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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 4, AUGUST 2001

A Hysteresis Model for a Vanadium Dioxide Transition-Edge Microbolometer Luiz Alberto Luz de Almeida, Member, IEEE, Gurdip Singh Deep, Senior Member, IEEE, Antonio Marcus Nogueira Lima, Member, IEEE, Helmut Franz Neff, and Raimundo Carlos Silvério Freire

Abstract—This paper presents the adaptation of the Preisach model, originally developed for magnetic hysteresis, to describe mathematically the hysteresis in the resistance-temperature characteristics of vanadium–dioxide (VO2 ) thin film radiation sensors. The necessary and sufficient conditions for the applicability of the Preisach model to a VO2 film sensor are experimentally verified. Experimentally measured characteristics are compared with those given by the model for minor and major loops. Index Terms—Bolometer, hysteresis modeling, metal-insulator transition, Preisach model, thermal radiation, vanadium–dioxide.

I. INTRODUCTION

T

HE vanadium-dioxide (VO ) thin film sensor has been employed recently as a bolometer in the nonhysteretic part of ) characteristics. The operthe resistance-temperature ( ating point was set at about 25 C using external cooling by a Peltier element [1]. However, this sensor can also be used as a bolometer within the transition region between 40 C to 60 C, thus exploiting its high resistance-temperature coefficient by adequately biasing the sensor and adjusting the operation point by Joule heating. VO is characterized by a temperature-driven, metal-insulator phase transition [2] from a low temperature semiconductor to a high temperature metallic phase. Fig. 1 shows the expericharacteristic curve of VO , when the film tempermental ature is raised from 20 C to 80 C and subsequently reduced to 20 C. This structural phase transition exhibits a crytallographic transformation that is accompanied by significant change of the electrical and optical film properties [2]. There are some difficulties in biasing and operating the sensor curve. The properties of in the hysteretic portion of the major hysteresis loops of VO thin film have been reported in the literature [3], but there is very little information available about minor loops, and the influence of the thermal history on the hysteresis trajectories [4]. Toward a theoretical evaluation of the VO bolometer performance, operating in the hysteretic transition region, a valid mathematical model is required to fully describe the major, as well as the minor hysteresis loops.

Fig. 1.

R 2 T characteristic of a VO

film.

There exist various physical and mathematical models to describe hysteretic phenomena. Some of these models are derived for magnetic hysteresis, like the Preisach model [5]. Others, such as the Jiles model [6], explain magnetization in terms of the movement of magnetic domain walls. However, due to its mathematical generality, the Preisach model is widely accepted as a suitable tool to formally describe hysteresis phenomena. In the following, we propose the adaptation of the Preisach model to describe the thermal hysteresis of a VO thin film that applies for a bolometric sensor. The original Preisach triangle and relay operator are redefined, along with adapted interpretations of its properties. Eventually, the experimentally obtained VO hysteresis curves were verified for the applicability of the Preisach model. A surface is proposed to fit a set of experimental first order descending (FOD) curves, in order to derive the Preisach distribution or weighting function. Finally, the model performance is evaluated by comparing experimental minor and major loops with those obtained from the model. II. PREISACH HYSTERESIS MODEL FOR VO

Manuscript received May 4, 2000; revised May 23, 2001. L. A. L. de Almeida is with the Departamento de Engenharia Elétrica, Universidade Federal da Bahia and with the Departamento de Engenharia Elétrica, Universidade Federal da Paraíba, Campina Grande, PB, Brazil (e-mail: [email protected]). G. S. Deep, A. M. N. Lima, H. F. Neff, and R. C. S. Freire are with the Departamento de Engenharia Elétrica, Universidade Federal da Paraíba, Campina Grande, PB, Brazil (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Publisher Item Identifier S 0018-9456(01)07891-3.

The original classical Preisach model was proposed in 1935 [7]. The model was conceived on a intuitive basis [5] and is based on some plausible hypotheses about the magnetization mechanism. Later, M. Krasnoselskii [8] showed that the Preisach model is quite general, and it is possible to adapt it to other than magnetic hysteretic phenomena. There is experimental evidence that a thin VO film is composed

0018–9456/01$10.00 © 2001 IEEE

DE ALMEIDA et al.: HYSTERESIS MODEL FOR A VANADIUM DIOXIDE TRANSITION-EDGE MICROBOLOMETER

Fig. 2. Graphical representation of the relay operator ^

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for VO .

of microcrystals, which individually exhibit a very sharp characteristic hysteretic transition [9]. The overall film is associated with the existence of an ensemble of either metallic or semiconducting microcrystals of VO . This led us to investigate the adaptability of Preisach model for thermal hysteresis to VO thin films. The classical Preisach model for as a sum of VO allows us to describe the film resistance , representing microcrystal states simple relay operators . and weighted by a statistical distribution function It is given by

Fig. 3.

Preisach triangle adapted for VO thermal hysteresis.

