Electrical Response of the Spinel ZnAl 2 O 4 and Its Application in the Detection of Propane Gas

: Nanoparticles of the semiconductor ZnAl 2 O 4 were prepared using a microwave-assisted wet chemistry method in the presence of ethylenediamine and calcination at 250 ◦ C. The material’s crystallinity and purity were veriﬁed by X-ray diffraction. The pure phase of the ZnAl 2 O 4 presented a cubic crystalline structure with cell parameters a = 8.087 Å and space group Fd-3m (227). Dynamic tests in propane atmospheres were carried out on pellets (~500 µ m in diameter) manufactured with ZnAl 2 O 4 powders. In the tests, the oxide showed variations with time in electrical resistance when injecting air-propane at an operating temperature of 250 ◦ C. The pellets showed good stability, high sensitivity, and an optimal dynamic response as a function of time. On the other hand, a mathematical model was proposed to describe the chemical sensor’s dynamic behavior based on the electrical response and linear systems theory. The sensor’s transient response was obtained with the model by exposing the oxide to air and propane gas; its stability was checked, and the stabilization time was calculated. Subsequently, an operating point was selected, and, with it, a propane gas detector was designed. The sensor operated ﬂawlessly at 250 ◦ C at a concentration of 1000 ppm, with a response time of three seconds. The developed device is inexpensive and easy to implement.


Introduction
For some years now, the literature has reported that spinel-type transition metal oxides conform to the general formula XM 2 O 4 [1,2], where X can be the divalent cations Fe 2+ , Subsequently, the solvent was evaporated using microwave radiation using a domestic oven (LG, model MS1147 X). The solution was radiated 40 times at 140 W during 70 s/cycle. After evaporation, the precursor material was dried at 200 • C for 8 h and calcined at 250 • C for 5 h, applying a heating ramp of 100 • C/h in air. The material's calcination process was completed using a programmable control muffle (Novatech, Tlaquepaque, Mexico).

Gas Sensing Tests
Before the electrical sensing tests, a crystallographic characterization of the ZnAl 2 O 4 powders calcined at 250 • C was carried out by X-ray powder diffraction using a Panalytical Empyren device with CuKα radiation (λ = 1.546 Å). The 2θ continuous scan range was from 10 to 90 • with steps of 0.026 • at a rate of 30 s/step.
The gas sensing tests were performed on the surface of pellets manufactured with the ZnAl 2 O 4 powders. For this, 0.3 g of the powders were compressed at a pressure of 11 tons for 5 min using a Simplex Ital Equip-25 tons hydraulic equipment. The dimensions of the pellets were 12 mm in diameter and 0.5 mm in thickness. Two ohmic contacts made from colloidal silver paint (Alfa Aesar, 99%, Mexico City, Mexico) were placed on the pellets' surface so that there was good contact between the surface and the electrons present during the experiments. The pellets were introduced into a small metal box (with a volume of 19 cm 3 ) located inside the measuring chamber (with a capacity of 10 −3 torr). For the dynamic tests, the small box had two holes for the inlet and outlet of the test gas. One hole also allowed the introduction of the pellets' electrodes. The gas discharged from the metal box was evacuated by a system installed in the vacuum chamber. A Leybold TM20 electronic detector monitored the test gas partial pressure. The variations in electrical resistance were recorded with a Keithley 2001 multimeter coupled to a control and data acquisition system using the LabView software (National Instruments, Mexico City, Mexico). Mass flow regulators (Brooks Instruments) with a capacity of 2600 cm 3 /min (model GF100CXXC-SH452.6L, Mexico City, Mexico) and 10 cm 3 /min (model GF100CXXC-SH40010C, Mexico City, Mexico) were used to control the propane flows. Figure 1 shows XRD results from the precursor material treated at 250 • C. As expected, at that temperature, the ZnAl 2 O 4 showed high-intensity peaks that suggested its high purity and crystallinity. The peaks' height and width were an indication that the material was made up of nanometric-sized particles [21]. The compound's pure phase was identified through PDF # 65-3104, indicating that the ZnAl 2 O 4 belongs to the family of spinel-type materials, with a cubic crystalline structure (a = 8.087 Å) and a spatial group Fd-3m (227) [2,3,5]. Our results were consistent with those reported by other research groups, which synthesized the same compound by different processes [5,6,[8][9][10].

