- Email: [email protected]
Pattern recognition from sensor arrays: Theoretical considerations
Sensors
and Actuators
A, 32 (1992) 507-518
Pattern recognition Fabrizio
A. M. Davide
Dipartimento
di lngegneria
Arnaldo
D’Amico
Dipartimento
di Ingegneria
Elettrica,
Elettronica,
507
from sensor arrays: theoretical Universitd
de L’Aquila,
Universitci
di Roma
Poggio
di, Roio,
Tor Vergata,
67100 L’AquiIa
via 0. Raimondo,
considerations
(Italy)
Rome (Italy)
Abstract A general approach to the design and the evaluation of multisensor measurement systems based upon real non-selective reproducible sensors is proposed. Generalized definitions of both sensitivity and precision of sensor arrays are introduced. By means of new graphical tools the possibility is analyzed of obtaining improved selectivity, sensitivity and precision from sensor arrays. Quantitative design criteria, allowing the optimal coupling of individual sensors to form the array as well as their selectivity and operating conditions, are introduced, taking into account the influences of technological processes.
1. Introduction
Most sensors are designed to be sensitive to one parameter only. Unfortunately, most sensors are sensitive to a number of measurands and it is often rather difficult to utilize the output signal of such sensors. When a sensor is sensitive to an undesirable parameter we usually speak of the cross-sensitivity of the sensor. The solution could be to measure the undesirable parameter and to generate a signal that can be used to coufieract the cross-sensitivity [ 11. On the other hand in many applications it is of interest not only to know one parameter but also to simultaneously obtain the values of a number of other parameters as well, e.g., in biomedical applications. Sensor arrays and pattern recognition techniques have been recognized to be a promising answer to both these problems. Figure 1 shows the principle of pattern recognition systems in which non-selective, reproducible gas sensors with crosssensitivity are used to determine the concentrations of different gases [2]. Much effort has been spent on pattern recognition techniques and sensor technological improvement [3]. In this paper we want to focus our attention on a different aspect. There is no doubt that when many sensors are operating together, the ensemble shows different features with respect 0924-4247/92/$5.00
Fig. 1. Multisensor for gas sensing.
pattern-recognition-based
measurement
system
to the ones shown when the sensors are used alone. Here we are mainly interested in considering arrays composed of non-selective low-cost sensors, that otherwise cannot be used in a measurement system but in controlled conditions. The cross-correlations among the sensor responses can be exploited to give acceptable measurements, and non-negligible improvements of selectivity, sensitivity and noise rejection characteristics. In this paper a theoretical frame has been constructed in order to help the array designer. @ 1992 -
Elsevier Sequoia. All rights reserved
508
Section 2 introduces some useful notations. A discussion of sensitivity and designer aid charts will be presented in Section 3. Noise analysis and related charts will be presented in Section 4. Section 5 will take into consideration as an example a solid-state gas sensors array in order to better explain the previous results. Technological process influences are also discussed in this Section.
2. Reference model
Let us consider n sensors whose outputs are dependent on m physical or chemical quantities (state variables) characterizing the environment where the sensors are located. Let zi be the output of the sensor i and z the output vector, Wj the j-state variable and w the state vector. Generally speaking we have: ZI =
Z,(Wl,. *. , %a>
(1)
i.e., a transformation z: W,, +Z. Let W be the state space, W,, the open subset of W, which represents the environmental conditions under which the sensors will be operating, and Z the outputs spiv;e*. According to Fig. 1 the sensor array is characterized by the partial derivative matrix (au}, with au = az,/aw,. Usually we say that the sensor i is selective if laiil > layI with i #j, however in the present paper we will not make a hypothesis on selectivity. Restricting our discussion to the case m = n = 2, the {au} matrix referring to a certain w can be geometrically visualized: Fig. 2 shows the vectors Vz, and Vz2, with the CYand /I angles, which are directly related to the selectivities of z1 and z2. For instance, the relation aII/aIz = cot u shows how a high selectivity for zr implies a is small in modulus. Similarly, the relation aZ2/ a,, = tan(cr + B) says that selectivity z2 is maximal for j3 close. to (7r/2) - a.
*In order to avoid mathematical inconsistencies we will consider W and Z spaces which have adimensional components.
!dwl
a,, -M-W)
a,,
Fig. 2. z, and z2 gradients state w,.
2.1. Notations
dW
grad Zz
with related angles, referred to a certain
2.2. State retrieval If the sensor i is not selective with respect to wi
then zi is non-negligibly dependent on the other state variables. This means that knowledge of z, only does not allow the exact knowledge of w, . If we compose an array of low-selective sensors, we can try to retrieve the state w from the n low-selective measurements of the w components (i.e., z components) exploiting the cross-correlations between them. The achievement of this result requires that the application (eqn. ( 1)) has to be invertible for all W,,. As is well known, this inversion turns out to be possible under the following conditions: (a) z must be infinitely and continuously differentiable in W,,; (b) the Jacobian of the transformation must never be 0 in W,,. In the following we will require the geometrical condition equivalent to condition (b): the zi gradient field must never be parallel to the z2 one in W,, i.e., B(W) # 0, VWE w,,.
3. Design approach
At design level our purpose is the optimization of the sensor array with respect to some given features. Some peculiar characteristics of a given sensor array can be expressed by an opportune functional in the integral form: dw
i,j=l,2
(2)
where F can be equivalently considered to depend on a(w), P(w), (IVzi I(, and y(w) defined as Y(W)= I)Vz2II/J/V&1).
509
Consequently our strategy is to optimize the functional under suitable continuity and end-point conditions, warning that a direct optimization would give two ideal sensors, which are by definition infinitely selective. In practice it is necessary to start with a given real sensor, characterized by some degree of selectivity and sensitivity, and to choose the other one by optimizing the functional. In the following we will define two possible functionals related to array sensitivity and precision, and we will consider the array design choices comparing the performances of the array with those of the sensors used alone.