(1) . The state Fig. 2 graphically represents the relay operator . of these operators depends on sensor temperature A simplified graphical interpretation of (1) can be achieved adapted for in terms of the so called Preisach triangle this sensor (Fig. 3). This right triangle is associated with only characteristics, lying bethe hysteretic portion of the tween 20 C and 80 C. In other words, the distribution funcis assumed to be zero outside triangle . tion and repThis right triangle can be divided into two parts: resenting the region where the relay operators are in the states 1 (semiconducting) and 0 (insulating), respectively. As an example, if the device temperature is reduced from 80 C to raised to , and again reduced to , the interface line beand is as shown in Fig. 3. The vertices tween the areas are related to the past extreme values of of interface line . It is thus evident that and depend on the history of thermal cycling. 1 for and 0 Considering , then (1) can be written as for (2)

III. PREISACH MODEL REPRESENTATION For the use of the Preisach model representation to describe a given hysteretic characteristic, some conditions must be fullfiled. These conditions can be stated in terms of the Preisach triangle presented in Fig. 3. Mayergoyz showed [5] that the wiping-out property and congruency property constitute the necessary and sufficient conditions to be satisfied for a hys-

Fig. 4. Experimental verification of the wiping-out property in VO : Inset: the thermal excitation waveform used in this verification.

teresis nonlinearity to be represented by the Preisach model for a set of piecewise monotonic excitations. Considering the Preisach triangle depicted in Fig. 3, we can redefine the wiping-out property, adapted for the VO case, as: wipes Wiping-Out Property: Each local minimum of , whose coordinates are above this out the vertices of wipes out the verminimum, and each local maximum of whose coordinates are below this maximum. tices of (inset Consider a piecewise monotonic excitation . This time interval Fig. 4), in an arbitrary time interval , during which the is composed by several subintervals temperature excitation is monotonic. Each subinterval has a and a local minimum in . Now we can local maximum in state the wiping-out property as: wipes Each new temperature maximum at the instant out any temperature maximum at an arbitrary previous inwhen , and each new temstant perature minimum at the instant wipes out any temper-

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 4, AUGUST 2001

(a)

(a)

(b) Fig. 6. (a) Experimental FOD curve and (b) the corresponding Preisach triangle.

(b) Fig. 5. Experimental verification of the congruency property for different minor loops: (a) R T characteristics and (b) log(R) T characteristics. Inset: the thermal excitation waveform used in this verification.

2

2

ature minimum at an arbitrary previous instant when . The wiping-out property was experimentally verified for the curves. From the VO film. Fig. 4 shows experimental and , wiping-out property, we should expect that points and be coincident. This happens in as well as the points a very close way suggesting that the wiping-out property holds. Fig. 5(a) depicts three pairs of minor loops and generated by corresponding three pairs of back-andand . Each pair forth temperature variations and of temperature variation has the same extremum values [inset zoom Fig. 5(a)], and occur around different temperature values. The variations and produce the minor loops and , respectively. The minor loops and are vertically plane due to the thermal history. The same shifted in the and . Based on occurs with the minor loop pairs these experimental observations, the congruency property can be redefined for the VO film as:

Congruency Property: All minor loops, vertically shifted and produced by a back-and-forth variation of temperature between the same maximum and minimum excitation values are congruent. The congruency property was also experimentally verified for the VO film. Fig. 5(a) presents the minor loops in the plane and corresponding excitation (shown in the inset). , We can see that the congruency property does not hold ( and ). This is primarily due to the asymmetric characteristic of the VO hysteresis curve. However, if plot is used, the minor loops can be seen more a distinctly over the whole range of resistance variation, and the , , and noncongruency effect is reduced, i.e., , as shown in Fig. 5(b).

IV. OBTAINING THE DISTRIBUTION FUNCTION A procedure, proposed by Mayergoyz for determining the , employs the experimental distribution function first-order descending transition (FOD) curves [5], where the term descending is related to the decreasing resistance. An FOD curve is generated by first increasing the temperature to its maximum value of 80 C, where the resistance value is . Next, the temperature is monotonically around with a resistance decreased until it reaches some value as shown in Fig. 6(a), with the corresponding Preisach triangle depicted in Fig. 6(b). Then, the temperature is increased

DE ALMEIDA et al.: HYSTERESIS MODEL FOR A VANADIUM DIOXIDE TRANSITION-EDGE MICROBOLOMETER

monotonically to some value where the sensor resistance is . to removes the region The increase of temperature from [Fig. 6(b)] from and adds it to the area . This region corresponds to the change in the sensor resistance given by (3)

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where represents the experimental FOD data and is the subset of the 3 dimensional space that defines the valid parameter search region. At the end of the first step, we have a set of tridimensional parameter vectors. The next step is to establish the dependence of the components of these vectors with . To characterize this relationship the following functions have been used (9)

The surface integral over region lowing double integral

can be written as the fol-

(10) (11)

