=Paper=
{{Paper
|id=Vol-533/paper-15
|storemode=property
|title=Applying Fuzzy Sets Intersection in the Sizing of Voltage Followers
|pdfUrl=https://ceur-ws.org/Vol-533/18_LANMR09_poster02.pdf
|volume=Vol-533
|dblpUrl=https://dblp.org/rec/conf/lanmr/BecerraCP09
}}
==Applying Fuzzy Sets Intersection in the Sizing of Voltage Followers==
https://ceur-ws.org/Vol-533/18_LANMR09_poster02.pdf
Applying Fuzzy Sets Intersection in the Sizing of
Voltage Followers
G. Flores–Becerra1, E. Tlelo–Cuautle2 , and S. Polanco–Martagón1
1
Instituto Tecnológico de Puebla.
Av. Tecnológico 420, Puebla. 72220 MEXICO
2
INAOE–Department of Electronics.
Luis Enrique Erro No. 1, Tonanzintla, Puebla. 72840 MEXICO
kremhilda@gmail.com,e.tlelo@ieee.org,polanco.s@ieee.org
Abstract. An automatic fuzzy–set–intersection–based approach is pre-
sented to compute the optimum sizes of Voltage Followers (VFs). Based
on Monte Carlo simluations and two fuzzy sets to represent the gains
closer to unity and higher bandwidths, the approach compute the opti-
mum sizes through the application of fuzzy sets intersection, subject to
a distance from the maximum gain defined by a threshold and a lower
bound to gains, both defined by the circuit design expert.
Key words: Fuzzy Sets Intersection, Circuit Optimization, Analog De-
sign Automation, Circuit Sizing
1 Introduction
Electronic systems, such as cellular telephones, magnetic disk drives and speech
recognition systems [1], require an interface with the external world. Since the
world is analog in nature all these kind of electronic systems require analog
circuits. In the analog design automation (ADA) tools, new techniques need to
be developed to improve the design of integrated circuits (ICs), in order to reduce
the costs of the production, to shorten the time to market [2], and to enhance
the quality and optimality of integrated circuits.
Different kinds of active elements that are used in analog signal processing
applications, such as to design chaotic oscillators [3], to design current conveyors
[4], to design filters [5] and to develop secure communication systems [6], are de-
signed by using unity-gain cells (UGCs) such as Voltage Followers (VFs) [7]. In
order to improve the performance of these applications, it is needed to compute
optimal sizes of the VFs [8]. In [8][9][10][11][12] some ADA procedures to auto-
mate the sizing process are given, but an open problem remains related to the
selection of an optimal sized topology whose parameters such as gain, bandwidth
(BW), input and output impedances, among others, need to be classified.
The automatic selection of an optimum sized VF topology is addressed in
this paper. Given a VF its gain is optimized to be closer to unity and with the
large BW. In this problem there are two linguistic variables: ”closer to unity”
and ”large”, then the fuzzy sets are well suited to represent the behavior of the
209
VF under several values of its parameters, such as width (W) and length (L) of
the transistors, and current bias, since the fuzzy sets allow formalizing linguistic
sentences to express ideas that are subjective and which can be interpreted in
different ways by various individuals [13]. Then, defining appropriate member-
ship functions it is possible to construct two fuzzy sets: a fuzzy set to represent
the higher BWs and a fuzzy set to represent the gains closer to unity.
In this manner, the proposed approach finds the optimum sizes of the VF
where both conditions (higher BW and gain closer to unity) are satisfied: first
selecting the parameters of the VF such that the gain is far from the maximum
gain in certain distance, that is defined by a threshold, and such that the gain is
greater than certain lower bound; second defining the fuzzy sets; then computing
the intersection of both fuzzy sets that takes the minimum between both mem-
bership values; finally, taking the maximum of intersection result. The threshold
and the lower bound, as well as the parameter values of the VF, are defined by
the circuit design expert.