s
a-y
(4)
where dw/lldw 11 z is the unit vector parallel to dw. 4.2. Average
sensitivity
As the sensor array is dependent on the dw orientation (determined by angle rr as in Fig. 2), it is necessary to consider an average sensitivity with cr in the range [ - n, rc]:
4. Sensor array sensitivity
(5) The closed form for eqn. (5) is:
4.1. Definitions Let us consider the infinitesimal change of the state dw around wo. The output vector change can be expressed as:
(6) where
dz = (Vz, - dw, VzZ - dw) We will define the sensor array around w. by the following ratio:
sensitivity
f(p,v)
=i(sinarctan(’
yrif18+ ‘>
-sinarctan()‘y:i!P
where )I- llaand 11 - 11are two norms respectively b
applied on W and Z spaces. The best choice for the II * lla is II * (Im, i.e., lldz Ila = lldz Iloo= max{dzi}
‘>)
(7)
It is worth noting that S(fi, y) is periodic in 8, with period x, and symmetric with respect to B = 0, n/2, considering the validity of the following expressions: f(BY Y) =f( - /%r)
i = l-2
for the following two properties exclusive for. this norm: (1) as is well known, IIx Ilmc IIx II0 for each x and whatever the norm is in the second member, and this leads to the worst-case evaluation of sensor array sensitivity; (2) for a degenerate array, i.e., an array composed of two equal sensors, we have /dz IJrn= Idz, 1, as is expected considering that practically the sensor array turns out to behave as one sensor. The norm IIdzllb is Euclidean in character with the advantage that in the case of degeneration of the array toward z1 (i.e., ,0 = 0) the array sensitivity reduces to the directional derivative of z1 with respect to dw. In this case we have:
f(;-AY)=f(;+AY) 4.3. Sensitivity ratio First we notice that in the presence of only sensor 1 the average sensitivity, as the average of the directional derivative eqn. (4), is given by:
Now we can rewrite eqn. (6) as the sensitivity ratio:
s
( >
array =rf dj &I
+f(B,
I9
Since yf(& l/y) + f(p, y) > 1 whatever p is, as can be easily verified, we can conclude that a
510
second sensor produces a local average array sensitivity always greater than the one obtained by a single sensor (also if ideal). 4.4. Array design Let us suppose the existence of the following integral, which represents the average value of S array (see eqn. (2)):
<&ray >wop
= -j+
s
&ray (4 dw
OP WOP
where Mop is the W,, measure. According to eqn. (6) &ray) w,_, can be considered as a functional dependent on p(w), y(w), [IVz,II, whose maximum point can be calculated with the bound of considering a given real z1 . In other words this means the selection of a z2 sensor complementary to z, with respect to the sensitivity. Since the function involves w only implicitly the procedure will be trivial. It is easy to verify that Sarray is a growing function with respect to y and that if y goes to + cc whatever p is, we have:
It turns out that the sensitivity nearly increases in proportion to IIVz211when y + co. On the other hand jIVzlII is . b ounded by the available technology and usually is as big as IIVz, 11.Furthermore it is strongly related to the cost of the sensor and for these reasons it cannot be increased as one would like. So the unique choice is to consider Vz, free only as far as the orientation is concerned. In this case the maximum of (Sarray)W,p can be found by solving the equation &,,,(/I, -,~)/a@= 0, which is satisfied in p = 0 and /I = 7~12 for each y value. Finally the function is maximized by: /&lax(W)= n/2
(9)
This result represents the constraint on the z2 sensor from sensitivity considerations. According to eqn. (9) the maximal ratio, y changing, is Sarlay/Szl = (1 + y*) ‘I*. Figure 3 shows Sarrar/SZ1 for /I = 7c/2, and also for other p values*. As a
*Note the similarities with the Bode diagram function.
of a one zero transfer
0.5
_....
;.
:..
..;
.
..‘.
...........
j
2
2.5
3
._
0 0
0.5
i
1.5
3.5
4
88mn8
Fig. 3. Sensitivity ratio as a function of y for some p values. For every fixed y the ratio assumes its maximum value when /3 = n/2.
numerical example when y = 1 and B = 7r/2 we i.e., an improvement have Sarray(w) = S,,(W)*, factor of 3 dB compared to what we can get from sensor 1 only. It is of some interest to note that for =&ify>l, B = 0, SarrayequalsS,,ify
(10)
which gives a A/3 of 0.28 rad, when E = 1% and y = 1. 4.5. Sensitivity ratio map Figure 4 represents constant sensitivity ratio curves on the (/I, y) plane. Dashed lines, obtained by eqn. (lo), are the boundaries of the region with a sensitivity close to the maximum value (E = 1%) for every fixed y. This sensitivity ratio map can serve a double purpose. First the array sensitivity can be easily read from the map: for instance the choice of the two sensors with B = 100” and y =0.33 induces a sensitivity increase ranging from 3.5% to 7% compared to sensor 1 only. Secondly the map can be utilized for the array design. In fact let us consider a technological process P which is able to produce sensors, and
511
-: : :: :: : : -+-i :: :: : : f
(_
: ..:..;._
:
: : : :
: : : : :_ i
: j
: j
.i
.._:._.i._
! : : . . .. j
: : : :_j
1% ., .: _ : : : : . ..1..1._ : : : F i ; IO’
Fig. 4. Constant sensitivity ratio curves on the (& y) plane constituting a sensitivity ratio map. Relative increments are indicated for each curve: e.g., + 15% means Sarray = 1.15X,. On the dash-dotted technological curves, referred to three different pz values, the current parameter is p,. Dashed lines are explained in the text.
which is characterized by two parameters p, and p2. For each couple (p, pz) we will have a particular array corresponding to a point in the (/I, y) plane. Dash-dotted lines in Fig. 4 correspond to some parameter values and clearly indicate the influence of technological choices on the array sensitivity. It is possible to conclude that array design, referring to the sensitivity constraints, means optimization of SarraY towards p, and p2. A simple practical example will be given in Section 6.
5. Noise analysis 5.1.
Uncertainty
range related
to w1
Measurements performed by sensors are always affected by noise. We are interested in determining the uncertainty range on the w estimation due to the array and comparing it with the case of a measurement performed by sensors working alone. Let us consider the sensor outputs as a vectorial stochastic process z(w, t) and its time average as the z(w) application (eqn. ( 1)). Assuming ergodicity to be valid let us consider in the Z space a square domain
of
Qz r side length, centred on z(wo), and containing with high probability (e.g., 99%) the random variable z(wo, to), with to the array sampling instant. As a consequence, the uncertainty range for the z components at the assigned probability is about large r and centred on zi(w,,). Figure 5 shows QZ on the t plane and its mapping Qw on the w plane. The uncertainty range Awl, related to the estimate of w,, will be the orthogonal projection of Qw on the w1 axis. Starting from a QZ definition, by means of the inverse transformation z-’ it is possible to determine the equations describing Qw and also to consider only the component related to the w, axis. Let us assume r to be sufficiently small. In this hypothesis z(w) can be locally linearized and easily inverted. Therefore the Qw domain is clearly a parallelogram and its projection on the w1 axis coincides with that one of a diagonal whose projection is larger. Indicating the uncertainty range by [Awl]_,,, we have: (11) where J is the Jacobian
of z(w).
