- Email: [email protected]
A new method to analyse signal transients in chemical sensors
308
Sensors and Actuators L?, 18-19 (1994) 308-312
A new method to analyse signal transients in chemical sensors J. Samitier, J.M. Lopez-Villegas, S. Marco, L. CBmara, A. Pardo, 0. Ruiz and J.R. Morante LCMM, Depariamento F&a Aplicada i Electrdnica, Universitatde Barcelona, Avenida Diagonal 645-647, 08028-BarceloM (Spain)
Abstract Many chemical sensors exhibit exponentially decaying responses to a step change in the concentration. The determination of the time constant can greatly improve the specificityof the sensor. A spectroscopic method to determine the time-constant distribution is theoretically presented and its application to the analysis of the response to alcohol sensors is illustrated.
1. Iatroduction Currently, great efforts are being devoted to the design and development of chemical sensors for gas specific detection. In order to improve the performance of a sensor system in the problem of multi-component analysis different principles can be used [I]: - An array of highly selective sensors, as many as components are to be detected in the mixture. - An array of sensors with partial overlapping of the selectivity. In this case, it is necessary to rely on pattern recognition methods. - Increase the speci&ity of the sensor by dynamic measurements. Then, a new set of parameters describing the sensor time response improves the selectivity of the sensor. Commonly, only the steady-state sensor response is used for the determination of the concentration of different species. However, the transient signal sometimes contains relevant information about the gas mixture, as reported by different authors. The transient is generated using an isothermal step change in the gas concentration or modulating the sensor response by changing the temperature of the reaction chamber [2-4]. For instance, in amperometric sensors based in a catalytic microreactor, the transient signal can be described by a sum of exponential decays [5], the time constants of which are selective for a specific gas. Then, the concentration of this gas can be extracted from the amplitude of the corresponding exponential term. Besides an improvement in the specificity of the sensor, the determination of the time constants is very important because it permits a better knowledge of the chemical and physical mechanisms involved in the sensor functionality. For example, in gas sensors based on
0925-4005/94/$7.00 0 1994 Elsevier
SSDI 0925-4005(93)01215-P
Sequoia. All
rights reserved
polycrystalline metal oxides, although the detailed conduction mechanism is not well known, some models, based on barrier limited conduction [6], show that the conductivity transient presents an exponential behaviour which can be directly related to the energetic localization of the grain boundary trap states [7]. This work is centred on the analysis of multi-exponential transient signals, as a method to improve the selectivity of chemical sensors in the analysis of gas mixtures. The proposed method, which we call METS (multi-exponential transient spectroscopy), is based on a numerical multi-differentiation of the transient and can be easily implemented in a discrete-time processing system. Noise and resolution power will be carefully analysed. Finally, as an example, we will apply METS to the analysis of the response of an alcohol sensor based on an electrochemical fuel cell.
2. Analysis of multi-exponentialdecays 2.1. IfltlodzKtion In very different physical and chemical processes the measured signal can be considered as a linear combination of exponentially decaying components: for example, gas relaxation kinetics [8], fluorescence [9] and radioactivity [lo], sedimentation [ll], magnetic nuclear resonance [12], defects in semiconductors [13, 141,etc. Accordingly, many authors have reported methods of decomposition of multi-component decays based on general mathematical techniques such as Fourier transforms [15], Laplace transform inversion [16] and nonlinear fitting [17]. The key problem is that exponentials, in contrast with sinusoidals, are not an orthogonal base of functions on the real axis. Thus, the
309
d~~ibution function of time insists cannot be unique if we only determine the transient during a finite time and with finite precision. A complete discussion about the rel&ve performances of these methods compared to METS will be published elsewhere.