XRD Analysis
Considering Figure 1, the crystal size t was calculated with Scherrer's equation [22]: where λ is the wavelength of the radiation (λ = 1518 nm), θ is the Bragg angle, and β is the full width at half maximum (FWHM) of the diffraction peak. In our case, all the diffractograms' reflections were considered, obtaining an average crystal size of~14.63 nm.  Considering Figure 1, the crystal size t was calculated with Scherrer's equation [22]: where λ is the wavelength of the radiation (λ = 1518 nm), θ is the Bragg angle, and β is the full width at half maximum (FWHM) of the diffraction peak. In our case, all the diffractograms' reflections were considered, obtaining an average crystal size of ~14.63 nm.

Analysis of the Dynamic Response of the ZnAl2O4
To prove the spinel ZnAl2O4′s ability to detect gas concentrations, dynamic direct current (DC) tests were performed with the prepared pellets (~0.5 mm thick), which were subjected to extra-dry air (500 mL/min) and propane (1000 ppm) flows for measuring the variations in electrical resistance. The experiments were performed in four steps: (1) the pellets were put inside the measurement system placing two electrodes (two-point method) on the ohmic contacts; (2) then, the measuring chamber was heated at a constant temperature of 250 °C, and 500 mL/min of extra-dry air (20% O2 and 80% N2) were injected for five minutes to stabilize the surface of the pellets; (3) after stabilization, 1000 ppm of propane were injected for five minutes, recording variations in the material's electrical resistance immediately; (4) after that, the propane was removed and extra-dry air was injected into the measuring chamber, which caused the electrical resistance to return to its initial values (when the pellets were stabilized in air, step 1). This process was repeated for several cycles until obtaining the results shown in Figure 2, where electrical resistance ( Figure 2a) and sensitivity ( Figure 2b) were plotted as a function of time. Using the results of Figure 2a, we calculated the sensitivity of the pellets with the formula = ( -)/ , where is the electrical resistance in extra dry air and is the electrical resistance in propane. By considering the reciprocal of the electrical resistance (1/electrical resistance), we calculate [1,23]: = ( -)/ , where and are the propane and air conductances, respectively.
From Figure 2a, when the propane molecules interacted with the surface of the pellets, a decrease in electrical resistance was recorded as the exposure time was extended. That was repeated on several occasions (cycles), which indicated that the pellets maintained a stable process of reversibility and stability in air-propane atmospheres [1]. The excellent response of the ZnAl2O4 was attributed mainly to the reaction that took place on