512
Fig. 5. The corresponding Q, and Q, domains Q, is consequently determined by L-‘(Z).
on the L and w planes. A square shape has been arbitrarily
5.2. Precision ratio
If the sensor z, is utilized alone in order to obtain the w1 measurement, we firstly need to know the value of the other state variable either through physical information (e.g., state changes) or measures taken in a different way. If in eqn. ( 1) we introduce the known wZOvalue, the result is that the uncertainty on z1 is reflected on w1 as follows:
(12)
[Awl,, =i+
Iall I Let us consider the ratio R,(w) between eqns. (11) and (12), namely the Aw, uncertainty ranges for both the array and the sensor 1 working alone, expressed by the quantities of Fig. 2:
RI(W)=
Isin(a + 8) 1+ jj lsin tcI) lcos GL l
(
IsinBI
(13)
We will call it the precision ratio (which is less than 1 if the array has a better precision) for w1 with respect to sensor 1. From eqn. ( 13) it can be seen that RI is periodic in M and fi, with period rc, then without any lack of generality we will assume CIand /3 in [0, rc]. If we evaluate eqn. ( 13) with p = rc - c( we will obtain R,(w) = lcos al/r, which is less than 1 for y > lcos CI]. The interesting result is that using a second sensor supporting zi we can reach a precision
chosen for Q, while the shape for
for w1 that is greater than the one associated sensor 1 alone.
to
5.3. Array design Let us suppose the existence of the following integral which represents the average value of R, on W,, (see eqn. (2)):
s
OP Ww
R,(w) dw
According to eqn. (13), (R,)WOP can be considered as a functional depending on a(w), p(w), y(w) whose minimum point can be calculated within boundary of a given real z, . As already done for sensitivity, we will look for sensor 2 complementary to a fixed sensor 1 from a precision point of view. In order to minimize (R,)wOp it is not possible to use the Euler-Lagrange equations. In fact, the equation referred to y, dRl(cr, /?, y)/ay = 0, is never satisfied because RI always decreases with y, whatever is the fixed /? value. As to the equation referred to p, it is inadequate because the integrand does not comply with the regularity hypothesis. We will consider, as we did in the case of sensitivity, the Vz2 orientation free and its modulus constant. We will search for the R, absolute minimum with respect to /3, considering y assigned.
513
01 0
20
40
60
80 beta
iDcl
i20
140
160
IN.
[degrsesl
Fig. 6. Precision ratio as a function of p for some y values and c( = 21”. For some values of the (8, y) couple the ratio is less than 1, i.e., the array shows a better precision with respect to W, than that shown by sensor 1 alone
Figure 6 shows a few sections of RI with y constant, a = 21” and fi E [0, ~1. It is clear that the absolute minimum is in /I = 71- c1when y > lcos CC] and it is coincident with the relative minimum when y -C (cos CC].In order to calculate the relative minimum of RI with respect to /I, we need to calculate its derivative and study the relative roots. The roots coincide with the solutions of 1 + y cos j3 = 0, i.e., Bmin= arccos( - 7)
if CLin [
0, n 2
1
and y -C lcos a] jImi,=71-arccos(-~)
if a in
ll
-,7c [2
1
(14)
and y c lcos aI Finally we find that the RI absolute minimum, y changing, is: _ lcos 4 Y Ri (G Bmin9Y)
when y > lcos a] (15)
= (Ices tx]+ [sin u]((l/y’)
- 1)“2)]cos aI
when y < Jcos a I
The case y > 1~0s~11is trivial because when /I = x - LXwe need Vz2 laid down on the w1 axis, which means z2 infinitely selective on wl. 5.4. Precision ratio map Figure 7 shows the R, constant curves for CI= 21” on the (/.I, y) plane. The precision increments and the locus /I = /Imin(y) are also indicated there. The hatched region is the set of the arrays which have a better precision on w,. As to the technological choices, remarks already made for sensitivity are still valid. The problem of optimizing the array with respect to the precision on w, is precisely symmetric when referred to w2. Figure 8 shows the precision ratio map for w2. The hatched H region indicates a better precision on w2 for the array: it is clear that we cannot simultaneously reach a precision improvement with respect to both w, and w2 as the relative regions do not have intersections. The trade-off choice which makes no distinction between the state variables is represented in the Figure by the T region. This choice shows a precision worsening of about 1.55 dB when y = 0.84 and /I = 105” (we must not forget that these numbers are meaningful when ci = 21”).
Fig, 7. Constant precision ratio curves referred to wl on the (p, y) plane, with OL= 21”, constituting a precision ratio map. Precision relative increments are expressed as 20 log,, R, ‘. In the hatched region arrays show higher precision. The dashed line shows the locus B = &,,i.(y),
Fig. 8. As in Fig. 7 but the precision ratio is referred to w,. Hatched a trade-off between W, and w2 precision requirements.