0.8
2.2. METS: theoretical develqment A general exponential decay can be written in terms of a ~nst~t-tie ~s~ibution:
0) We define the first-order METS signal as
The function G(r) contains the amplitude for an exponential decay of time constant T. Making the variable change y =ln(t) and z =ln(r), which is equivalent to changing the time axis from linear to logarithmic, then METS,(t)==A j i G(T) exp( -
i) dT
becomes +m exp(y-z- exp(vz))G(exp(z)) exp(.z)dz l -0D
(4) If we define the functions: h(v) = expcV- exp(Y)) TG(r) = exp(r)G(exp(r))
(5)
the METS, signal is the convolution product of these functions: ~E~S,~~~~~~G~~~~~-~~ -II)
dz=ATG(z)!$y)
TABLE l.Evolutionofthemain Order
1 2 3 4 5
0
MET&(v)=A
Fig. 1. Functions ~~~~(f)). The time scales have been modified in each case. to keep the maxima in the same position.
(6)
parametersfor
Peak position (time)
Intensity
1 2 3 4 5
0.368 0.541 3.65 1.34 21.04
tbeh&)functions
(decades)
MY)==P(~Y- expW
1.042 0.737 0.598 0.516 0.461
(7)
These functions (Fig. 1) present a peak at y=ln(n) and they have a peak value of h_=n”/exp( -n) which increases. The resolution power of this method is directly related to the FWHM of the peak. Table 1 shows how the FWHM diminishes as the order of the METS signal is increased and, in consequence, how the spectral power increases. Now we can define the n-order METS signal as ~~S~~~
=A’s ~G~~~~~ -z) dr =~~G(z~~~)
(8)
-w
Figure 1 shows the shape of the h(y) function. The peak is located at y=O (t= 1) and has a maximum intensity of l/e. A special feature of this method resides on the constant FWHM expressed in decades of time. Theoretically, this FWHM has a value of 1.062 decades. From this expression it is clear that if the h(y) signal were substituted by a Dirac impulse then the METS signal would give us the time constant ~t~~ution. 2.3. Genemtization to order n In order to reach this objective, we substitute the h(v) function in the convolution product by
Differentiation of the convolution product allows us to obtain a recurrent formula for the METS signals:
(9) This formula facilitates the calculation of an n-order signal from eleventh data. From a purely mathematical point of view, it would probably be desirable to change the convolution fmrction in order to prevent peak shii as the order increases. This can be done using the modified function:
310
g&9= exp(v- exp( - nr)>
(10)
However, this new function does not keep an easy recurrent formula for the METS signal and, in consequence, it is not worth the substitution. 2.4. Analysti of a discrete distribution of time constants To illustrate the power and capacity of the METS signal we will consider now the analysis of a signal composed by the addition of different exponential decays: W) = $rA1 exp( - f/r,)
2.5. Discrete-time processing of the METS signal The availability of low cost, high performance DSP processors, and microprocessors in general, permits us to include a great signal processing power in a smallvolume instrumentation system. In the following, we will discuss the discrete-time implementation of the METS signal. A very convenient way to calculate the numerical differentiation involved in eqn. (2) is to arrange, by interpolation and averaging, the sample points (obtained from the continuous transient by means of an A/D converter) in a geometrical sequence of ratio k. Then, the first-order METS signal may be expressed as
(11) Z(tklR) -Z
The n-order METS signal will be given by
It can be observed that each discrete time constant contributes a peak to the final spectrum, the maxima of which are located at y = ln(nrJ. In order to compare the results of different order METS signals, it is interesting to shift backwards the results in the y-axis by In(n) in order to keep the maxima in the same position for the different orders. As we have already commented, the resolution power of this method is directly related to the FWHM of the peak. This is even clearer in Fig. 2 where the spectra corresponding to different orders have been simulated for a transient composed of two exponential decays, the time constants of which are separated by a factor tive. The time scales have been normalized in each case to keep the peaks in the same position.
METS
(a.u.)