(4) for each FOD curve Considering that there is a fixed and by differentiating (4) twice, the weighting function at any can be determined [5] as point

In this case, the parameter vector is also determined by solving the following parameter optimization problem

(5) Applying the logarithmic transformation on the resistance data, the weighting or distribution function is defined by (6)

V. CHOICE OF THE FOD SURFACE Initial attempts to fit a low-order polynomial surface to the experimentally obtained FOD curves did not yield satisfactory results. This is primarily due to the highly nonlinear and asymmetric characteristics of the sensor hysteresis. Polynomial surfaces of higher order match the measured data points, but exhibit highly oscillatory behavior between the measured data points. This is in conflict with the smooth hysteresis characteristics observed experimentally. Some intuitive attempts have indicated that each FOD surface may be adequately fitted by using a two step parameter optimization procedure. In the first step it is assumed that the th FOD curve can be described by

(12) , , and represent the values obtained where is from the solution of the first optimization problem and that defines the valid the subset of the 9 dimensional space parameter search region. With these two steps, the FOD curves can be represented by a surface given as

(13) whose second derivative (6) yields the estimated distribution , thus concluding the hysteresis modeling function procedure. Also, considering the high temperature saturation reand , (2) can be sistance rewritten as (14)

A. Numerical Procedure C

(7)

with

The upper temperature limit 80 C is indicated in the Preisach triangle shown in Fig. 3. The parameter vector for the th FOD can be determined by solving the following parameter optimization problem

(8)

In order to fit the experimental data, the Nelder–Mead algorithm [10] was employed to solve both parameter optimization problems described in the previous section. This is a relatively simple nonlinear minimization algorithm that tends to converge to a local minimum. Consequently, this algorithm requires a relatively good initial guess. Both parameter optimization problems were solved in the MATLAB environment by using the function that implements the Nelder–Mead method. VI. EXPERIMENTAL RESULTS Fig. 7(a)–(c) shows the fit of the functions , to the estimated value sets , and , and , respectively. The estimation of each one of the eighteen parameter vectors

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 4, AUGUST 2001

(a)

(a)

(b)

(c) Fig. 7. Functions (a) f (T ), (b) g (T ), and (c) h(T ) estimated from the vectors issued from the first parameter optimization step fa ; a ; . . . ; a g, fb ; b ; . . . ; b g, and fc ; c ; . . . ; c g respectively.

(b) Fig. 8. Comparison between the proposed model and experimental data for (a) minor loops and (b) major loop.

has employed at least 200 resistance-temperature measurement pairs collected over each one of the eighteen and FOD curves. A good agreement in the fit of is visually observable in Fig. 7(b) and (c), respectively. for Fig. 7(a) shows a good agreement in the fit of temperatures up to 45 C, but after this point, the fit is poor. Thus, the estimate of the parameters for the surface described , , , by (13) are the following: , , , , , , and . , an attempt has been made In order to improve the fit of to directly to FOD experimental data. to fit parameters Several other cost and penalty functions, along with different types of norms, have been tested, but the simplex algorithm did not converge to a better result. From the estimated surface given by (13), the distribution was obtained using (6). Employing a procefunction can be then calculated dure developed in [5], the resistance using (14). To verify the model for an arbitrary excitation quality, two minor and a major loop were experimentally obtained and compared with those generated from the model, as

we can see in Fig. 8(a) and (b), respectively. The data generated from the proposed model agrees very well with experimental data for the minor loops [Fig. 8(a)]. Furthermore, we also observe good agreement for a major loop over the range of 35 C to 80 C [Fig. 8(b)]. VII. CONCLUSIONS The applicability of the classical Preisach model for the thermal hysteresis in VO thin films has been investigated. hysteresis in the phase transition We focused on the region of the sensor characteristics. For the necessary and sufficient conditions to be satisfied for the applicability of characteristic was found the Preisach model, a characteristic curve, to be more suitable than the which is asymmetrical. A procedure for determining the model parameters has been described. Model simulations have shown good agreement with experimental data for the minor loops and major loops over a limited temperature range. However, due to the asymmetric behavior of the sensor hysteresis (Fig. 1), experimental data at low temperatures do not agree with the model. The model is thus valid in the transition region where the resistance hysteresis is relevant. ACKNOWLEDGMENT The authors thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and CAPES (Fundação Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) for the award of research and study fellowship during the course of these investigations.