6. Analysis
of a real case
6.1. Solid-state gas sensors It is well known that SnOZ thin films doped by indium, through thermal evaporating or RF sput-
region H corresponds
to better array precision on w2_ Region T concerns
tering, are sensitive to many gases, such as NOz, Hz, CO, SO,. The response of this material to these gases is temperature dependent. As an example we will consider two identical SnO,(In) sensors (taken from ref. 4) operating at different temperatures,
515
such as 310 “C (sensor 2) and 450 “C (sensor l), and an ambient where CO and SO1 gases are present with w1 and w2 as concentration values respectively (ppm). Assuming that z, is the relative variation of conductivity of the i sensor we get: z, = o.15w,“.28 + o.11w2°.20 i ~2 = 0.022~1’.~’ + O.O66~2’.~~ The above equations represent a very simplified model, which is assumed to be valid with concentrations ranging between 10 and 1000 ppm. Figure 9 shows the characteristics of the two sensors. The limited selectivity suggests the use of a sensor array, which is theoretically possible in virtue of the validity of inversion conditions. Let us evaluate sensitivity and precision at a concentration of 100 ppm for both gases. 6.2. Sensitivity evaluation Figure 10 reports the non-averaged sensitivities of the two separately utilized sensors, as functions of the dtv orientation and their envelope that represents the non-averaged array sensitivity. The envelope average value is greater than the averages of the two component sensor sensitivities, as expected according to Section 4.3. The inversion condition p # 0 implies that the nonaveraged array sensitivity is never 0 for each 0 value; on the contrary this is not true for the components sensitivities. Table 1 shows that in this case the sensitivity only improves by 1.8%; this is mainly due to the low y value and to a p value too far from 90”, as can be directly deduced by the sensitivity map in Fig. 4. Figure 11 shows the sensitivity ratio map in a polar diagram form. Curves with the same sensitivity are ellipses with major axes parallel to Vz,. Furthermore the minimum of IIV.QIIwith a fixedarray sensitivity value would correspond to p = 90”. In our example this is not feasible because both gradients are physically forced to lie on the same quadrant. For the dash-dotted curves, referring to two different dopings utilized for the fabrication of z2, the current parameter is the operating temperature of z2. At 310 “C the representative vectors of Vz, are A and A ‘, while B and B’ refer to 227 “C. More precisely, the curve which A and B belong to refers to arrays composed by two identical sensors operating at different tempera-
tures. On the contrary the other curve refers to arrays composed of sensor 1 already introduced, and of a differently doped sensor 2, operating at a different temperature too. As the technological curves are nearly parallel to the constant sensitivity ratio curves it turns out that there are not sensitivity variations between A and B arrays or A ’ and B’ ones. 6.3. Precision evaluation Let us suppose that both sensors are supplied by a 1 uA current and that the output noise causes maximal relative conductivity fluctuations of 1 x 10e3, then we obtain the values shown on Table 2. These values correspond to 2% and 3.5% errors for the CO and SO2 concentrations. From the polar map referred to in Fig. 12, we can read the same worsening of 10 dB on w, precision for both A and B cases. The conclusion is that only by changing the temperature conditions can we obtain from two identical sensors the possibility to simultaneously measure the CO and SO2 concentrations, but the sensitivity and the precision of the single sensors are not likely to be improved. This can be understood by looking at the compensation of /? and y variations on the technological curves. The reason for this compensation will not be taken into consideration in this paper and will represent a subject for further investigations. The doping variations, on the contrary, bring a certain improvement, as we can see from the comparison between A and A’ arrays, for instance. Nevertheless in this case we have an increased difficulty in using a different process for the fabrication of the sensors.
7. Conclusions
The above considerations may be useful for many practical applications: -Reduction in disturbance. Silicon pressure sensors, for instance, are sensitive to the temperature, and magnetic field sensors to the pressure. In order to reduce the cross-sensitivity it is convenient to consider a second sensor of a poor quality which plays a slave role. In this case the precision with respect to w, only will be optimized, while any other feature with respect to w2 will be neglected.
516
: 0.A
(a)
(b) Fig. 9. Response characteristics for a SnO,(In) concentrations, i.e., w, and w2.
thin film sensor when operating
-Multisensing. It is the case taken into consideration in this paper, where no privilege was given to any state variable. For instance it is a typical problem in the chemical-sensing field. -Increased W,, domain. This problem too can be
at 450 “C and 310 “C respectively, as functions
of SO2 and CO
expressed from the function optimization point of view, and studied independently of the particular sensors utilized. Some of the obtained results can be eminently used with silicon planar technology, offering a
517
0.2
.,....,.
j
.,..........,.
Z?j\ .i I
!
’
.\
Qi
/-
i
:
;: , ,:
t
““‘7
’ ./
Sioma+Alpha Ulegreesl
Fig. 10. Sensitivities (not averaged) of sensors working alone, dashed and dotted lines, as functions of dw orientation (see Fig. 2): the envelope (solid line) of single sensors sensitivities represents the array sensitivity (not averaged) according to the I(* 11.definition (see Section 4.1).
TABLE 1. Sensitivities and other related quantities 450 “C, respectively
IIVZI II
B
Y
(deg)
(ppm-‘1 1.70 x 10-s
for the array composed
21
35
0.31
number of important advantages when applied to the fabrication of sensors. One important feature of silicon technology is that it is a process which can be strictly controlled and which is capable of producing huge numbers of devices all with equal and reproducible characteristics. Furthermore silicon sensors tend to be rather expensive devices when only a small number of them is required. By making a sensor chip that contains an array of different sensing elements the problem could be alleviated. If the sensor chip contains n different sensors, n sensor types could be simultaneously fabricated at much lower cost, assuming compatibility with regard to the processing steps [ 11. Until now the complete characterization of every sensing element, even if experimental only, has been assumed. As a consequence we have focused our attention on the extraction of the information
by two SnO,(In)
sensors operating
L
Sarray
(PPm-‘)
(ppm-‘)
1.08 x 10-s
1.10 x 10-J
at temperatures
of 310 “C and
1.8%
from the sensor array. In practice this is not always possible. For instance silicon-based chemical sensors have displayed many serious problems with respect to reliability, reproducibility, stability. This is partly due to the lack of a complete theory able to explain the correlation between the superflcial ad-desorption processes and the electrical sensor response. In addition chemical sensors, and many other sensors, are highly dependent on both the fabrication and operating parameters, which are often not well known. However, an analysis limited to a certain state, e.g., concentrations equal to 100 ppm in Section 6, may sometimes be enough when linked to an approximate knowledge of parameters influence. Furthermore for specific applications it is possible to generalize the results for more than two sensors, employing a matrix description, to define new
518
TABLE 2. Uncertainty
IVw,lr,
ranges and precision
20 log R,
(PPm) 0.62
IO
ratios related to w, and w2 for the SnO,(In) [VW,
[Vw21:2
lEUT,,
(ppm)
(ppm)
1.96
2.22
-0.5
array 20 log R,
[vw*l.T~.y (ppm)
4
3.51
-
Fig. 1l. The sensitivity ratio map in polar form for the SnO,(In) sensor array (normalized modulus with respect to /JVz, /I). On the dash-dotted technological curves, related to differently doped sensors, the current parameter is the operating temperature.
Fig. 12. The precision ratio map for w, with respect to sensor 1 in polar form for the SnO,(In) sensor array. The same considerations of Fig. 1I are still valid.
functionals, to plot new charts or even to assume just a statistical characterization for the sensor array. Finally it is worth stressing the validity of the described theory and of its main results for future 8pplications involving more complicated array characteristics.
D’Ostilio and Dr A. Liparoti assistance.