METS&
k) =
(13)
In k
In the limit k+l, this signal converges to the exact differentiation, eqn. (2). The discretization of the signal involves a modification of the signal shape that depends on the value of the parameter k. It can be easily calculated that the position and maximum of the peak (for a signal containing a unique exponential term) have been shifted to k’9n k tInax=k_lc max(METS,(t, k))=A, ‘2
k-kl(C-l)
(14)
It can be shown that the FWHM of the peak increases as k increases. Thus, to achieve the maximum resolution power a k value near unity should be chosen. On the other hand, the METS signal is more sensitive to signal noise as k approaches unity. Thus, a balance between both effects must be found. In most practical cases k=2l”= 1.26 is a good choice. In order to define the n-order METS signal considering the discretization of the sensor signal we define:
&
l)f~(tknn-i,
S.(t, k) E
(15)
ln”k
Then the METS,, signal is METS,(t, k) = (- l)“S,(t, k) n-1 + ,F; (- lY-ian,n-iMETSn-&, 0
1
2
k)
(16)
where the anmmcan be calculated from
log (t) Fig. 2. Simulated METS,, signals for a transient containing hvo exponential decays, with time constants separated by a factor five: (a) n-l, (b) n-2, (c) n=3, (d) n=4.
an.m=~~,-l,,+~,-l,m-l a 1.1--1
(17)
311
- The experimental implementation of the method is very easy.
3. Analysis of experimental transients
The alcohol sensor is based on an electrochemical fuel cell using H,SO, as an electrolyte absorbed by a porous PVC and Pt electrodes that serve also as a catalytic material. The alcohol sensor response to a step change in the gas concentration can be empirically described by the sum of two exponential decays: Z(t) =A[exp( - t/q) - exp( - fh)]
t Fig. 3. Influence of the discretization factor k in the position and intensity of the peaks for the convolution functions $(v).
Figure 3 shows how the peak position and maxima are altered due to numerical differentiation for several k and n values. If a real time processing of the n-order METS signal is going to be implemented using the above scheme, it is worth noticing that there will be, at least, a delay of (k”- I4 - 1)r due to the necessary knowledge of some points in advance to calculate the successive differentiations.
2.6. Advantages of the METS analysis The METS analysis shares the advantages of the spectroscopies: - It permits us to deal with discrete and continuous time-constant distributions. This feature isvery adequate in the analysis of decays of unknown nature. - The diierent exponential components of the decay are determined simultaneously. This is advantageous over iterative methods based on the sequential extraction of the time constants. In the later methods the errors made on the first parameters influence the subsequent results. - The appearance of wider peaks, compared to the FWHM for an exponential component, suggests the existence of more than one component that the signal has not been able to resolve. Moreover, in the METS, the user can adjust the spectroscopic power by changing the signal order. Other advantages are summarized in the following: - The required computation power is much less than that required for alternative methods based on the calculation of FFTs.
(18)
with TV> TV.Typical transients are shown in Fig. 4. Although the fastest time constant is probably influenced by the non-ideal gas injection, the slower exponential permits the application of the presented method. Three different alcohols (ethanol, methanol and 2propanol) were used in the study. First, the individual response of the sensor to each gas was recorded and analysed by METS. It was observed that the exponential amplitude was directly proportional to the gas concentration, while the time constant was specific to a certain gas. The variations of time constant with gas concentration were negligible, although small shifts were found in the comparison among different devices. The selectivity of this parameter to the gas type should permit the identification and quantification of a gas mixture of different alcohols. The resolution of the different time constants will very much depend on their closeness and the signal-to-noise ratio. For example, Fig. 5 shows the first- and third-order METS
‘.2I 1.0
-
-
ethanol 2-proponol methanol
-0.6 3 -
0.4
0.2
0.0
0
20
Fig. 4. Normalized transient
signals of the alcohol sensor.
312
-0.62 j i
\
-0.78 -I
-0.86
!
-0.94
-
P 2
-1.02 -
-1.10
11
10
time
(set)
Fig. 5. Normalized first- and third-order METS signals for a mixture of methanol (5.8 ppm) and Z-propanol (0.9 ppm).
signals for a mixture of methanol and 2-propanol. While the first-order signal presents a peak much wider than that corresponding to a single exponential decay, the third-order signal is able to separate both contributions.