DE ALMEIDA et al.: HYSTERESIS MODEL FOR A VANADIUM DIOXIDE TRANSITION-EDGE MICROBOLOMETER

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REFERENCES [1] A. P. Gruzdeva, V. V. Zerov, and O. P. Konovalova, “Bolometric and noise properties of elements for uncooled IR arrays based on vanadium dioxide films,” J. Opt. Technol., vol. 64, pp. 1110–1113, Dec. 1997. [2] H. Jerominek, F. Picard, and D. Vincent, “vanadium dioxide films for optical switching and detection,” Opt. Eng., vol. 32, pp. 2092–2099, Sept. 1993. [3] H. S. Choi, J. S. Ahm, J. H. Jung, T. W. Noh, and D. H. Kim, “Midinfrared properties of a VO film near the metal-insulator transition,” Phys. Rev. B, Condens. Matter, vol. 54, pp. 4621–4628, Aug. 1996. [4] J. R. Sun, G. H. Rao, and J. K. Liang, “Thermal history dependent electronic transport and magnetic properties in bulk La Nd Ca MnO ,” Physica Status Solidi, vol. 163, pp. 141–147, Sept. 1997. [5] I. D. Mayergoyz, Mathematical Models of Hysteresis. New York: Springer-Verlag, 1991. [6] D. C. Jiles and D. L. Atherton, “Ferromagnetic hysteresis,” IEEE Trans. Magn., vol. 19, pp. 2183–2185, Sept. 1983. [7] F. Preisach, “ber Die Magnetische Nachwirkung,” Zeitschrift fr Physik, vol. 94, p. 277, 1935. [8] M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis. Moscow: Nauka, 1983. [9] I. A. Khakhaev, F. A. Ghudnovskii, and E. B. Shadrin, “Martensitic effects in the metal-insulator phase transition in a vanadium dioxide film,” Phys. Solid State, vol. 36, June 1994. [10] J. A. Nelder, “Simplex method for function minimization,” Eng. Technol. Appl. Sci., vol. 15, no. 15, pp. 12–12, 1979.

Gurdip Singh Deep (M’76–SM’84) was born December 12, 1937. He received the B. Tech. (Hons.) degree in electrical eng. from Indian Institute of Technology (IIT), Kharagpur, India, in 1959, the M.E. degree in power engineering (electrical) from the Indian Institute of Science, Bangalore, India, in 1961, and the Ph.D. in electrical engineering from IIT Kanpur, India, in 1971. From 1961 to 1965, he worked as an Assistant Professor in Guru Nanak Engineering College Ludhiana, and from 1965 to 1972, he was with the IIT, Kanpur, as a Lecturer/Assistant Professor. Since July 1972, he has been a titular Professor at the Centre of Science and Technology of Federal University of Paraíba, Campina Grande, Brazil. Presently, he is the coordinator of the Electronic Instrumentation and Control Laboratory of the University. He was a Consultant for Encardio-rite Electronics (Pvt) Ltd., India, during 1969–1970. His research interests are electronic instrumentation and microcomputer-based process control.

Antonio Marcus Nogueira Lima (S’77–M’89) was born in Recife, Pernambuco, Brazil, in 1958. He received the B.S. and M.S. degrees in electrical engineering from Federal University of Paraíba, Campina Grande, Paraíba, Brazil in 1982 and 1985, respectively. He received his doctoral degree in 1989 from Institut National Polytechnique de Toulouse, Toulouse, France. Since September 1983, he has been a Professor in the Electrical Engineering Department, Federal University of Paraíba. His research interests are in the fields of electrical machine drives, power electronics, electronic instrumentation, control systems, and system identification.

Helmut Franz Neff was born on May 10, 1948. He received the doctoral degree in physics from University of Berlin, Berlin, Germany in 1981. He is a solid state physicist, with long standing research on micro-structured transition edge devices and sensors. He is presently working at VIR-TECH, Denmark, on the development of biochemical sensors, using surface plasma resonance principles.

Luiz Alberto Luz de Almeida (M’00) was born October 17, 1962. He received the M.Sc. degree in electrical engineering from Universidade Federal da Bahia, Salvador, Brazil, in 1995. He is currently a Ph.D. student at the Department of Electrical Engineering, Universidade Federal da Paraíba, Campina Grande, Brazil. From 1986 to 1993, he worked at industry where he gained experience in the design of analog/digital instrumentation and control systems. He joined the Department of Electrical Engineering, Universidade Federal da Bahia, Salvador, Brazil, in 1996, where he has been an Assistant Professor for telecommunication systems. His current research interests include signal processing and modeling of sensors and actuators.

Raimundo Carlos Silvério Freire was born on October 10, 1955, in Poço de Pedra-RN, Brazil. He received the B.S. degree in electrical engineering from the Federal University of Maranhão, São Luis, Brazil, in 1980, and the M.S. degree in electrical engineering from Federal University of Paraíba, Campina Grande, Brazil, in 1982. He received his doctoral degree in electronics, automation, and measurements at the National Polytechnical Institute of Lorraine, Nancy, France, in 1988. He worked as an Electrical Engineer for Maranhão Educational Television, Brazil, from 1980 to 1983. He was a Professor of electrical engineering at Federal University of Maranhão from 1982 to 1985. Since December 1989, he has been on the faculty of the Electrical Engineering Department of the Federal University of Paraíba. His research interests include electronic intrumentation and sensors, and microcomputer-based process control.