Acknowledgements
We wish to thank Mr M. Collacciani graphical releases; and Dr P. Coccia,
for the Dr M.
for the technical
References 1 S. Middelhoek and S. A. Audet, Silicon Sensors, Academic, London, 1st edn., 1989, pp. 331-359. 2 W. GBpel, Solid-state chemical sensors: atomistic models and research trends, Sensors and Actuators, 16 (1989) 167- 193. 3 Sensors and Actuators A, 25-27 (1991-d Sensors and Actuators L?, 4 (1991). 4 G. Sberbeglieri, Fisica dei sensori per gas a semiconduttori, Fi.sica Tecnof. 12 (2) (1989) 88-111.
and Actuators
A, 32 (1992) 507-518
Pattern recognition Fabrizio
A. M. Davide
Dipartimento
di lngegneria
Arnaldo
D’Amico
Dipartimento
di Ingegneria
Elettrica,
Elettronica,
507
from sensor arrays: theoretical Universitd
de L’Aquila,
Universitci
di Roma
Poggio
di, Roio,
Tor Vergata,
67100 L’AquiIa
via 0. Raimondo,
considerations
(Italy)
Rome (Italy)
Abstract A general approach to the design and the evaluation of multisensor measurement systems based upon real non-selective reproducible sensors is proposed. Generalized definitions of both sensitivity and precision of sensor arrays are introduced. By means of new graphical tools the possibility is analyzed of obtaining improved selectivity, sensitivity and precision from sensor arrays. Quantitative design criteria, allowing the optimal coupling of individual sensors to form the array as well as their selectivity and operating conditions, are introduced, taking into account the influences of technological processes.
1. Introduction
Most sensors are designed to be sensitive to one parameter only. Unfortunately, most sensors are sensitive to a number of measurands and it is often rather difficult to utilize the output signal of such sensors. When a sensor is sensitive to an undesirable parameter we usually speak of the cross-sensitivity of the sensor. The solution could be to measure the undesirable parameter and to generate a signal that can be used to coufieract the cross-sensitivity [ 11. On the other hand in many applications it is of interest not only to know one parameter but also to simultaneously obtain the values of a number of other parameters as well, e.g., in biomedical applications. Sensor arrays and pattern recognition techniques have been recognized to be a promising answer to both these problems. Figure 1 shows the principle of pattern recognition systems in which non-selective, reproducible gas sensors with crosssensitivity are used to determine the concentrations of different gases [2]. Much effort has been spent on pattern recognition techniques and sensor technological improvement [3]. In this paper we want to focus our attention on a different aspect. There is no doubt that when many sensors are operating together, the ensemble shows different features with respect 0924-4247/92/$5.00
Fig. 1. Multisensor for gas sensing.
pattern-recognition-based
measurement
system
to the ones shown when the sensors are used alone. Here we are mainly interested in considering arrays composed of non-selective low-cost sensors, that otherwise cannot be used in a measurement system but in controlled conditions. The cross-correlations among the sensor responses can be exploited to give acceptable measurements, and non-negligible improvements of selectivity, sensitivity and noise rejection characteristics. In this paper a theoretical frame has been constructed in order to help the array designer. @ 1992 -
Elsevier Sequoia. All rights reserved
508
Section 2 introduces some useful notations. A discussion of sensitivity and designer aid charts will be presented in Section 3. Noise analysis and related charts will be presented in Section 4. Section 5 will take into consideration as an example a solid-state gas sensors array in order to better explain the previous results. Technological process influences are also discussed in this Section.
2. Reference model
Let us consider n sensors whose outputs are dependent on m physical or chemical quantities (state variables) characterizing the environment where the sensors are located. Let zi be the output of the sensor i and z the output vector, Wj the j-state variable and w the state vector. Generally speaking we have: ZI =
Z,(Wl,. *. , %a>
(1)
i.e., a transformation z: W,, +Z. Let W be the state space, W,, the open subset of W, which represents the environmental conditions under which the sensors will be operating, and Z the outputs spiv;e*. According to Fig. 1 the sensor array is characterized by the partial derivative matrix (au}, with au = az,/aw,. Usually we say that the sensor i is selective if laiil > layI with i #j, however in the present paper we will not make a hypothesis on selectivity. Restricting our discussion to the case m = n = 2, the {au} matrix referring to a certain w can be geometrically visualized: Fig. 2 shows the vectors Vz, and Vz2, with the CYand /I angles, which are directly related to the selectivities of z1 and z2. For instance, the relation aII/aIz = cot u shows how a high selectivity for zr implies a is small in modulus. Similarly, the relation aZ2/ a,, = tan(cr + B) says that selectivity z2 is maximal for j3 close. to (7r/2) - a.
*In order to avoid mathematical inconsistencies we will consider W and Z spaces which have adimensional components.
!dwl
a,, -M-W)
a,,
Fig. 2. z, and z2 gradients state w,.
2.1. Notations
dW
grad Zz
with related angles, referred to a certain
2.2. State retrieval If the sensor i is not selective with respect to wi
then zi is non-negligibly dependent on the other state variables. This means that knowledge of z, only does not allow the exact knowledge of w, . If we compose an array of low-selective sensors, we can try to retrieve the state w from the n low-selective measurements of the w components (i.e., z components) exploiting the cross-correlations between them. The achievement of this result requires that the application (eqn. ( 1)) has to be invertible for all W,,. As is well known, this inversion turns out to be possible under the following conditions: (a) z must be infinitely and continuously differentiable in W,,; (b) the Jacobian of the transformation must never be 0 in W,,. In the following we will require the geometrical condition equivalent to condition (b): the zi gradient field must never be parallel to the z2 one in W,, i.e., B(W) # 0, VWE w,,.
3. Design approach
At design level our purpose is the optimization of the sensor array with respect to some given features. Some peculiar characteristics of a given sensor array can be expressed by an opportune functional in the integral form: dw
i,j=l,2
(2)
where F can be equivalently considered to depend on a(w), P(w), (IVzi I(, and y(w) defined as Y(W)= I)Vz2II/J/V&1).
509
Consequently our strategy is to optimize the functional under suitable continuity and end-point conditions, warning that a direct optimization would give two ideal sensors, which are by definition infinitely selective. In practice it is necessary to start with a given real sensor, characterized by some degree of selectivity and sensitivity, and to choose the other one by optimizing the functional. In the following we will define two possible functionals related to array sensitivity and precision, and we will consider the array design choices comparing the performances of the array with those of the sensors used alone.