2 The Fuzzy Sets Intersection Method
Let X be the universe set of all sizes combinations of a VF and their perfor-
mances. Since in this paper a VF is characterized by its length and width, and
the current bias used, it is defined a conventional set on X as follows: Let P be
a set of parameters defined by [14] [15]
P = {x |x = {L(µm), I(µA), W (µm) }}
where L is the length of all transistors, W is the width of each transistor, and
I is the current bias. Each x associates sizes to perform a SPICE simulation,
which results are introduced in two fuzzy sets defined on X as follows: The BWs
are collected in A, the fuzzy set of large bandwidths, defined as
e
( )
µA (x) x
A= e x is a large bandwidth and µA (x) = , , (1)
e x e max bandwidth
where max bandwidth is the maximum BW of all P sizes and the membership
value of each BW, µA (x), depends on how much its value is large. The gains are
e
collected in B , the fuzzy set of gains close to unity, defined as
e
( )
µB (x)
B= e x is a gain close to unity and µB (x) = x , (2)
e x e
where the membership value of each gain, µB (x), depends on how much its value
e
is closer to unity. Then, there is a correspondence among elements in P , A and
B. e
e The proposed method allows to the circuit design expert to define the P set
through the definition of L, I and W values of a VF, and the expert can request
210
that the gain obtained from SPICE simulation is greater than a certain lower
bound, low bou, then if
gain ≥ low bou, (3)
the gain, that is represented with an element of B , along with the correspond-
ing elements in P and A are desired values, else e
they are eliminated. Also, the
expert may request that e the gain is far from maximum gain in certain desired
des dis×∆gain
distance given through a threshold, defined as thr = max gain − 100 ,
where max gain is the maximum gain of all P configurations, des dis is the
desired distance from maximum gain (measured in a percentage), and ∆gain =
max gain − min gain, where min gain is the minimum gain of all P configura-
tions. Then if
gain ≥ thr, (4)
the gain (represented with an element of B ) along with the corresponding ele-
e
ments in P and A are desired values, else they are elminated.
e
Once the low bou and thr are defined, and the gains (with the corresponding
bandwidths and parameters) are selected under the two restrictions (3) and (4),
the proposed method builds the fuzzy sets A and B in accordance with (1) and
e
(2) and computes the intersection of these fuzzy e as follows
sets
( )
\ µC (x)
C =A B= e µC (x) = min{µA (x), µB (x)} .
e e e x e e e
In this case, as A is the fuzzy set of large bandwidth and B is the fuzzy set of gains
e intersection of A and B represents the
close to unity, the e set of P configurations
e
where the gain is close to unitynand the e
o BW is large. Then the optimum sizes are
computed by OptV F = max µC (x) , where the correspondient P element of
this maximum is the optimum size e of the given VF. All these steps to compute
the optimum sizes of a VF automatically are collected in the following algorithm.
ALGORITHM FuzzySetIntersecMethod
IN: VF:file;Lvalues,Ivalues,Wvalues:Set;desDis:int;lowBou:real
OUT: optLvalue,optIvalue,optWvalue,optGain,optBandwidth:real
BEGIN
/* Define P set of the VF parameters */
t = 1
FOR i = 1, 2, ..., cardinality of Lvalues
FOR j = 1, 2, ..., cardinality of Ivalues
FOR k = 1, 2, ..., cardinality of Wvalues BEGIN
P(t) = [Lvalues(i), Ivalues(j), Wvalues(k)]
t = t+1
END
/* Collect gains and bandwidths using Montecarlo simulation */
FOR i = 1, 2, ..., cardinality of P BEGIN
[GAINS(i), BANDWIDTH(i)] = SPICE(VF with P(i) parameters)