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 4, AUGUST 2001

A Hysteresis Model for a Vanadium Dioxide Transition-Edge Microbolometer Luiz Alberto Luz de Almeida, Member, IEEE, Gurdip Singh Deep, Senior Member, IEEE, Antonio Marcus Nogueira Lima, Member, IEEE, Helmut Franz Neff, and Raimundo Carlos Silvério Freire

Abstract—This paper presents the adaptation of the Preisach model, originally developed for magnetic hysteresis, to describe mathematically the hysteresis in the resistance-temperature characteristics of vanadium–dioxide (VO2 ) thin film radiation sensors. The necessary and sufficient conditions for the applicability of the Preisach model to a VO2 film sensor are experimentally verified. Experimentally measured characteristics are compared with those given by the model for minor and major loops. Index Terms—Bolometer, hysteresis modeling, metal-insulator transition, Preisach model, thermal radiation, vanadium–dioxide.

I. INTRODUCTION

T

HE vanadium-dioxide (VO ) thin film sensor has been employed recently as a bolometer in the nonhysteretic part of ) characteristics. The operthe resistance-temperature ( ating point was set at about 25 C using external cooling by a Peltier element [1]. However, this sensor can also be used as a bolometer within the transition region between 40 C to 60 C, thus exploiting its high resistance-temperature coefficient by adequately biasing the sensor and adjusting the operation point by Joule heating. VO is characterized by a temperature-driven, metal-insulator phase transition [2] from a low temperature semiconductor to a high temperature metallic phase. Fig. 1 shows the expericharacteristic curve of VO , when the film tempermental ature is raised from 20 C to 80 C and subsequently reduced to 20 C. This structural phase transition exhibits a crytallographic transformation that is accompanied by significant change of the electrical and optical film properties [2]. There are some difficulties in biasing and operating the sensor curve. The properties of in the hysteretic portion of the major hysteresis loops of VO thin film have been reported in the literature [3], but there is very little information available about minor loops, and the influence of the thermal history on the hysteresis trajectories [4]. Toward a theoretical evaluation of the VO bolometer performance, operating in the hysteretic transition region, a valid mathematical model is required to fully describe the major, as well as the minor hysteresis loops.

Fig. 1.

R 2 T characteristic of a VO

film.

There exist various physical and mathematical models to describe hysteretic phenomena. Some of these models are derived for magnetic hysteresis, like the Preisach model [5]. Others, such as the Jiles model [6], explain magnetization in terms of the movement of magnetic domain walls. However, due to its mathematical generality, the Preisach model is widely accepted as a suitable tool to formally describe hysteresis phenomena. In the following, we propose the adaptation of the Preisach model to describe the thermal hysteresis of a VO thin film that applies for a bolometric sensor. The original Preisach triangle and relay operator are redefined, along with adapted interpretations of its properties. Eventually, the experimentally obtained VO hysteresis curves were verified for the applicability of the Preisach model. A surface is proposed to fit a set of experimental first order descending (FOD) curves, in order to derive the Preisach distribution or weighting function. Finally, the model performance is evaluated by comparing experimental minor and major loops with those obtained from the model. II. PREISACH HYSTERESIS MODEL FOR VO

Manuscript received May 4, 2000; revised May 23, 2001. L. A. L. de Almeida is with the Departamento de Engenharia Elétrica, Universidade Federal da Bahia and with the Departamento de Engenharia Elétrica, Universidade Federal da Paraíba, Campina Grande, PB, Brazil (e-mail: [email protected]). G. S. Deep, A. M. N. Lima, H. F. Neff, and R. C. S. Freire are with the Departamento de Engenharia Elétrica, Universidade Federal da Paraíba, Campina Grande, PB, Brazil (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Publisher Item Identifier S 0018-9456(01)07891-3.

The original classical Preisach model was proposed in 1935 [7]. The model was conceived on a intuitive basis [5] and is based on some plausible hypotheses about the magnetization mechanism. Later, M. Krasnoselskii [8] showed that the Preisach model is quite general, and it is possible to adapt it to other than magnetic hysteretic phenomena. There is experimental evidence that a thin VO film is composed

0018–9456/01$10.00 © 2001 IEEE

DE ALMEIDA et al.: HYSTERESIS MODEL FOR A VANADIUM DIOXIDE TRANSITION-EDGE MICROBOLOMETER

Fig. 2. Graphical representation of the relay operator ^

1031

for VO .

of microcrystals, which individually exhibit a very sharp characteristic hysteretic transition [9]. The overall film is associated with the existence of an ensemble of either metallic or semiconducting microcrystals of VO . This led us to investigate the adaptability of Preisach model for thermal hysteresis to VO thin films. The classical Preisach model for as a sum of VO allows us to describe the film resistance , representing microcrystal states simple relay operators . and weighted by a statistical distribution function It is given by

Fig. 3.

Preisach triangle adapted for VO thermal hysteresis.