Analysis of the Dynamic Response of the ZnAl 2 O 4
To prove the spinel ZnAl 2 O 4 s ability to detect gas concentrations, dynamic direct current (DC) tests were performed with the prepared pellets (~0.5 mm thick), which were subjected to extra-dry air (500 mL/min) and propane (1000 ppm) flows for measuring the variations in electrical resistance. The experiments were performed in four steps: (1) the pellets were put inside the measurement system placing two electrodes (two-point method) on the ohmic contacts; (2) then, the measuring chamber was heated at a constant temperature of 250 • C, and 500 mL/min of extra-dry air (20% O 2 and 80% N 2 ) were injected for five minutes to stabilize the surface of the pellets; (3) after stabilization, 1000 ppm of propane were injected for five minutes, recording variations in the material's electrical resistance immediately; (4) after that, the propane was removed and extra-dry air was injected into the measuring chamber, which caused the electrical resistance to return to its initial values (when the pellets were stabilized in air, step 1). This process was repeated for several cycles until obtaining the results shown in Figure 2 From Figure 2a, when the propane molecules interacted with the surface of the pellets, a decrease in electrical resistance was recorded as the exposure time was extended. That was repeated on several occasions (cycles), which indicated that the pellets maintained a stable process of reversibility and stability in air-propane atmospheres [1]. The excellent response of the ZnAl 2 O 4 was attributed mainly to the reaction that took place on the pellets' surface between the oxygen and the test gas because of the operating temperature (250 • C) [15,23,24]. The temperature provoked the activation of the charge carriers (electrons) [1], making them move faster on the sensor's surface [23,25], causing an almost immediate change in electrical resistance and an increase in the sensitivity of the pellets [1]. In addition, the temperature favored the diffusion of the test gas [26], which contributed to an increase in the oxide's sensitivity (Figure 2b). The variation in electrical resistance (∆R) was calculated in the range of 1520.04-647.17 kΩ, with an average of 816.85 kΩ (Figure 2a). The sensitivity variations (∆S) were estimated in the range of 0.11-1.33, with maximum peaks recorded at~1353 (Figure 2b). The response time (200 s) was estimated considering 90% of the response in propane; the recovery time (189 s) was calculated by considering 10% of the value when the material was exposed to air [1]. The results depicted in Figure 2a,b were consistent with similar semiconductors studied as gas sensors [1,9,[26][27][28].
The excellent dynamic response and good sensitivity shown in Figure 2a,b indicate that the oxide is an n-type semiconductor [1]. The oxide's behavior was due to the reaction between the ionsorbed oxygen [29] on the material's surface and the propane gas since there was an imbalance between them, leading to the variations in conductivity and, therefore, to the increase in sensitivity. The oxygen species that mostly appeared at 250 • C were the O − and O 2− [30] ionic forms, more reactive than species that predominate below 150 • C [23,30]. That led to the propane's oxidation on the material's surface [1], favoring the release of charge carriers that caused the increase in conductivity and sensitivity [1,25,31,32].
immediate change in electrical resistance and an increase in the sensitivity of the pellets [1]. In addition, the temperature favored the diffusion of the test gas [26], which contributed to an increase in the oxide's sensitivity (Figure 2b). The variation in electrical resistance (ΔR) was calculated in the range of 1520.04-647.17 kΩ, with an average of 816.85 kΩ (Figure 2a). The sensitivity variations (ΔS) were estimated in the range of 0.11-1.33, with maximum peaks recorded at ~1353 (Figure 2b). The response time (200 s) was estimated considering 90% of the response in propane; the recovery time (189 s) was calculated by considering 10% of the value when the material was exposed to air [1]. The results depicted in Figure 2a,b were consistent with similar semiconductors studied as gas sensors [1,9,[26][27][28].
The excellent dynamic response and good sensitivity shown in Figure 2a,b indicate that the oxide is an n-type semiconductor [1]. The oxide's behavior was due to the reaction between the ionsorbed oxygen [29] on the material's surface and the propane gas since there was an imbalance between them, leading to the variations in conductivity and, therefore, to the increase in sensitivity. The oxygen species that mostly appeared at 250 °C were the and [30] ionic forms, more reactive than species that predominate below 150 °C [23,30]. That led to the propane's oxidation on the material's surface [1], favoring the release of charge carriers that caused the increase in conductivity and sensitivity [1,25,31,32]. The excellent results were attributed to the obtained crystallite size. It has been reported that semiconductors with very fine particles or crystallite sizes smaller than 100 nm showed high electrical variations with short response and recovery times [1,23,25,26,29,31,32]. In agreement with references [1,26], the excellent electrical response and sensitivity of our ZnAl2O4 were due to the rapid and effective propane diffusion over the entire material's surface, composed of nanometric crystallites (~14.63 nm) that created a favorable porous structure [29]. Therefore, we consider that the ZnAl2O4 is a strong candidate to be applied as a sensor of toxic gases, mainly propane. Comparing the results in Figure 2 with those reported in the references [9,15,23], we found that our sensor ZnAl2O4 had better sensitivity, excellent dynamic response, good reproducibility, shorter response and recovery times at 250 °C. In addition, the curves obtained from the experiments performed at 1000-ppm propane concentrations (see Figure 2), were consistent with those reported in reference [1]. The excellent results were attributed to the obtained crystallite size. It has been reported that semiconductors with very fine particles or crystallite sizes smaller than 100 nm showed high electrical variations with short response and recovery times [1,23,25,26,29,31,32]. In agreement with references [1,26], the excellent electrical response and sensitivity of our ZnAl 2 O 4 were due to the rapid and effective propane diffusion over the entire material's surface, composed of nanometric crystallites (~14.63 nm) that created a favorable porous structure [29]. Therefore, we consider that the ZnAl 2 O 4 is a strong candidate to be applied as a sensor of toxic gases, mainly propane. Comparing the results in Figure 2 with those reported in the references [9,15,23], we found that our sensor ZnAl 2 O 4 had better sensitivity, excellent dynamic response, good reproducibility, shorter response and recovery times at 250 • C. In addition, the curves obtained from the experiments performed at 1000-ppm propane concentrations (see Figure 2), were consistent with those reported in reference [1].