END
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/* Compute the two fuzzy sets */
max_bandwidht = maximum(BANDWIDTHS)
[max_gain,min_gain] = [maximum(GAINS), minimum(GAINS)]
thres = max_gain - (desDis*(max_gain - min_gain))/100
t = 1
FOR i = 1, 2, ..., cardinality of GAINS BEGIN
IF (GAINS(i)>=thres) AND (GAINS(i)>=lowBou) BEGIN
Afuzzy(t) = BANDWIDTHS(i) / max_bandwidth
Bfuzzy(t) = GAINS(i)
Q(t) = P(i)
t = t+1
END
END
/* Compute intersection of the fuzzy sets */
FOR i = 1, 2, ..., cardinality of Afuzzy BEGIN
Cfuzzy(i) = minimum(Afuzzy(i),Bfuzzy(i))
END
/* Compute VF optimum size (intersection maximum) */
optc = Cfuzzy(1)
j = 1
FOR i = 2, ..., cardinality of Cfuzzy BEGIN
IF (optc < Cfuzzy(i)) THEN BEGIN
optc = Cfuzzy(i)
j = i
END
END
[optLvalue, optIvalue, optWvalue] = [Q(j).Lvalue,
Q(j).Ivalue, Q(j).Wvalue]
[optGain, optBandwidth] = [GAINS(j), BANDWIDTHS(j)]
END
This algorithm allows to compute the optimum parameters of a VF such that
its bandwidth is large and its gain is close to unity at the same time through a
single operation (the fuzzy sets intersection), based on the proper definition of
the fuzzy sets, meanwhile other methods compute the optimum using different
stages. For example, the approach [8] takes a few VF parameters to simulate the
VF, first computing the VF parameters where the gains are close to unity, then
selecting from these computed gains the larger bandwidth.
3 Experimental Results
In Fig. 1 are shown the VFs used to compute their optimum sizes by using SPICE
and under several P set, des dis and low bou values. These VFs are synthesized
by P–MOSFETs and N–MOSFETs [9]. For each VF of Fig. 1 the circuit design
expert has defined the following P set
P = {L, I, {WA , WB , WC }} , (5)
212
Fig. 1. Voltage Followers under test.
1 1
Optimum with des_dis
not used
2%
0.95 5% 0.995
10%
20%
0.99
0.9
0.985
0.85
Optimum with des_dis
not used
0.98 2%
Gain
Gain
0.8 5%
10%
0.975 20%
0.75
0.97
0.7
0.965
0.65 0.96
0.6 0.955
0 20 40 60 80 100 120 140 160 180 0 50 100 150 200 250 300 350
Bandwidth (MHz) Bandwidth (MHz)
Fig. 2. Optimums computed by the proposed fuzzy set intersection method for the VF
of Fig. 1(a)–(b).
213
where L = {0.4, 0.7, 1.0} µm, I = {10, 20, ..., 100} µA, WA = {10, 20, ..., 100}
µm, WB = {300, 310, ..., 400} µm, and WC = {600, 610, ..., 700} µm. Then,
the VFs were sized using three different lengths (L), ten different currents for
biasing (I) and thirty different VF widths (WA , WB , WC ). This leads us to 4950
combinations in sizing when the P–MOSFETs widths are greater than or equal
to the N–MOSFETs widths, that means that WM1 ,M2 ≥ WM3 ,M4 for VF of Fig.
1(a), WM1 ,M4 ≥ WM2 ,M3 for VF of Fig. 1(b), and WM3 ,M4 ≥ WM1 ,M2 for VFs
of Fig. 1(c) and Fig. 1(d).
The results obtained with the proposed method to size the VFs of Fig. 1,
under (5) parameters, are given in Fig. 2–3. In all cases, the optimums computed
by the proposed method are marked under several des dis values, des dis =
{2%, 5%, 10%, 20%}, and low bou = 0.5. It is easy to see that the proposed
method gives good optimums applying the fuzzy sets intersection, sice although
there are some results where the BW is greater, the method has selected the
sizes–combinations where the BW is greater but AV is closer to unity.
1 1
Optimum with des_dis
not used
0.95 2%
5%
10%
20% 0.995
0.9
0.85
Optimum with des_dis
0.99 not used
2%
0.8 5%
Gain
Gain
10%
20%
0.75
0.985
0.7
0.65
0.98
0.6
0.55 0.975
0 100 200 300 400 500 600 0 50 100 150 200 250 300 350 400
Bandwidth (MHz) Bandwidth (MHz)
Fig. 3. Optimums computed by the proposed fuzzy set intersection method for the VF
of Fig. 1(c)–(d).
