## 3 Results
### Theoretical Model
#### 3.3.1 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 \(V(s)=\frac{1}{s}\)). So, we obtained:  
\[R_{air}(s)=\frac{K_{air}}{s(\tau_{air}s+1)}e^{-t_{0}s}+\frac{\tau_{air}R_{air} (0)}{\tau_{air}s+1}, \tag{10}\]  
where \(t_{o}\) is the displacement time of the resistive signal when the sensor was exposed to air, \(\tau_{air}\) is the characteristic time, \(K_{air}\) is the gain, and \(R_{air}(0)\) are the initial conditions. Rearranging Equation (10):  
\[R_{air}(s)=\frac{\frac{K_{air}}{\tau_{air}}}{s\Big{(}s+\frac{1}{\tau_{air}} \Big{)}}e^{-t_{0}s}+\frac{R_{air}(0)}{s+\frac{1}{\tau_{air}}}. \tag{11}\]  
The calculation process was as follows: as a first step, we determined the free response in the time domain through  
\[L^{-1}\{R_{Free}(s)\}=L^{-1}\Bigg{\{}\frac{R_{air}(0)}{s+\frac{1}{\tau_{air}} }\Bigg{\}}\;\;\rightarrow\;\;R_{Free}(t)=R_{air}(0)e^{-\frac{t}{\tau_{air}}}, \tag{12}\]where \(L^{-1}\{\cdot\}\) is the inverse Laplace transform operator; as a second step, we calculated the forced response with  
\[L^{-1}\{R_{Force}(s)\}=L^{-1}\left\{\frac{\frac{K_{air}}{\tau_{air}}}{s\Big{(}s+ \frac{1}{\tau_{air}}\Big{)}}e^{-t_{0}s}\right\}=L^{-1}\left\{K_{air}\left(\frac{ 1}{s}-\frac{1}{s+\frac{1}{\tau_{air}}}\right)e^{-t_{0}s}\right\}, \tag{13}\]  
which, solving for \(R_{Force}\), took the form:  
\[R_{Force}(t)=K_{air}\bigg{(}1-e^{-(\frac{t}{\tau_{air}}+t_{0})}\bigg{)}u(t-t_{0 }), \tag{14}\]  
where the Heaviside function is due to system drift; as a third step, both responses were added:  
\[L^{-1}\{R_{air}(s)\}=L^{-1}\{R_{Force}(s)\}+L^{-1}\{R_{Free}(s)\}\rightarrow\ R_{air}(t)=R_{Force}(t)+R_{Free}(t). \tag{15}\]  
Substituting expressions (12) and (14) in (15), we obtained:  
\[R_{air}(t)=K_{air}\bigg{(}1-e^{-(\frac{t}{\tau_{air}}+t_{0})}\bigg{)}u(t-t_{0 })+R_{air}(0)e^{-\frac{t}{\tau_{air}}}. \tag{16}\]  
In the limit, when time tends to infinity (and the decay term \(e^{-\frac{t}{\tau_{air}}}\) tends to zero), expression (16) can be approximated to:  
\[R_{air}(t)=K_{air}\bigg{(}e^{-(\frac{t}{\tau_{air}}+t_{0})}\bigg{)}u(t-t_{0})+ R_{air}(0), \tag{17}\]  
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:  
\[R_{PG}(s)=\frac{\frac{K_{PG}}{\tau_{PG}}}{s+\frac{1}{\tau_{PG}}}e^{-(t_{0}+ \frac{T}{2})s}+\frac{R_{PG}(0)}{s+\frac{1}{\tau_{PG}}}, \tag{18}\]  
where \(t_{0}+\frac{T}{2}\) is the delay time when the sensor was exposed to propane gas, \(K_{PG}\) is the system gain, \(\tau_{PG}\) is the characteristic time, and \(R_{PG}(0)\) is the initial condition. Taking its inverse Laplace transform,  
\[L^{-1}\{R_{PG}(s)\}=L^{-1}\left\{\frac{\frac{K_{PG}}{\tau_{PG}}}{s+\frac{1}{ \tau_{PG}}}e^{-(t_{0}+\frac{T}{2})s}\right\}+L^{-1}\left\{\frac{R_{PG}(0)}{s+ \frac{1}{\tau_{PG}}}\right\}, \tag{19}\]  
we calculated  
\[R_{air}(s)=\frac{\frac{K_{air}}{\tau_{air}}}{s\Big{(}s+\frac{1}{\tau_{air}} \Big{)}}e^{-t_{0}s}+\frac{R_{air}(0)}{s+\frac{1}{\tau_{air}}}. \tag{11}\]0  
where, again, the contribution of the exponential term \(e^{-\frac{t}{\tau_{PG}}}\) was negligible. Then, Equation (20) could be approximated to  
\[R_{air}(s)=\frac{\frac{K_{air}}{\tau_{air}}}{s\Big{(}s+\frac{1}{\tau_{air}} \Big{)}}e^{-t_{0}s}+\frac{R_{air}(0)}{s+\frac{1}{\tau_{air}}}. \tag{11}\]1  
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:  
\[