(1) . The state Fig. 2 graphically represents the relay operator . of these operators depends on sensor temperature A simplified graphical interpretation of (1) can be achieved adapted for in terms of the so called Preisach triangle this sensor (Fig. 3). This right triangle is associated with only characteristics, lying bethe hysteretic portion of the tween 20 C and 80 C. In other words, the distribution funcis assumed to be zero outside triangle . tion and repThis right triangle can be divided into two parts: resenting the region where the relay operators are in the states 1 (semiconducting) and 0 (insulating), respectively. As an example, if the device temperature is reduced from 80 C to raised to , and again reduced to , the interface line beand is as shown in Fig. 3. The vertices tween the areas are related to the past extreme values of of interface line . It is thus evident that and depend on the history of thermal cycling. 1 for and 0 Considering , then (1) can be written as for (2)

III. PREISACH MODEL REPRESENTATION For the use of the Preisach model representation to describe a given hysteretic characteristic, some conditions must be fullfiled. These conditions can be stated in terms of the Preisach triangle presented in Fig. 3. Mayergoyz showed [5] that the wiping-out property and congruency property constitute the necessary and sufficient conditions to be satisfied for a hys-

Fig. 4. Experimental verification of the wiping-out property in VO : Inset: the thermal excitation waveform used in this verification.

teresis nonlinearity to be represented by the Preisach model for a set of piecewise monotonic excitations. Considering the Preisach triangle depicted in Fig. 3, we can redefine the wiping-out property, adapted for the VO case, as: wipes Wiping-Out Property: Each local minimum of , whose coordinates are above this out the vertices of wipes out the verminimum, and each local maximum of whose coordinates are below this maximum. tices of (inset Consider a piecewise monotonic excitation . This time interval Fig. 4), in an arbitrary time interval , during which the is composed by several subintervals temperature excitation is monotonic. Each subinterval has a and a local minimum in . Now we can local maximum in state the wiping-out property as: wipes Each new temperature maximum at the instant out any temperature maximum at an arbitrary previous inwhen , and each new temstant perature minimum at the instant wipes out any temper-

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 4, AUGUST 2001

(a)

(a)

(b) Fig. 6. (a) Experimental FOD curve and (b) the corresponding Preisach triangle.

(b) Fig. 5. Experimental verification of the congruency property for different minor loops: (a) R T characteristics and (b) log(R) T characteristics. Inset: the thermal excitation waveform used in this verification.

2

2

ature minimum at an arbitrary previous instant when . The wiping-out property was experimentally verified for the curves. From the VO film. Fig. 4 shows experimental and , wiping-out property, we should expect that points and be coincident. This happens in as well as the points a very close way suggesting that the wiping-out property holds. Fig. 5(a) depicts three pairs of minor loops and generated by corresponding three pairs of back-andand . Each pair forth temperature variations and of temperature variation has the same extremum values [inset zoom Fig. 5(a)], and occur around different temperature values. The variations and produce the minor loops and , respectively. The minor loops and are vertically plane due to the thermal history. The same shifted in the and . Based on occurs with the minor loop pairs these experimental observations, the congruency property can be redefined for the VO film as:

Congruency Property: All minor loops, vertically shifted and produced by a back-and-forth variation of temperature between the same maximum and minimum excitation values are congruent. The congruency property was also experimentally verified for the VO film. Fig. 5(a) presents the minor loops in the plane and corresponding excitation (shown in the inset). , We can see that the congruency property does not hold ( and ). This is primarily due to the asymmetric characteristic of the VO hysteresis curve. However, if plot is used, the minor loops can be seen more a distinctly over the whole range of resistance variation, and the , , and noncongruency effect is reduced, i.e., , as shown in Fig. 5(b).

IV. OBTAINING THE DISTRIBUTION FUNCTION A procedure, proposed by Mayergoyz for determining the , employs the experimental distribution function first-order descending transition (FOD) curves [5], where the term descending is related to the decreasing resistance. An FOD curve is generated by first increasing the temperature to its maximum value of 80 C, where the resistance value is . Next, the temperature is monotonically around with a resistance decreased until it reaches some value as shown in Fig. 6(a), with the corresponding Preisach triangle depicted in Fig. 6(b). Then, the temperature is increased

DE ALMEIDA et al.: HYSTERESIS MODEL FOR A VANADIUM DIOXIDE TRANSITION-EDGE MICROBOLOMETER

monotonically to some value where the sensor resistance is . to removes the region The increase of temperature from [Fig. 6(b)] from and adds it to the area . This region corresponds to the change in the sensor resistance given by (3)

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where represents the experimental FOD data and is the subset of the 3 dimensional space that defines the valid parameter search region. At the end of the first step, we have a set of tridimensional parameter vectors. The next step is to establish the dependence of the components of these vectors with . To characterize this relationship the following functions have been used (9)

The surface integral over region lowing double integral

can be written as the fol-

(10) (11)