Some details of the results obtained in each graphic of Fig. 2–3 are given in
Table 1. For VF of Fig. 1(a), good results are given when des dis = {5%, 10%}
(Av =0.9723, BW =69.98M Hz and Av =0.9526, BW =86.1M Hz, respectively).
Meanwhile a good result is obtained for VF of Fig. 1(b) with des dis = 20%,
where the bandwidth is almost larger among all results (BW =248.3M Hz) and
the gain is good enough (Av =0.9891). When des dis = {5%, 10%} good opti-
mums are obtained (BW =199.5 M Hz, Av =0.9942 and BW =201.8M Hz, Av =
0.9935, respectively).
A good behavior for VF of Fig. 1(c) in Table 1 is provided when des dis =
10%, where Av =0.9684 and BW =260M Hz. However, an acceptable result is
given when des dis = 20% (Av =0.9192, BW =363.1M Hz). Finally, a good result
is obtained for VF of Fig. 1(d) when des dis is not used, since the gain is good
enough, Av =0.9831, and BW =393.6M Hz is largest of all experiment results.
214
Table 1. VF optimum sizes of Fig. 1(a)–(d).
VF of des dis L I1,2 I3,4 WM1 ,M2 WM3 ,M4 Av BW
Fig. 1(a) without 0.4 100 50 30 20 0.7789 134.9
5% 1 100 50 70 70 0.9723 69.98
10% 0.7 100 50 60 60 0.9526 86.1
20% 0.7 100 50 40 40 0.9353 86.1
des dis L I1,2 WM1 ,M4 WM2 ,M3 Av BW
Fig. 1(b) without 0.4 100 90 90 0.9701 305.5
5% 1 100 90 70 0.9942 199.5
10% 1 100 60 60 0.9935 201.8
20% 0.7 100 60 60 0.9891 248.3
des dis L I1,2,3,4 WM3 ,M4 WM1 ,M2 Av BW
Fig. 1(c) without 0.4 100 30 30 0.82 407.4
5% 0.7 100 90 90 0.9771 251.2
10% 0.7 100 60 60 0.9684 260
20% 0.4 100 90 90 0.9192 363.1
des dis L I1,2,3,4 WM3 ,M4 WM1 ,M2 Av BW
Fig. 1(d) withou 0.4 100 90 90 0.9832 393.6
5% 1 100 80 70 0.997 218.8
10% 1 100 50 50 0.9966 221.3
20% 0.7 100 60 60 0.9945 281.8
4 Conclusions
An automatic method based on intersection of fuzzy sets has been introduced
to solve the open problem of sizing Voltage Followers (VFs), unity–gain cells
that are used to design active elements used in several analog signal processing
applications. The proposed method formalize the representation of the behavior
of VFs under several values of its parameters (width and length of the MOS-
FET, and current bias) using two fuzzy sets: a fuzzy set to represent the large
bandwidths and a fuzzy set to represent the gains close to unity of a VF. The
bandwidths and the gains are obtanied of Monte Carlo simulations using several
combinations and varying the curren biases, the widths and the length of the
MOSFETs, that are parameters defined by the circuit design expert. Through
the fuzzy sets intersection the optimum zised of a given VF is computed, since
the intersection is a natural manner to represent the VF configurations where
the gain is close to unity and the bandwidth is large at the same time. The suc-
cessful of the results obtained with the intersection, as has been seen in previous
section, lies in the definition of the fuzzy sets and their appropriate membership
functions, that are the key of the proposed method. The expert can select a
desired distance from the maximum gain obtained from SPICE simulations to
compute the optimum VF behavior. Also, in order to eliminate non–significant
results obtained from Monte Carlo simulations, the proposed method allows to
215
expert to define a lower bound to gain. These additional restrictions help to
compute several optimums that can be conveniently selected by the expert.
Acknowledgments. This work is partially supported by CONACyT under the
project number 48396–Y and DGEST under the project number PUE–MI–2008–
206.
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