s
a-y
(4)
where dw/lldw 11 z is the unit vector parallel to dw. 4.2. Average
sensitivity
As the sensor array is dependent on the dw orientation (determined by angle rr as in Fig. 2), it is necessary to consider an average sensitivity with cr in the range [ - n, rc]:
4. Sensor array sensitivity
(5) The closed form for eqn. (5) is:
4.1. Definitions Let us consider the infinitesimal change of the state dw around wo. The output vector change can be expressed as:
(6) where
dz = (Vz, - dw, VzZ - dw) We will define the sensor array around w. by the following ratio:
sensitivity
f(p,v)
=i(sinarctan(’
yrif18+ ‘>
-sinarctan()‘y:i!P
where )I- llaand 11 - 11are two norms respectively b
applied on W and Z spaces. The best choice for the II * lla is II * (Im, i.e., lldz Ila = lldz Iloo= max{dzi}
‘>)
(7)
It is worth noting that S(fi, y) is periodic in 8, with period x, and symmetric with respect to B = 0, n/2, considering the validity of the following expressions: f(BY Y) =f( - /%r)
i = l-2
for the following two properties exclusive for. this norm: (1) as is well known, IIx Ilmc IIx II0 for each x and whatever the norm is in the second member, and this leads to the worst-case evaluation of sensor array sensitivity; (2) for a degenerate array, i.e., an array composed of two equal sensors, we have /dz IJrn= Idz, 1, as is expected considering that practically the sensor array turns out to behave as one sensor. The norm IIdzllb is Euclidean in character with the advantage that in the case of degeneration of the array toward z1 (i.e., ,0 = 0) the array sensitivity reduces to the directional derivative of z1 with respect to dw. In this case we have:
f(;-AY)=f(;+AY) 4.3. Sensitivity ratio First we notice that in the presence of only sensor 1 the average sensitivity, as the average of the directional derivative eqn. (4), is given by:
Now we can rewrite eqn. (6) as the sensitivity ratio:
s
( >
array =rf dj &I
+f(B,
I9
Since yf(& l/y) + f(p, y) > 1 whatever p is, as can be easily verified, we can conclude that a
510
second sensor produces a local average array sensitivity always greater than the one obtained by a single sensor (also if ideal). 4.4. Array design Let us suppose the existence of the following integral, which represents the average value of S array (see eqn. (2)):
<&ray >wop
= -j+
s
&ray (4 dw
OP WOP
where Mop is the W,, measure. According to eqn. (6) &ray) w,_, can be considered as a functional dependent on p(w), y(w), [IVz,II, whose maximum point can be calculated with the bound of considering a given real z1 . In other words this means the selection of a z2 sensor complementary to z, with respect to the sensitivity. Since the function involves w only implicitly the procedure will be trivial. It is easy to verify that Sarray is a growing function with respect to y and that if y goes to + cc whatever p is, we have:
It turns out that the sensitivity nearly increases in proportion to IIVz211when y + co. On the other hand jIVzlII is . b ounded by the available technology and usually is as big as IIVz, 11.Furthermore it is strongly related to the cost of the sensor and for these reasons it cannot be increased as one would like. So the unique choice is to consider Vz, free only as far as the orientation is concerned. In this case the maximum of (Sarray)W,p can be found by solving the equation &,,,(/I, -,~)/a@= 0, which is satisfied in p = 0 and /I = 7~12 for each y value. Finally the function is maximized by: /&lax(W)= n/2
(9)
This result represents the constraint on the z2 sensor from sensitivity considerations. According to eqn. (9) the maximal ratio, y changing, is Sarlay/Szl = (1 + y*) ‘I*. Figure 3 shows Sarrar/SZ1 for /I = 7c/2, and also for other p values*. As a
*Note the similarities with the Bode diagram function.
of a one zero transfer
0.5
_....
;.
:..
..;
.
..‘.
...........
j
2
2.5
3
._
0 0
0.5
i
1.5
3.5
4
88mn8
Fig. 3. Sensitivity ratio as a function of y for some p values. For every fixed y the ratio assumes its maximum value when /3 = n/2.
numerical example when y = 1 and B = 7r/2 we i.e., an improvement have Sarray(w) = S,,(W)*, factor of 3 dB compared to what we can get from sensor 1 only. It is of some interest to note that for =&ify>l, B = 0, SarrayequalsS,,ify
(10)
which gives a A/3 of 0.28 rad, when E = 1% and y = 1. 4.5. Sensitivity ratio map Figure 4 represents constant sensitivity ratio curves on the (/I, y) plane. Dashed lines, obtained by eqn. (lo), are the boundaries of the region with a sensitivity close to the maximum value (E = 1%) for every fixed y. This sensitivity ratio map can serve a double purpose. First the array sensitivity can be easily read from the map: for instance the choice of the two sensors with B = 100” and y =0.33 induces a sensitivity increase ranging from 3.5% to 7% compared to sensor 1 only. Secondly the map can be utilized for the array design. In fact let us consider a technological process P which is able to produce sensors, and
511
-: : :: :: : : -+-i :: :: : : f
(_
: ..:..;._
:
: : : :
: : : : :_ i
: j
: j
.i
.._:._.i._
! : : . . .. j
: : : :_j
1% ., .: _ : : : : . ..1..1._ : : : F i ; IO’
Fig. 4. Constant sensitivity ratio curves on the (& y) plane constituting a sensitivity ratio map. Relative increments are indicated for each curve: e.g., + 15% means Sarray = 1.15X,. On the dash-dotted technological curves, referred to three different pz values, the current parameter is p,. Dashed lines are explained in the text.
which is characterized by two parameters p, and p2. For each couple (p, pz) we will have a particular array corresponding to a point in the (/I, y) plane. Dash-dotted lines in Fig. 4 correspond to some parameter values and clearly indicate the influence of technological choices on the array sensitivity. It is possible to conclude that array design, referring to the sensitivity constraints, means optimization of SarraY towards p, and p2. A simple practical example will be given in Section 6.
5. Noise analysis 5.1.
Uncertainty
range related
to w1
Measurements performed by sensors are always affected by noise. We are interested in determining the uncertainty range on the w estimation due to the array and comparing it with the case of a measurement performed by sensors working alone. Let us consider the sensor outputs as a vectorial stochastic process z(w, t) and its time average as the z(w) application (eqn. ( 1)). Assuming ergodicity to be valid let us consider in the Z space a square domain
of
Qz r side length, centred on z(wo), and containing with high probability (e.g., 99%) the random variable z(wo, to), with to the array sampling instant. As a consequence, the uncertainty range for the z components at the assigned probability is about large r and centred on zi(w,,). Figure 5 shows QZ on the t plane and its mapping Qw on the w plane. The uncertainty range Awl, related to the estimate of w,, will be the orthogonal projection of Qw on the w1 axis. Starting from a QZ definition, by means of the inverse transformation z-’ it is possible to determine the equations describing Qw and also to consider only the component related to the w, axis. Let us assume r to be sufficiently small. In this hypothesis z(w) can be locally linearized and easily inverted. Therefore the Qw domain is clearly a parallelogram and its projection on the w1 axis coincides with that one of a diagonal whose projection is larger. Indicating the uncertainty range by [Awl]_,,, we have: (11) where J is the Jacobian
of z(w).