4. Conclusions A novel method, METS, to find the time-constant distribution in an exponential decay has been presented. Its features make METS a valuable alternative for the analysis of chemical sensor transient responses. A practical example in alcohol sensors demonstrates its usefulness and versatility.
References 1 S. Vaihmger and W. Gopel, Multi-component anaIysis in chemical sensing, in W. Gopel, J. Hesse and J.N. Zemel (eds.), Sensors: A Coqmhensive Survey, Vol. II, VCH, Weinheim, 1991, pp. 192-234.
2 S. Wiodek, K. Colbow and F. Consadori, Signal shape analysis of a thermally cycled tin-oxide gas sensor, Senses ondActua:ors B, 3 (1991) &68. 3 G. Jordan Maclay, J.R. Stetter and S. Christesen, Use of time-dependent chemical sensor signals for selective identification, Senrors and Actuators, 20 (1989) 277-285. 4 T. Otawaga and J.R. Stetter, A chemical concentration modulation sensor for selective detection of air-borne chemicals, Sensors and Actuators, II (1987) 251-264. 5 S. Vaihinger, W. Gopel and J.R. Stetter, Detection of halogenated and other hydrocarbons in air: response functions of catalyst/electrochemical sensor systems, Sermxs and Acmtors B, 4 (1991) 337-343. 6 J.W. Gardner, Detection of vapours and odours from a multisensor array using pattern recognition: Part. 1. Principal component and cluster analysis, Sensors und Actuators B, 4 (1991) 109-115. 7 N. Butta, M. Mell and S. Pizzini, Influence of surface parameters and doping on the selectivity and on the response times of tin-oxide sensors, Sensors and Achutots B, 2 (1990) 151-161. 8 S.M. Pizer, A.B. Ashare, A.B. Callahan and G.L. Borovmel, Fourier transform analysis on tracer data, in F. Heinmets (ed.), Concepts and Mod& ofBiom&ematic~, Marcel Dekker, New York, 1969. 9 I. Iscnberg, R.D. Dyson and R. Hanson, Studies on the analysis of fluorescence decay data by the method of moments, 3. Biophys., 13 (1973) 1090-1097. 10 S.L. Laiken and M.P. Printz, Kinetic class analysis of hydrogen _ exchange data, Biochemky, 9 (1970) 1547-1553. 11 R.D. Dyson and I. Isenberg, analysis of exponential curves by a method of moments with special attention to sedimcntation equilibrium and fluorescence decay, Biochcmisy, 10 (1971) 3233-3242. 12 M.R. Smith, S. Cohn-Sfetcu and I-LA. Buckmaster, Decomposition of multicomponent exponential decays by spectra1 analytic techniques, Technometrics, 28 (1976) 467-482. 13 J. Samitier, J.R. Moran& L. Giraudet and S. Gourrier, Optical behavior of the U band in relation to EL2 and EL6 levels in boron-implanted GaAs, Appi. Phys. Len., # (1986) 113b1140. 14 J. Samitier, S. Marco, A. Perez-Rodriguez, J.R. Morante, P. Boher and M. Renaud, Anomalous optical and electrical recovery process of photoquenched EL2 defect produced by oxygen and boron implantation in GaAs, I. Appl. Phys, 71 (1992) 252-259. 15 D.G. Gardner, C. Gardner, G. Laush and W.W. Meinke, Method for the analysis of multicomponent exponential decay curves, I. C&n. Phys., 31 (1959) 978-986. 16 C. Lanczos,AppkedAna~s& Prentice-Hall, Englewwd Cliffs, NJ, 1964. 17 J.W. Evans, W.B. Gregg and R.J. Leveque, On least squares exponential sum approximation with positive coefficients, Math.comput,34 (1980) 203-211.
Sensors and Actuators L?, 18-19 (1994) 308-312
A new method to analyse signal transients in chemical sensors J. Samitier, J.M. Lopez-Villegas, S. Marco, L. CBmara, A. Pardo, 0. Ruiz and J.R. Morante LCMM, Depariamento F&a Aplicada i Electrdnica, Universitatde Barcelona, Avenida Diagonal 645-647, 08028-BarceloM (Spain)
Abstract Many chemical sensors exhibit exponentially decaying responses to a step change in the concentration. The determination of the time constant can greatly improve the specificityof the sensor. A spectroscopic method to determine the time-constant distribution is theoretically presented and its application to the analysis of the response to alcohol sensors is illustrated.
