine arithmetic concepts 
and overcomes the difficulties mentioned above. An affine 
arithmetic number can be defined as, 
 0 1 1 2 2 m mx x x x xε ε ε= + ⋅ + ⋅ + + …  (1) 
where, x0 is defined as the central value, {x1, x2, .. xm} are 
the partial deviations or errors which, when added to central 
value, result in increased accuracy, and εi is the noise symbol 
whose value is assumed to be in the range {-1,+1}. 
The symbolic analysis toolbox used in the analysis 
presented in this paper contains MOSFET small signal affine 
models of different levels of complexity, shown in Fig. 2. 
These models were built from conventional device models by 
redistributing and regrouping the circuit elements using the 
concept of affine arithmetic numbers as illustrated in equation 
(1). The details of the transformation and equations governing 
the elements in the affine MOSFET models can be found in 
[3]. 
B. Log Domain Circuits 
The exponential relationship between voltage and current, 
when a transistor is in weak inversion, has led to the 
foundation of a group of analog circuits classified as log-
domain or current-mode circuits. The relationship between 
terminal voltages and drain current in weak inversion [4] is 
given in by, 
 ( )0V V n U V U V UG T T S T D TD SI I e e e
− ⋅
− −
= ⋅ ⋅ ⎡ − ⎤⎣ ⎦  (2) 
where, VG is the gate voltage, VD is the drain voltage, VS is 
the source voltage (all referred to the bulk voltage), VT0 is the 
zero source threshold voltage, UT is the thermal voltage, and n 
is the gate reduction factor. Equation (2) gives the weak 
inversion drain current (ID) when it is small, i.e., much less 
than the specific current (IS). The specific current is dependent 
on the physical characteristics of the transistor such as its 
width and length, carrier mobility, gate oxide capacitance and 
thermal voltage. The drain current reaches saturation when the 
drain voltage becomes larger than 4UT. In log-domain circuits 
current, rather than voltage, is the main signal carrier. As a 
result, they have a wider dynamic range to exploit than 
voltage domain circuits [5]. Research into log-domain circuits 
has revealed that issues with noise and matching make it very 
difficult for these circuits to fulfil their potential [6]. Some of 
these issues are made worse by the need for sub-threshold bias 
currents in these circuits. 
C. Translinear Loops 
Many current-mode circuits use translinear loops. 
Translinear loops exist where transistors create closed loops of 
gate-source junctions [7]. The translinear principle can be 
summarized as follows for MOS transistors operating in weak 
inversion: in a closed loop containing an equal number of 
gate-source junctions of one polarity and an equal number of 
gate-source junctions of the opposite polarity, the product of 
the currents of one polarity is equal to the product of currents 
of the opposite polarity [7]. 
D. The Tau Cell 
The Tau Cell [1] is a type of log-domain filter designed for 
complete programmability via two parameters: the time 
constant, τ, and the current feedback gain, Ai. The Tau Cell is 
designed to be a building block to create much more 
complicated filters. The Tau Cell is shown in Fig. 1. Its 
transfer function is given by, 
 [ ] 1
0
1( )
1
where, 
i
i i i i
i T
T s
s A A T
C U I
τ
τ
+
=
⋅ + − + ⋅
= ⋅
 (3) 
The Tau Cell utilizes the principle of translinear loops. In 
Fig. 1 the closed loop of gate-source junctions is created by 
transistors M1 to M4. Hence, by the translinear principle, 
 0 2IN OUTI I I I⋅ = ⋅  (4) 
This relationship has become a convenient way of 
analysing and representing current-mode circuits. The 
principle of translinear loops is also used to create the current 
feedback gain, Ai, however, for a first order log-domain filter, 
Ai is equal to 0, so no multiplier is needed for these. 
In this paper we analyse the first order Tau Cell using 
symbolic small signal analysis and show that it allows us to 
deduce more about the circuit than standard analysis with 
translinear loops. 
III. SYMBOLIC SMALL SIGNAL ANALYSIS USING AFFINE 
MOSFET MODELS 
Small signal analysis is a valid method of analysing log-
domain filters since the voltage signal swing is normally small 
in these circuits, independent of the region of operation 
(strong or weak inversion) and the transistors can be 
approximated with a linear transconductance. When the small 
signal condition is not met the analysis presented here is not 
valid. 
 