Theoretical Model
According to the dynamic response of the ZnAl 2 O 4 , depicted in Figure 2, the oxide behaved like a first-order system. Then, based on systems theory [33], its dynamic behavior could be described by the following differential equation: where R(t + t 0 ) is the resistance produced by the oxide due to its exposure to air and propane gas, t 0 is the shifting time due to the equipment stabilization, τ is the system's characteristic time, K is the system's gain, v(t) is the applied voltage, and t is time. Taking the Laplace Transform of the differential Equation (2), we calculated: where R(s) is the resistance in the complex plane s, e t 0 s is the phase produced by the shift, R(0) are the initial conditions of the differential, and L{·} = ∞ 0 ·e st dt is the Laplace transform operator where s = σ + ωi is a complex number: i is the complex rotation operator, ω is the angular frequency rad/s, and σ is a real number. From this last expression, we obtained: and finally, the signal will be According to Equation (6), the resistive response consisted of free and forced parts. The free resistive response R Free (s) depended on the sensor's initial conditions, and it did not depend on the input signal: The forced resistive response R Force (s) depended on the sensor's input signal V(s) but it did not depend on the initial conditions: That is, the total resistive response of the Pellet is formed by the sum of responses R Free (s) and R Force (s),

Transient Response
To obtain the sensor's transient response, we applied input signals V(s) [34,35]. The input signal was a step v(t) = 1 V (whose Laplace transform is V(s) = 1 s ). So, we obtained: where t o is the displacement time of the resistive signal when the sensor was exposed to air, τ air is the characteristic time, K air is the gain, and R air (0) are the initial conditions. Rearranging Equation (10): The calculation process was as follows: as a first step, we determined the free response in the time domain through where L −1 {·} is the inverse Laplace transform operator; as a second step, we calculated the forced response with which, solving for R Force , took the form: where the Heaviside function is due to system drift; as a third step, both responses were added: Substituting expressions (12) and (14) in (15), we obtained: In the limit, when time tends to infinity (and the decay term e − t τ air tends to zero), expression (16) can be approximated to: which corresponded to the resistance behavior when the oxide was exposed to air. On the other hand, to obtain the resistive dynamic response of the ZnAl 2 O 4 due to its exposure to propane gas, we considered the input signal as an impulse (whose Laplace transform is a unit constant). Then, the resistive dynamic behavior was given by: where t 0 + T 2 is the delay time when the sensor was exposed to propane gas, K PG is the system gain, τ PG is the characteristic time, and R PG (0) is the initial condition. Taking its inverse Laplace transform, we calculated where, again, the contribution of the exponential term e − t τ PG was negligible. Then, Equation (20) could be approximated to Equation (21) corresponded to the resistance when the oxide was exposed to propane gas. According to Figure 2, the oxide's dynamic resistive response was periodic because Appl. Sci. 2021, 11, 9488 8 of 15 the designed chemical sensor had repeatability. The first half of the period corresponded to the measured air and the second half of the period corresponded to the measured propane gas. Therefore, its total transient response could be expressed as: Equation (21) corresponded to the resistance when the oxide was exposed to propane gas. According to Figure 2, the oxide's dynamic resistive response was periodic because the designed chemical sensor had repeatability. The first half of the period corresponded to the measured air and the second half of the period corresponded to the measured propane gas. Therefore, its total transient response could be expressed as:   From Figure 3, the theoretical result was close to the experimental measurements when the sensor detected air and propane gas. The minor variations can be attributed to experimental errors, system noise, or contamination in the measurement chamber.