1. Iatroduction Currently, great efforts are being devoted to the design and development of chemical sensors for gas specific detection. In order to improve the performance of a sensor system in the problem of multi-component analysis different principles can be used [I]: - An array of highly selective sensors, as many as components are to be detected in the mixture. - An array of sensors with partial overlapping of the selectivity. In this case, it is necessary to rely on pattern recognition methods. - Increase the speci&ity of the sensor by dynamic measurements. Then, a new set of parameters describing the sensor time response improves the selectivity of the sensor. Commonly, only the steady-state sensor response is used for the determination of the concentration of different species. However, the transient signal sometimes contains relevant information about the gas mixture, as reported by different authors. The transient is generated using an isothermal step change in the gas concentration or modulating the sensor response by changing the temperature of the reaction chamber [2-4]. For instance, in amperometric sensors based in a catalytic microreactor, the transient signal can be described by a sum of exponential decays [5], the time constants of which are selective for a specific gas. Then, the concentration of this gas can be extracted from the amplitude of the corresponding exponential term. Besides an improvement in the specificity of the sensor, the determination of the time constants is very important because it permits a better knowledge of the chemical and physical mechanisms involved in the sensor functionality. For example, in gas sensors based on
0925-4005/94/$7.00 0 1994 Elsevier
SSDI 0925-4005(93)01215-P
Sequoia. All
rights reserved
polycrystalline metal oxides, although the detailed conduction mechanism is not well known, some models, based on barrier limited conduction [6], show that the conductivity transient presents an exponential behaviour which can be directly related to the energetic localization of the grain boundary trap states [7]. This work is centred on the analysis of multi-exponential transient signals, as a method to improve the selectivity of chemical sensors in the analysis of gas mixtures. The proposed method, which we call METS (multi-exponential transient spectroscopy), is based on a numerical multi-differentiation of the transient and can be easily implemented in a discrete-time processing system. Noise and resolution power will be carefully analysed. Finally, as an example, we will apply METS to the analysis of the response of an alcohol sensor based on an electrochemical fuel cell.
2. Analysis of multi-exponentialdecays 2.1. IfltlodzKtion In very different physical and chemical processes the measured signal can be considered as a linear combination of exponentially decaying components: for example, gas relaxation kinetics [8], fluorescence [9] and radioactivity [lo], sedimentation [ll], magnetic nuclear resonance [12], defects in semiconductors [13, 141,etc. Accordingly, many authors have reported methods of decomposition of multi-component decays based on general mathematical techniques such as Fourier transforms [15], Laplace transform inversion [16] and nonlinear fitting [17]. The key problem is that exponentials, in contrast with sinusoidals, are not an orthogonal base of functions on the real axis. Thus, the
309
d~~ibution function of time insists cannot be unique if we only determine the transient during a finite time and with finite precision. A complete discussion about the rel&ve performances of these methods compared to METS will be published elsewhere.
0.8
2.2. METS: theoretical develqment A general exponential decay can be written in terms of a ~nst~t-tie ~s~ibution:
0) We define the first-order METS signal as
The function G(r) contains the amplitude for an exponential decay of time constant T. Making the variable change y =ln(t) and z =ln(r), which is equivalent to changing the time axis from linear to logarithmic, then METS,(t)==A j i G(T) exp( -
i) dT
becomes +m exp(y-z- exp(vz))G(exp(z)) exp(.z)dz l -0D
(4) If we define the functions: h(v) = expcV- exp(Y)) TG(r) = exp(r)G(exp(r))
(5)
the METS, signal is the convolution product of these functions: ~E~S,~~~~~~G~~~~~-~~ -II)
dz=ATG(z)!$y)
TABLE l.Evolutionofthemain Order
1 2 3 4 5
0
MET&(v)=A
Fig. 1. Functions ~~~~(f)). The time scales have been modified in each case. to keep the maxima in the same position.