Fig 2. Affine MOSFET small-signal model at different accuracy levels 
1096
 
The Matlab toolbox used for this analysis directly reads the 
circuit netlist and DC operating point parameters from SPICE 
simulation results. Using this toolbox, the transfer function for 
small and medium size circuits can be derived using the 
Determinant technique or the Sequence-of-Expressions 
technique as shown in [2]. The Tau Cell was designed using 
1.6µm technology and simulated using ELDO SPICE to 
obtain the circuit parameter values from the DC operating 
point results. The transfer function, IOUT/IIN, for the Tau Cell 
was derived using Determinant technique. 
The symbolic transfer functions were computed after 
substituting the MOSFETs with affine level-2 and affine 
level-4 models respectively, as shown in Fig. 2. The 
numerator and denominator of the transfer function obtained 
from our Matlab toolbox in the case of level-2 model is shown 
in equations (5) and (6). 
 4 2 3mN mN dsNnum g g G= × ×  (5) 
 3 1 2 2( )dsN dsN dsN mNden G G s C G g= × × ⋅ + +  (6) 
As can be seen from the equations (5) and (6), the drain 
conductance of M3 cancels out and the (trans) conductance of 
M2 influences the time constant in transfer function. The 
transfer function can be represented in the simplified form as, 
 ( )
4 2
1 2 2
( ) mN mNi
dsN dsN mN
g gT s
G s C G g
= ×
⋅ + +  (7) 
As can be seen from equation (7) the parameters of 
transistors M4 and M1 contribute only to the gain in the 
expression and do not affect the time constant in the transfer 
function. This simplified analysis provides a quick insight into 
the circuit operation of the Tau Cell with transistors M1 and 
M4 determining the current gain and transistor M2 dominating 
the time constant. 
A thorough analysis including the parasitic capacitances 
can be performed using the affine level-4 model for all the 
transistors, shown in Fig. 2 (d). It is expected that the 
parameters of the transistors M1 and M4 will dominate the 
gain while M2 will dominate the time constant. Transistor M3 
may not cancel out, but remain equally present in both 
numerator and denominator. The numerator and denominator 
of the transfer function obtained from the Matlab toolbox 
using the level-4 model is shown in equations (8) and (9). 
 
( ) ( )
( )
4 4 3 3
2 2 2       
new
mN gdN dsN dsN
new
mN gdN gsN
num g s C G s C
g s C s C
= − ⋅ × + ⋅
× − ⋅ + ⋅
 (8) 
 
( )
( )
1 3 3 4
2 3 3      
new new
dsN dsN gsN
new
dsN dsN
den sub G s C C
sub G s C
= × + ⋅ +
+ × + ⋅
 (9) 
where, 
( ) ( )
( ) ( )
( )
2 2 1 1 2
1 2 2 2 1 1
2 2 2
2
new new new
dsN dsN dsN dsN gsN
new new
mN gsN gdN dsN dsN
gdN mN gdN
G s C s C G s C s C
sub g s C s C G s C
s C g s C
⎧ ⎫+ ⋅ + ⋅ × + ⋅ + ⋅⎪ ⎪⎪ ⎪
= + + ⋅ − × ⋅ × + ⋅⎨ ⎬⎪ ⎪
− ⋅ × − ⋅⎪ ⎪⎩ ⎭
(10) 
 ( )2 4 1 1 4new new newgsN dsN dsN gsNsub s C G s C s C= ⋅ × + ⋅ + ⋅  (11) 
Equations (8) and (9) show that, the parameters of M3 are 
equally present in both the numerator and the denominator. 
Therefore changing the parameters of M3 will not 
significantly influence the frequency response characteristics 
of the filter. The “sub”-expressions, sub1 and sub2, are 
dominated by (GdsN1+s⋅CdsN1), in other words by transistor M1, 
while (gmN4-s⋅CgdN4) is a factor in the numerator. Therefore the 
gain is still strongly dependent on M1 and M4, despite the fact 
that their parameters cannot be completely extracted out as 
common expressions. The denominator expression, sub1, 
shows that the parameters of transistor M2 continue to 
dominate the time constant. 
The expectation about the circuit behaviour from the low 
accuracy symbolic analysis (equation (7)) correlates well with 
the results obtained from the high accuracy symbolic analysis 
(equations (8) – (11)). This is because the use of affine 
arithmetic models for symbolic simulation helps keep track of 
errors as the level of accuracy improves. Therefore, our 
analysis provides a more intuitive understanding of circuit 
behaviour when compared with analysis using the 
conventional MOSFET small signal models. 
The results from the symbolic analysis of the Tau Cell 
using the level-4 affine MOSFET model were validated with 
numerical computation using 1.6µm transistor BSIM3v3 
parameters. Fig. 3 shows the frequency response of the first 
order Tau Cell shown in Fig. 1. The input DC current was 
varied over two orders of magnitude from 47nA to 6.42uA. 
101 102 103 104 105
-35
-30
-25
-20
-15
-10
-5
0
5
M
ag
ni
tu
de
 (d
B
)
Frequency  (Hz)
Idc_47nA
Idc_79nA
Idc_640nA
Idc_896nA
Idc_6420nA
 
Fig 3. First Order Tau Cell Bode Plot from Symbolic Analysis Toolbox 
101 102 103 104
-35
-30
-25
-20
-15
-10
-5
0
5
M
ag
ni
tu
de
 (d
B
)
Frequency  (Hz)
Idc_47nA
Idc_79nA
Idc_640nA
Idc_896nA
Idc_6420nA
 