Establishment Time
The stability of the ZnAl 2 O 4 sensor could be determined by knowing the position of the poles belonging to the transfer function. The establishment time could be known with the sensor's "response speed" or its characteristic time.
Since our interest was to detect propane gas atmospheres, we focused on the propane gas transfer function G(s): Substituting values from the first cycle, we calculated: According to the denominator of Equation (25), the sensor's transfer function had one pole at position s = − 1 τ PG = −0.0209, located in the negative half-plane of the complex plane s [36-38] (see Figure 4a). It indicated that the sensor was stable since taking the inverse Laplace transform of the transfer function, a decaying process was obtained (as Equation (23) shows). If the establishment time is defined as 4τ PG [36], the sensor reached its stability at 4τ PG = 191.2 s, with the exit signal at 1.83%. The stability time is shown in Figure 4b.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 16 From Figure 3, the theoretical result was close to the experimental measurements when the sensor detected air and propane gas. The minor variations can be attributed to experimental errors, system noise, or contamination in the measurement chamber.

Establishment Time
The stability of the ZnAl2O4 sensor could be determined by knowing the position of the poles belonging to the transfer function. The establishment time could be known with the sensor's "response speed" or its characteristic time.
Since our interest was to detect propane gas atmospheres, we focused on the propane gas transfer function ( ): Substituting values from the first cycle, we calculated: According to the denominator of Equation (25), the sensor's transfer function had one pole at position = − = −0.0209, located in the negative half-plane of the complex plane [36][37][38] (see Figure 4a). It indicated that the sensor was stable since taking the inverse Laplace transform of the transfer function, a decaying process was obtained (as Equation (23) shows). If the establishment time is defined as 4 [36], the sensor reached its stability at 4 = 191.2 , with the exit signal at 1.83%. The stability time is shown in Figure 4b. Based on Figure 4, the sensor was stable, which allowed us to state that the ZnAl2O4 could be safely applied to detect propane gas (for example, to localize explosion-safe zones).

Frequency Response
In this section, using Bode diagrams, a study is made of the behavior at the sensor frequency [36]. With this type of diagram, the behavior of magnitude vs. angular frequency and phase vs. angular frequency and from these graphs it is possible to determine the frequencies where the sensor operates effectively. Based on Figure 4, the sensor was stable, which allowed us to state that the ZnAl 2 O 4 could be safely applied to detect propane gas (for example, to localize explosion-safe zones).

Frequency Response
In this section, using Bode diagrams, a study is made of the behavior at the sensor frequency [36]. With this type of diagram, the behavior of magnitude vs. angular frequency and phase vs. angular frequency and from these graphs it is possible to determine the frequencies where the sensor operates effectively.
First, we performed a change of variable s = ωi in the transfer function (25) [36,38] and obtained (after some algebraic manipulations): Removing the complex term from the denominator We have: From expression (28), the magnitude was calculated as: and the phase with where −1129.34ω was the offset angle due to the shift time. The shift time only affected the phase and did not affect the magnitude, as shown in Equations (29) and (30). Figure 5 shows the Bode diagram of the transfer function.
First, we performed a change of variable = in the transfer function (25) From expression (28), the magnitude was calculated as: where −1129.34 was the offset angle due to the shift time. The shift time only affected the phase and did not affect the magnitude, as shown in Equations (29) and (30). Figure 5 shows the Bode diagram of the transfer function. According to Figure 5, there was a cutoff frequency of approximately 1.91 kHz. Consequently, the propane gas sensor would operate adequately within the interval from 1 Hz to 1.91 KHz, while for higher frequencies its resistance would decay, as Figure 5 shows. According to Figure 5, there was a cutoff frequency of approximately 1.91 kHz. Consequently, the propane gas sensor would operate adequately within the interval from 1 Hz to 1.91 KHz, while for higher frequencies its resistance would decay, as Figure 5 shows.