512
Fig. 5. The corresponding Q, and Q, domains Q, is consequently determined by L-‘(Z).
on the L and w planes. A square shape has been arbitrarily
5.2. Precision ratio
If the sensor z, is utilized alone in order to obtain the w1 measurement, we firstly need to know the value of the other state variable either through physical information (e.g., state changes) or measures taken in a different way. If in eqn. ( 1) we introduce the known wZOvalue, the result is that the uncertainty on z1 is reflected on w1 as follows:
(12)
[Awl,, =i+
Iall I Let us consider the ratio R,(w) between eqns. (11) and (12), namely the Aw, uncertainty ranges for both the array and the sensor 1 working alone, expressed by the quantities of Fig. 2:
RI(W)=
Isin(a + 8) 1+ jj lsin tcI) lcos GL l
(
IsinBI
(13)
We will call it the precision ratio (which is less than 1 if the array has a better precision) for w1 with respect to sensor 1. From eqn. ( 13) it can be seen that RI is periodic in M and fi, with period rc, then without any lack of generality we will assume CIand /3 in [0, rc]. If we evaluate eqn. ( 13) with p = rc - c( we will obtain R,(w) = lcos al/r, which is less than 1 for y > lcos CI]. The interesting result is that using a second sensor supporting zi we can reach a precision
chosen for Q, while the shape for
for w1 that is greater than the one associated sensor 1 alone.
to
5.3. Array design Let us suppose the existence of the following integral which represents the average value of R, on W,, (see eqn. (2)):
s
OP Ww
R,(w) dw
According to eqn. (13), (R,)WOP can be considered as a functional depending on a(w), p(w), y(w) whose minimum point can be calculated within boundary of a given real z, . As already done for sensitivity, we will look for sensor 2 complementary to a fixed sensor 1 from a precision point of view. In order to minimize (R,)wOp it is not possible to use the Euler-Lagrange equations. In fact, the equation referred to y, dRl(cr, /?, y)/ay = 0, is never satisfied because RI always decreases with y, whatever is the fixed /? value. As to the equation referred to p, it is inadequate because the integrand does not comply with the regularity hypothesis. We will consider, as we did in the case of sensitivity, the Vz2 orientation free and its modulus constant. We will search for the R, absolute minimum with respect to /3, considering y assigned.
513
01 0
20
40
60
80 beta
iDcl
i20
140
160
IN.
[degrsesl
Fig. 6. Precision ratio as a function of p for some y values and c( = 21”. For some values of the (8, y) couple the ratio is less than 1, i.e., the array shows a better precision with respect to W, than that shown by sensor 1 alone
Figure 6 shows a few sections of RI with y constant, a = 21” and fi E [0, ~1. It is clear that the absolute minimum is in /I = 71- c1when y > lcos CC] and it is coincident with the relative minimum when y -C (cos CC].In order to calculate the relative minimum of RI with respect to /I, we need to calculate its derivative and study the relative roots. The roots coincide with the solutions of 1 + y cos j3 = 0, i.e., Bmin= arccos( - 7)
if CLin [
0, n 2
1
and y -C lcos a] jImi,=71-arccos(-~)
if a in
ll
-,7c [2
1
(14)
and y c lcos aI Finally we find that the RI absolute minimum, y changing, is: _ lcos 4 Y Ri (G Bmin9Y)
when y > lcos a] (15)
= (Ices tx]+ [sin u]((l/y’)
- 1)“2)]cos aI
when y < Jcos a I
The case y > 1~0s~11is trivial because when /I = x - LXwe need Vz2 laid down on the w1 axis, which means z2 infinitely selective on wl. 5.4. Precision ratio map Figure 7 shows the R, constant curves for CI= 21” on the (/.I, y) plane. The precision increments and the locus /I = /Imin(y) are also indicated there. The hatched region is the set of the arrays which have a better precision on w,. As to the technological choices, remarks already made for sensitivity are still valid. The problem of optimizing the array with respect to the precision on w, is precisely symmetric when referred to w2. Figure 8 shows the precision ratio map for w2. The hatched H region indicates a better precision on w2 for the array: it is clear that we cannot simultaneously reach a precision improvement with respect to both w, and w2 as the relative regions do not have intersections. The trade-off choice which makes no distinction between the state variables is represented in the Figure by the T region. This choice shows a precision worsening of about 1.55 dB when y = 0.84 and /I = 105” (we must not forget that these numbers are meaningful when ci = 21”).
Fig, 7. Constant precision ratio curves referred to wl on the (p, y) plane, with OL= 21”, constituting a precision ratio map. Precision relative increments are expressed as 20 log,, R, ‘. In the hatched region arrays show higher precision. The dashed line shows the locus B = &,,i.(y),
Fig. 8. As in Fig. 7 but the precision ratio is referred to w,. Hatched a trade-off between W, and w2 precision requirements.