Fig 4. Second Order Tau Cell Bode Plot from Symbolic Analysis 
Toolbox
1097
 
 
The Bode plot obtained from Matlab shows that the gain 
changes by 2-3dB between the lowest and highest DC currents. 
This is expected from the symbolic expressions (8) – (11) 
since transistors M1 and M4 were shown to dominate the gain. 
Neglecting the gain, the shape of the frequency response curve 
remains constant across the 4 decade input current range. This 
is also expected from our symbolic analysis as only transistor 
M2 was shown to dominate the time constant of the circuit. 
IV. EXPERIMENTAL RESULTS 
To confirm our theoretical analysis of the Tau Cell a test 
chip was used. The chip was fabricated using MOSIS AMI 
1.6µm technology. The chip was originally designed to test 
matching in second order Tau Cell filters in the audio 
frequency range [8], however, it did allow us to vary the input 
DC current over a limited range. Each chip contained six 
filters that were configured as second order low pass filters. 
Given the structure of the second order filter, however, which 
consists of a cascade of first order filters [1], the DC 
independence hypothesized in section III still holds. This can 
be seen in simulation in Fig. 4. 
Our experimental set up allowed us to vary the input DC 
current reliably from 47 nA to 6.4 µA. The frequency 
response of the filter was measured by injecting a current with 
a constant DC-to-signal amplitude ratio of 11.5% into the 
filter and measuring the output amplitude as the frequency 
was swept from 10 Hz to 2 kHz. Over the two orders of 
magnitude of DC currents that we measured only slight 
differences in the frequency response of the filter were 
observed, as shown in Fig. 5. There was a change of about 2 
dB in gain between the highest DC current, 6.4 µA, and the 
lowest current, 47 nA. 
V. CONCLUSIONS 
The behaviour of the Tau Cell log-domain filter has been 
analysed in detail in this paper. The intuitive understanding of 
the filter’s behaviour developed using symbolic circuit 
expressions has been corroborated with measurements from a 
fabricated chip. This, to the best of our knowledge, is the first 
time that the conclusions and predictions from the symbolic 
analysis of a novel circuit have been verified with fabricated 
chip results. 
According to symbolic analysis, the frequency response of 
the Tau Cell is independent of the DC offset. This was 
demonstrated over a two decade range on a fabricated chip. 
The symbolic analysis also predicted a gain dependant on the 
transconductance of input and output transistors. This was 
measured to be 2dB over the two decade DC offset range and 
matched well with the numerical simulation result. 
We can therefore conclude that the Tau Cell is not 
restricted to sub-threshold input DC offset currents. This 
means that matching may improve and the signal swing can be 
large. These are very important features for filters used in 
audio applications with potentially high Q values, illustrating 
the utility of analysis using symbolic methods even for the 
most simple of circuits. 
REFERENCES 
[1] A. van Schaik and C. Jin, "The tau-cell: a new method for the 
implementation of arbitrary differential equations," in Circuits and 
Systems, 2003. ISCAS '03. Proceedings of the 2003 International 
Symposium on, 2003, pp. I-569-I-572 vol.1. 
[2] F. V. Fernandez, J. L. Huertas, A. Rodriguez-Vazquez, and G. G. E. 
Gielen, "Symbolic analysis techniques: applications to analog design 
automation," New York: IEEE Press, 1998. 
[3] B. Thanigaivelan, A. Postula, and T. J. Hamilton, "A modified 
MOSFET small-signal model based on Affine Arithmetic concepts," in 
PRIME Asia Shangai, China, 2009. 
[4] E. A. Vittoz, "Micropower Techniques," in Design of analog-digital 
VLSI circuits for telecommunications and signal processing: Prentice-
Hall, Inc., 1994, pp. 53-96. 
[5] D. Haigh, C. Toumazou, and F. J. Lidgey, Analogue IC design : the 
current - mode approach. London :: Peregrinus on behalf of the 
Institution of Electrical Engineers, 1990. 
[6] H. Schmid, "Why 'Current Mode' Does Not Guarantee Good 
Performance," Analog Integr. Circuits Signal Process., vol. 35, pp. 79-
90, 2003. 
[7] B. Gilbert, "Current-mode circuits from a translinear viewpoint: a 
tutorial," Analogue IC Design: the current-mode approach, pp. 11-91, 
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[8] T. J. Hamilton, C. Jin, and A. van Schaik, "An analysis of matching in 
the Tau cell log-domain filter," in Circuits and Systems, 2006. ISCAS 
2006. Proceedings. 2006 IEEE International Symposium on, 2006. 
 
 
Fig 5. Tau Cell frequency response measured from a fabricated chip 
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Symbolic analysis of tau cell log-domain filter using affine MOSFET models

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Symbolic analysis of tau cell log-domain filter using affine MOSFET models

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