Application to the Detection of Propane Gas
According to Figures 2a and 3, the sensor's resistance changed as a function of time. Therefore, if our goal was to apply it in a new propane gas detector, the sensor had to operate at a specific point in the dynamic response, which we called the "operating point". Now, if the sensor's response time was selected at three seconds, the operating point had the coordinates (≈1151.9 [s] ≈ 1,518,215 [KΩ]) (see Figure 6). Then, we designed the electronic circuit from that operating point.

Application to the Detection of Propane Gas
According to Figures 2a and 3, the sensor's resistance changed as a function of time. Therefore, if our goal was to apply it in a new propane gas detector, the sensor had to operate at a specific point in the dynamic response, which we called the "operating point." Now, if the sensor's response time was selected at three seconds, the operating point had the coordinates (≈1151.9 [s] ≈ 1,518,215 [KΩ]) (see Figure 6). Then, we designed the electronic circuit from that operating point.  Figure 7a shows the electronic diagram of the propane gas detector. The detector was based on a Wheatstone bridge, an instrumentation amplifier with two operational amplifiers, a comparator circuit, and a voltage source of ±12 Volts. Its operating temperature was 250 °C, it detected concentrations of 1000 ppm, and its response time was three seconds.
Its working principle consisted of calibration and detection stages [39]. In the calibration stage, the chemical sensor (the ZnAl2O4) was placed in an air atmosphere, its terminals were connected to one arm of the Wheatstone bridge, and the variable resistance was changed until the bridge calibration was achieved, such that the voltage difference (output voltage of the Wheatstone Bridge) was − = 0. and were compared with the instrumentation amplifier, and their difference was multiplied by the gain. The output voltage was defined through: where is the amplifier gain, , , , , are precision resistors, and = = .
When − = 0 was satisfied, the voltage was zero. This voltage may be due to the fact that the comparator circuit had negative saturation and therefore, the diode did not conduct, eliminating any negative signal. Therefore, the alarm signal had a value of zero, = 0. That is, the device would not detect the presence of propane gas in the  Figure 7a shows the electronic diagram of the propane gas detector. The detector was based on a Wheatstone bridge, an instrumentation amplifier with two operational amplifiers, a comparator circuit, and a voltage source of ±12 Volts. Its operating temperature was 250 • C, it detected concentrations of 1000 ppm, and its response time was three seconds.
Its working principle consisted of calibration and detection stages [39]. In the calibration stage, the chemical sensor (the ZnAl 2 O 4 ) was placed in an air atmosphere, its terminals were connected to one arm of the Wheatstone bridge, and the variable resistance R C was changed until the bridge calibration was achieved, such that the voltage difference (output voltage of the Wheatstone Bridge) was V B − V A = 0. V A and V B were compared with the instrumentation amplifier, and their difference was multiplied by the gain. The output voltage was defined through: where R g is the amplifier gain, R 1 , R 2 , R 3 , R 4 , are precision resistors, and K = R 4 R 3 = R 2 R 1 . When V B − V A = 0 was satisfied, the voltage V o was zero. This voltage may be due to the fact that the comparator circuit had negative saturation and therefore, the diode did not conduct, eliminating any negative signal. Therefore, the alarm signal had a value of zero, V Alarm = 0. That is, the device would not detect the presence of propane gas in the atmosphere. In the detection stage, when the sensor surface came into contact with propane gas, the sensor had a surface current and suffered a resistance variation. Consequently, the Wheatstone test was unbalanced, and the voltage V A was less than the voltage V B . Subsequently, the instrumentation amplifier compared both voltages and amplified 10 times, the difference: with R 1 = R 2 = R 3 = R 4 = 1000 Ω, K = 10, and R g = 250 Ω (see Equation (31) and Figure 7). Since the voltage V o was greater than zero, the comparator had positive saturation. Therefore, the alarm signal was equal to the positive saturation voltage of the operational amplifier minus the voltage drop of the rectifier diode (V d = 0.7 V), V Alarm = V sat ≈ 11.3 V. Therefore, the device detected the presence of propane gas in the monitored atmosphere. atmosphere. In the detection stage, when the sensor surface came into contact with propane gas, the sensor had a surface current and suffered a resistance variation. Consequently, the Wheatstone test was unbalanced, and the voltage was less than the voltage . Subsequently, the instrumentation amplifier compared both voltages and amplified 10 times, the difference: with = = = = 1000 Ω, = 10, and = 250 Ω (see Equation (31) and Figure 7). Since the voltage was greater than zero, the comparator had positive saturation. Therefore, the alarm signal was equal to the positive saturation voltage of the operational amplifier minus the voltage drop of the rectifier diode ( = 0.7 V), = ≈ 11.3 . Therefore, the device detected the presence of propane gas in the monitored atmosphere.  Figure 7 shows the propane gas detector's design obtained with proteus ® software. The detector was 10 cm × 10 cm, and its main features were low construction cost, high working temperatures, short response time, high sensitivity, easy repair, and a chemical sensor based on the oxide ZnAl2O4.  Figure 7 shows the propane gas detector's design obtained with proteus ® software. The detector was 10 cm × 10 cm, and its main features were low construction cost, high working temperatures, short response time, high sensitivity, easy repair, and a chemical sensor based on the oxide ZnAl 2 O 4 .
In particular, we performed a comparison of our results obtained from the dynamic response and the designed device of the ZnAl 2 O 4 with other semiconducting oxides that have been investigated as potential gas sensors. For example, in references [1,9,27] they report changes in electrical resistance as a function of time, concentration of propane atmospheres and sensitivity humidity on the ZnAl 2 O 4 and ZnO. According to these authors, its optimum temperature to detect propane is from 200 to 300 • C. On the other hand, in reference [26] they acquired the detection of CO 2 and O 2 over the CoSb 2 O 6 applying an operating temperature of 400 • C. In this work, it was found that the ZnAl 2 O 4 shows excellent dynamic response and sensitivity at 250 • C (1000 ppm of propane), which is a lower temperature than those reported in the previously cited references. In addition, we found that the electrical response of our material was uniform and stable. This is reflected in the number of cycles obtained, which show the thermal stability, and reversibility of the test gas detection process. These parameters and the excellent electrical response, as well as the dynamic sensitivity obtained in the conditions applied during the measurements (250 • C y 1000 ppm of propane) are positive. Theoretical and experimental models were developed for the electronic design of the device based on ZnAl 2 O 4 in dynamic propane atmospheres. Therefore, with the obtained results and the operating conditions of our sensor, these are the optimal parameters to obtain a sensitive, efficient, reliable, easy to implement sensor that presents a prompt response in conditions of propane contamination in the atmosphere.