(4) for each FOD curve Considering that there is a fixed and by differentiating (4) twice, the weighting function at any can be determined [5] as point

In this case, the parameter vector is also determined by solving the following parameter optimization problem

(5) Applying the logarithmic transformation on the resistance data, the weighting or distribution function is defined by (6)

V. CHOICE OF THE FOD SURFACE Initial attempts to fit a low-order polynomial surface to the experimentally obtained FOD curves did not yield satisfactory results. This is primarily due to the highly nonlinear and asymmetric characteristics of the sensor hysteresis. Polynomial surfaces of higher order match the measured data points, but exhibit highly oscillatory behavior between the measured data points. This is in conflict with the smooth hysteresis characteristics observed experimentally. Some intuitive attempts have indicated that each FOD surface may be adequately fitted by using a two step parameter optimization procedure. In the first step it is assumed that the th FOD curve can be described by

(12) , , and represent the values obtained where is from the solution of the first optimization problem and that defines the valid the subset of the 9 dimensional space parameter search region. With these two steps, the FOD curves can be represented by a surface given as

(13) whose second derivative (6) yields the estimated distribution , thus concluding the hysteresis modeling function procedure. Also, considering the high temperature saturation reand , (2) can be sistance rewritten as (14)

A. Numerical Procedure C

(7)

with

The upper temperature limit 80 C is indicated in the Preisach triangle shown in Fig. 3. The parameter vector for the th FOD can be determined by solving the following parameter optimization problem

(8)

In order to fit the experimental data, the Nelder–Mead algorithm [10] was employed to solve both parameter optimization problems described in the previous section. This is a relatively simple nonlinear minimization algorithm that tends to converge to a local minimum. Consequently, this algorithm requires a relatively good initial guess. Both parameter optimization problems were solved in the MATLAB environment by using the function that implements the Nelder–Mead method. VI. EXPERIMENTAL RESULTS Fig. 7(a)–(c) shows the fit of the functions , to the estimated value sets , and , and , respectively. The estimation of each one of the eighteen parameter vectors

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 4, AUGUST 2001

(a)

(a)

(b)

(c) Fig. 7. Functions (a) f (T ), (b) g (T ), and (c) h(T ) estimated from the vectors issued from the first parameter optimization step fa ; a ; . . . ; a g, fb ; b ; . . . ; b g, and fc ; c ; . . . ; c g respectively.

(b) Fig. 8. Comparison between the proposed model and experimental data for (a) minor loops and (b) major loop.

has employed at least 200 resistance-temperature measurement pairs collected over each one of the eighteen and FOD curves. A good agreement in the fit of is visually observable in Fig. 7(b) and (c), respectively. for Fig. 7(a) shows a good agreement in the fit of temperatures up to 45 C, but after this point, the fit is poor. Thus, the estimate of the parameters for the surface described , , , by (13) are the following: , , , , , , and . , an attempt has been made In order to improve the fit of to directly to FOD experimental data. to fit parameters Several other cost and penalty functions, along with different types of norms, have been tested, but the simplex algorithm did not converge to a better result. From the estimated surface given by (13), the distribution was obtained using (6). Employing a procefunction can be then calculated dure developed in [5], the resistance using (14). To verify the model for an arbitrary excitation quality, two minor and a major loop were experimentally obtained and compared with those generated from the model, as

we can see in Fig. 8(a) and (b), respectively. The data generated from the proposed model agrees very well with experimental data for the minor loops [Fig. 8(a)]. Furthermore, we also observe good agreement for a major loop over the range of 35 C to 80 C [Fig. 8(b)]. VII. CONCLUSIONS The applicability of the classical Preisach model for the thermal hysteresis in VO thin films has been investigated. hysteresis in the phase transition We focused on the region of the sensor characteristics. For the necessary and sufficient conditions to be satisfied for the applicability of characteristic was found the Preisach model, a characteristic curve, to be more suitable than the which is asymmetrical. A procedure for determining the model parameters has been described. Model simulations have shown good agreement with experimental data for the minor loops and major loops over a limited temperature range. However, due to the asymmetric behavior of the sensor hysteresis (Fig. 1), experimental data at low temperatures do not agree with the model. The model is thus valid in the transition region where the resistance hysteresis is relevant. ACKNOWLEDGMENT The authors thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and CAPES (Fundação Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) for the award of research and study fellowship during the course of these investigations.