(6)
parametersfor
Peak position (time)
Intensity
1 2 3 4 5
0.368 0.541 3.65 1.34 21.04
tbeh&)functions
(decades)
MY)==P(~Y- expW
1.042 0.737 0.598 0.516 0.461
(7)
These functions (Fig. 1) present a peak at y=ln(n) and they have a peak value of h_=n”/exp( -n) which increases. The resolution power of this method is directly related to the FWHM of the peak. Table 1 shows how the FWHM diminishes as the order of the METS signal is increased and, in consequence, how the spectral power increases. Now we can define the n-order METS signal as ~~S~~~
=A’s ~G~~~~~ -z) dr =~~G(z~~~)
(8)
-w
Figure 1 shows the shape of the h(y) function. The peak is located at y=O (t= 1) and has a maximum intensity of l/e. A special feature of this method resides on the constant FWHM expressed in decades of time. Theoretically, this FWHM has a value of 1.062 decades. From this expression it is clear that if the h(y) signal were substituted by a Dirac impulse then the METS signal would give us the time constant ~t~~ution. 2.3. Genemtization to order n In order to reach this objective, we substitute the h(v) function in the convolution product by
Differentiation of the convolution product allows us to obtain a recurrent formula for the METS signals:
(9) This formula facilitates the calculation of an n-order signal from eleventh data. From a purely mathematical point of view, it would probably be desirable to change the convolution fmrction in order to prevent peak shii as the order increases. This can be done using the modified function:
310
g&9= exp(v- exp( - nr)>
(10)
However, this new function does not keep an easy recurrent formula for the METS signal and, in consequence, it is not worth the substitution. 2.4. Analysti of a discrete distribution of time constants To illustrate the power and capacity of the METS signal we will consider now the analysis of a signal composed by the addition of different exponential decays: W) = $rA1 exp( - f/r,)
2.5. Discrete-time processing of the METS signal The availability of low cost, high performance DSP processors, and microprocessors in general, permits us to include a great signal processing power in a smallvolume instrumentation system. In the following, we will discuss the discrete-time implementation of the METS signal. A very convenient way to calculate the numerical differentiation involved in eqn. (2) is to arrange, by interpolation and averaging, the sample points (obtained from the continuous transient by means of an A/D converter) in a geometrical sequence of ratio k. Then, the first-order METS signal may be expressed as
(11) Z(tklR) -Z
The n-order METS signal will be given by
It can be observed that each discrete time constant contributes a peak to the final spectrum, the maxima of which are located at y = ln(nrJ. In order to compare the results of different order METS signals, it is interesting to shift backwards the results in the y-axis by In(n) in order to keep the maxima in the same position for the different orders. As we have already commented, the resolution power of this method is directly related to the FWHM of the peak. This is even clearer in Fig. 2 where the spectra corresponding to different orders have been simulated for a transient composed of two exponential decays, the time constants of which are separated by a factor tive. The time scales have been normalized in each case to keep the peaks in the same position.
METS
(a.u.)
METS&
k) =
(13)
In k
In the limit k+l, this signal converges to the exact differentiation, eqn. (2). The discretization of the signal involves a modification of the signal shape that depends on the value of the parameter k. It can be easily calculated that the position and maximum of the peak (for a signal containing a unique exponential term) have been shifted to k’9n k tInax=k_lc max(METS,(t, k))=A, ‘2
k-kl(C-l)
(14)
It can be shown that the FWHM of the peak increases as k increases. Thus, to achieve the maximum resolution power a k value near unity should be chosen. On the other hand, the METS signal is more sensitive to signal noise as k approaches unity. Thus, a balance between both effects must be found. In most practical cases k=2l”= 1.26 is a good choice. In order to define the n-order METS signal considering the discretization of the sensor signal we define:
&
l)f~(tknn-i,
S.(t, k) E
(15)
ln”k
Then the METS,, signal is METS,(t, k) = (- l)“S,(t, k) n-1 + ,F; (- lY-ian,n-iMETSn-&, 0
1
2
k)
(16)
where the anmmcan be calculated from
log (t) Fig. 2. Simulated METS,, signals for a transient containing hvo exponential decays, with time constants separated by a factor five: (a) n-l, (b) n-2, (c) n=3, (d) n=4.
an.m=~~,-l,,+~,-l,m-l a 1.1--1
(17)
311
- The experimental implementation of the method is very easy.