Discussion
Our experimental results for propane detection were consistent with those reported in references [9,[26][27][28]. The characteristics of the designed propane gas detector were: 1.
11.3 V alarm voltage due to the presence of propane gas.

4.
Short response times when detecting the presence of propane gas (e.g., three seconds).
Supply voltage of 120 Volts. 10. The proposed device finds practical application in boiler safety systems where high temperatures and high concentrations are frequent. 11. If the device is placed in an atmosphere with different conditions, the chemical sensor based on the oxide ZnAl 2 O 6 will have a different electrical response, causing the Wheatstone bridge to decalibrate, and therefore, the device will send an erroneous alarm signal. To solve such a situation, the Wheatstone bridge must be calibrated for the new operating conditions.
Our proposal has a wide range of application where there is interest in detecting areas with a high risk of explosion due to fuel leaks, its construction is economical, and it has excellent functional characteristics. Our future work is to develop gas detectors applying programmable electronic devices and develop an error analysis.

Conclusions
The semiconductor oxide ZnAl 2 O 4 was synthesized by a wet chemistry method and characterized by X-ray diffraction, identifying its pure phase and crystalline structure. For electrical characterization, pellets were made to record their electrical resistance as a function of time, obtaining excellent results.
A mathematical model based on linear systems was proposed, considering the chemical sensor as a first-order system with shift and initial conditions. We obtained the sensor's transient response when exposed to air and propane gas by combining the mathematical model and the experimental results. With the models, its stabilization time was calculated, and its frequency response was verified. We selected an operating point, and a novel device was proposed for working as a propane gas detector. Our proposal has many applications, especially detecting areas with a high explosion risk due to fuel leaks. Its construction is economical and has excellent functional features. Our next aim is to develop gas detectors applying programmable electronic devices.