6. Analysis
of a real case
6.1. Solid-state gas sensors It is well known that SnOZ thin films doped by indium, through thermal evaporating or RF sput-
region H corresponds
to better array precision on w2_ Region T concerns
tering, are sensitive to many gases, such as NOz, Hz, CO, SO,. The response of this material to these gases is temperature dependent. As an example we will consider two identical SnO,(In) sensors (taken from ref. 4) operating at different temperatures,
515
such as 310 “C (sensor 2) and 450 “C (sensor l), and an ambient where CO and SO1 gases are present with w1 and w2 as concentration values respectively (ppm). Assuming that z, is the relative variation of conductivity of the i sensor we get: z, = o.15w,“.28 + o.11w2°.20 i ~2 = 0.022~1’.~’ + O.O66~2’.~~ The above equations represent a very simplified model, which is assumed to be valid with concentrations ranging between 10 and 1000 ppm. Figure 9 shows the characteristics of the two sensors. The limited selectivity suggests the use of a sensor array, which is theoretically possible in virtue of the validity of inversion conditions. Let us evaluate sensitivity and precision at a concentration of 100 ppm for both gases. 6.2. Sensitivity evaluation Figure 10 reports the non-averaged sensitivities of the two separately utilized sensors, as functions of the dtv orientation and their envelope that represents the non-averaged array sensitivity. The envelope average value is greater than the averages of the two component sensor sensitivities, as expected according to Section 4.3. The inversion condition p # 0 implies that the nonaveraged array sensitivity is never 0 for each 0 value; on the contrary this is not true for the components sensitivities. Table 1 shows that in this case the sensitivity only improves by 1.8%; this is mainly due to the low y value and to a p value too far from 90”, as can be directly deduced by the sensitivity map in Fig. 4. Figure 11 shows the sensitivity ratio map in a polar diagram form. Curves with the same sensitivity are ellipses with major axes parallel to Vz,. Furthermore the minimum of IIV.QIIwith a fixedarray sensitivity value would correspond to p = 90”. In our example this is not feasible because both gradients are physically forced to lie on the same quadrant. For the dash-dotted curves, referring to two different dopings utilized for the fabrication of z2, the current parameter is the operating temperature of z2. At 310 “C the representative vectors of Vz, are A and A ‘, while B and B’ refer to 227 “C. More precisely, the curve which A and B belong to refers to arrays composed by two identical sensors operating at different tempera-
tures. On the contrary the other curve refers to arrays composed of sensor 1 already introduced, and of a differently doped sensor 2, operating at a different temperature too. As the technological curves are nearly parallel to the constant sensitivity ratio curves it turns out that there are not sensitivity variations between A and B arrays or A ’ and B’ ones. 6.3. Precision evaluation Let us suppose that both sensors are supplied by a 1 uA current and that the output noise causes maximal relative conductivity fluctuations of 1 x 10e3, then we obtain the values shown on Table 2. These values correspond to 2% and 3.5% errors for the CO and SO2 concentrations. From the polar map referred to in Fig. 12, we can read the same worsening of 10 dB on w, precision for both A and B cases. The conclusion is that only by changing the temperature conditions can we obtain from two identical sensors the possibility to simultaneously measure the CO and SO2 concentrations, but the sensitivity and the precision of the single sensors are not likely to be improved. This can be understood by looking at the compensation of /? and y variations on the technological curves. The reason for this compensation will not be taken into consideration in this paper and will represent a subject for further investigations. The doping variations, on the contrary, bring a certain improvement, as we can see from the comparison between A and A’ arrays, for instance. Nevertheless in this case we have an increased difficulty in using a different process for the fabrication of the sensors.
7. Conclusions
The above considerations may be useful for many practical applications: -Reduction in disturbance. Silicon pressure sensors, for instance, are sensitive to the temperature, and magnetic field sensors to the pressure. In order to reduce the cross-sensitivity it is convenient to consider a second sensor of a poor quality which plays a slave role. In this case the precision with respect to w, only will be optimized, while any other feature with respect to w2 will be neglected.
516
: 0.A
(a)
(b) Fig. 9. Response characteristics for a SnO,(In) concentrations, i.e., w, and w2.
thin film sensor when operating
-Multisensing. It is the case taken into consideration in this paper, where no privilege was given to any state variable. For instance it is a typical problem in the chemical-sensing field. -Increased W,, domain. This problem too can be
at 450 “C and 310 “C respectively, as functions
of SO2 and CO
expressed from the function optimization point of view, and studied independently of the particular sensors utilized. Some of the obtained results can be eminently used with silicon planar technology, offering a
517
0.2
.,....,.
j
.,..........,.
Z?j\ .i I
!
’
.\
Qi
/-
i
:
;: , ,:
t
““‘7
’ ./
Sioma+Alpha Ulegreesl
Fig. 10. Sensitivities (not averaged) of sensors working alone, dashed and dotted lines, as functions of dw orientation (see Fig. 2): the envelope (solid line) of single sensors sensitivities represents the array sensitivity (not averaged) according to the I(* 11.definition (see Section 4.1).
TABLE 1. Sensitivities and other related quantities 450 “C, respectively
IIVZI II
B
Y
(deg)
(ppm-‘1 1.70 x 10-s
for the array composed
21
35
0.31
number of important advantages when applied to the fabrication of sensors. One important feature of silicon technology is that it is a process which can be strictly controlled and which is capable of producing huge numbers of devices all with equal and reproducible characteristics. Furthermore silicon sensors tend to be rather expensive devices when only a small number of them is required. By making a sensor chip that contains an array of different sensing elements the problem could be alleviated. If the sensor chip contains n different sensors, n sensor types could be simultaneously fabricated at much lower cost, assuming compatibility with regard to the processing steps [ 11. Until now the complete characterization of every sensing element, even if experimental only, has been assumed. As a consequence we have focused our attention on the extraction of the information
by two SnO,(In)
sensors operating
L
Sarray
(PPm-‘)
(ppm-‘)
1.08 x 10-s
1.10 x 10-J
at temperatures
of 310 “C and
1.8%
from the sensor array. In practice this is not always possible. For instance silicon-based chemical sensors have displayed many serious problems with respect to reliability, reproducibility, stability. This is partly due to the lack of a complete theory able to explain the correlation between the superflcial ad-desorption processes and the electrical sensor response. In addition chemical sensors, and many other sensors, are highly dependent on both the fabrication and operating parameters, which are often not well known. However, an analysis limited to a certain state, e.g., concentrations equal to 100 ppm in Section 6, may sometimes be enough when linked to an approximate knowledge of parameters influence. Furthermore for specific applications it is possible to generalize the results for more than two sensors, employing a matrix description, to define new
518
TABLE 2. Uncertainty
IVw,lr,
ranges and precision
20 log R,
(PPm) 0.62
IO
ratios related to w, and w2 for the SnO,(In) [VW,
[Vw21:2
lEUT,,
(ppm)
(ppm)
1.96
2.22
-0.5
array 20 log R,
[vw*l.T~.y (ppm)
4
3.51
-
Fig. 1l. The sensitivity ratio map in polar form for the SnO,(In) sensor array (normalized modulus with respect to /JVz, /I). On the dash-dotted technological curves, related to differently doped sensors, the current parameter is the operating temperature.
Fig. 12. The precision ratio map for w, with respect to sensor 1 in polar form for the SnO,(In) sensor array. The same considerations of Fig. 1I are still valid.
functionals, to plot new charts or even to assume just a statistical characterization for the sensor array. Finally it is worth stressing the validity of the described theory and of its main results for future 8pplications involving more complicated array characteristics.
D’Ostilio and Dr A. Liparoti assistance.
Acknowledgements
We wish to thank Mr M. Collacciani graphical releases; and Dr P. Coccia,
for the Dr M.
for the technical
References 1 S. Middelhoek and S. A. Audet, Silicon Sensors, Academic, London, 1st edn., 1989, pp. 331-359. 2 W. GBpel, Solid-state chemical sensors: atomistic models and research trends, Sensors and Actuators, 16 (1989) 167- 193. 3 Sensors and Actuators A, 25-27 (1991-d Sensors and Actuators L?, 4 (1991). 4 G. Sberbeglieri, Fisica dei sensori per gas a semiconduttori, Fi.sica Tecnof. 12 (2) (1989) 88-111.