DE ALMEIDA et al.: HYSTERESIS MODEL FOR A VANADIUM DIOXIDE TRANSITION-EDGE MICROBOLOMETER

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REFERENCES [1] A. P. Gruzdeva, V. V. Zerov, and O. P. Konovalova, “Bolometric and noise properties of elements for uncooled IR arrays based on vanadium dioxide films,” J. Opt. Technol., vol. 64, pp. 1110–1113, Dec. 1997. [2] H. Jerominek, F. Picard, and D. Vincent, “vanadium dioxide films for optical switching and detection,” Opt. Eng., vol. 32, pp. 2092–2099, Sept. 1993. [3] H. S. Choi, J. S. Ahm, J. H. Jung, T. W. Noh, and D. H. Kim, “Midinfrared properties of a VO film near the metal-insulator transition,” Phys. Rev. B, Condens. Matter, vol. 54, pp. 4621–4628, Aug. 1996. [4] J. R. Sun, G. H. Rao, and J. K. Liang, “Thermal history dependent electronic transport and magnetic properties in bulk La Nd Ca MnO ,” Physica Status Solidi, vol. 163, pp. 141–147, Sept. 1997. [5] I. D. Mayergoyz, Mathematical Models of Hysteresis. New York: Springer-Verlag, 1991. [6] D. C. Jiles and D. L. Atherton, “Ferromagnetic hysteresis,” IEEE Trans. Magn., vol. 19, pp. 2183–2185, Sept. 1983. [7] F. Preisach, “ber Die Magnetische Nachwirkung,” Zeitschrift fr Physik, vol. 94, p. 277, 1935. [8] M. Krasnoselskii and A. Pokrovskii, Systems with Hysteresis. Moscow: Nauka, 1983. [9] I. A. Khakhaev, F. A. Ghudnovskii, and E. B. Shadrin, “Martensitic effects in the metal-insulator phase transition in a vanadium dioxide film,” Phys. Solid State, vol. 36, June 1994. [10] J. A. Nelder, “Simplex method for function minimization,” Eng. Technol. Appl. Sci., vol. 15, no. 15, pp. 12–12, 1979.

Gurdip Singh Deep (M’76–SM’84) was born December 12, 1937. He received the B. Tech. (Hons.) degree in electrical eng. from Indian Institute of Technology (IIT), Kharagpur, India, in 1959, the M.E. degree in power engineering (electrical) from the Indian Institute of Science, Bangalore, India, in 1961, and the Ph.D. in electrical engineering from IIT Kanpur, India, in 1971. From 1961 to 1965, he worked as an Assistant Professor in Guru Nanak Engineering College Ludhiana, and from 1965 to 1972, he was with the IIT, Kanpur, as a Lecturer/Assistant Professor. Since July 1972, he has been a titular Professor at the Centre of Science and Technology of Federal University of Paraíba, Campina Grande, Brazil. Presently, he is the coordinator of the Electronic Instrumentation and Control Laboratory of the University. He was a Consultant for Encardio-rite Electronics (Pvt) Ltd., India, during 1969–1970. His research interests are electronic instrumentation and microcomputer-based process control.

Antonio Marcus Nogueira Lima (S’77–M’89) was born in Recife, Pernambuco, Brazil, in 1958. He received the B.S. and M.S. degrees in electrical engineering from Federal University of Paraíba, Campina Grande, Paraíba, Brazil in 1982 and 1985, respectively. He received his doctoral degree in 1989 from Institut National Polytechnique de Toulouse, Toulouse, France. Since September 1983, he has been a Professor in the Electrical Engineering Department, Federal University of Paraíba. His research interests are in the fields of electrical machine drives, power electronics, electronic instrumentation, control systems, and system identification.

Helmut Franz Neff was born on May 10, 1948. He received the doctoral degree in physics from University of Berlin, Berlin, Germany in 1981. He is a solid state physicist, with long standing research on micro-structured transition edge devices and sensors. He is presently working at VIR-TECH, Denmark, on the development of biochemical sensors, using surface plasma resonance principles.

Luiz Alberto Luz de Almeida (M’00) was born October 17, 1962. He received the M.Sc. degree in electrical engineering from Universidade Federal da Bahia, Salvador, Brazil, in 1995. He is currently a Ph.D. student at the Department of Electrical Engineering, Universidade Federal da Paraíba, Campina Grande, Brazil. From 1986 to 1993, he worked at industry where he gained experience in the design of analog/digital instrumentation and control systems. He joined the Department of Electrical Engineering, Universidade Federal da Bahia, Salvador, Brazil, in 1996, where he has been an Assistant Professor for telecommunication systems. His current research interests include signal processing and modeling of sensors and actuators.

Raimundo Carlos Silvério Freire was born on October 10, 1955, in Poço de Pedra-RN, Brazil. He received the B.S. degree in electrical engineering from the Federal University of Maranhão, São Luis, Brazil, in 1980, and the M.S. degree in electrical engineering from Federal University of Paraíba, Campina Grande, Brazil, in 1982. He received his doctoral degree in electronics, automation, and measurements at the National Polytechnical Institute of Lorraine, Nancy, France, in 1988. He worked as an Electrical Engineer for Maranhão Educational Television, Brazil, from 1980 to 1983. He was a Professor of electrical engineering at Federal University of Maranhão from 1982 to 1985. Since December 1989, he has been on the faculty of the Electrical Engineering Department of the Federal University of Paraíba. His research interests include electronic intrumentation and sensors, and microcomputer-based process control.

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