3. Analysis of experimental transients
The alcohol sensor is based on an electrochemical fuel cell using H,SO, as an electrolyte absorbed by a porous PVC and Pt electrodes that serve also as a catalytic material. The alcohol sensor response to a step change in the gas concentration can be empirically described by the sum of two exponential decays: Z(t) =A[exp( - t/q) - exp( - fh)]
t Fig. 3. Influence of the discretization factor k in the position and intensity of the peaks for the convolution functions $(v).
Figure 3 shows how the peak position and maxima are altered due to numerical differentiation for several k and n values. If a real time processing of the n-order METS signal is going to be implemented using the above scheme, it is worth noticing that there will be, at least, a delay of (k”- I4 - 1)r due to the necessary knowledge of some points in advance to calculate the successive differentiations.
2.6. Advantages of the METS analysis The METS analysis shares the advantages of the spectroscopies: - It permits us to deal with discrete and continuous time-constant distributions. This feature isvery adequate in the analysis of decays of unknown nature. - The diierent exponential components of the decay are determined simultaneously. This is advantageous over iterative methods based on the sequential extraction of the time constants. In the later methods the errors made on the first parameters influence the subsequent results. - The appearance of wider peaks, compared to the FWHM for an exponential component, suggests the existence of more than one component that the signal has not been able to resolve. Moreover, in the METS, the user can adjust the spectroscopic power by changing the signal order. Other advantages are summarized in the following: - The required computation power is much less than that required for alternative methods based on the calculation of FFTs.
(18)
with TV> TV.Typical transients are shown in Fig. 4. Although the fastest time constant is probably influenced by the non-ideal gas injection, the slower exponential permits the application of the presented method. Three different alcohols (ethanol, methanol and 2propanol) were used in the study. First, the individual response of the sensor to each gas was recorded and analysed by METS. It was observed that the exponential amplitude was directly proportional to the gas concentration, while the time constant was specific to a certain gas. The variations of time constant with gas concentration were negligible, although small shifts were found in the comparison among different devices. The selectivity of this parameter to the gas type should permit the identification and quantification of a gas mixture of different alcohols. The resolution of the different time constants will very much depend on their closeness and the signal-to-noise ratio. For example, Fig. 5 shows the first- and third-order METS
‘.2I 1.0
-
-
ethanol 2-proponol methanol
-0.6 3 -
0.4
0.2
0.0
0
20
Fig. 4. Normalized transient
signals of the alcohol sensor.
312
-0.62 j i
\
-0.78 -I
-0.86
!
-0.94
-
P 2
-1.02 -
-1.10
11
10
time
(set)
Fig. 5. Normalized first- and third-order METS signals for a mixture of methanol (5.8 ppm) and Z-propanol (0.9 ppm).
signals for a mixture of methanol and 2-propanol. While the first-order signal presents a peak much wider than that corresponding to a single exponential decay, the third-order signal is able to separate both contributions.
4. Conclusions A novel method, METS, to find the time-constant distribution in an exponential decay has been presented. Its features make METS a valuable alternative for the analysis of chemical sensor transient responses. A practical example in alcohol sensors demonstrates its usefulness and versatility.
References 1 S. Vaihmger and W. Gopel, Multi-component anaIysis in chemical sensing, in W. Gopel, J. Hesse and J.N. Zemel (eds.), Sensors: A Coqmhensive Survey, Vol. II, VCH, Weinheim, 1991, pp. 192-234.
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