## 3 Results and Discussion
### Propane Analysis
Therefore, if we consider an operation point at \([\overline{\mathbf{t}},\ \mathbf{z}(\overline{\mathbf{t}})]\) and \([\overline{\mathbf{t}},\ \mathbf{\theta}(\overline{\mathbf{t}})]\)[67], the impedance would be as follows:  
\[\mathbf{z}_{\mathbf{s}}(\mathbf{s})=\mathbf{z}_{\mathbf{c}}+\mathbf{z}_{\mathbf{d}}(\mathbf{s})=\mathbf{z}_{\mathbf{ c}}+\frac{\mathbf{r}_{\mathbf{ds}}}{\mathbf{r}_{\mathbf{ds}}+\mathbf{X}_{\mathbf{ds}}}. \tag{6}\]  
Substituting the identity \(\mathbf{X}_{\mathbf{ds}}=\frac{1}{\mathbf{s}\mathbf{c}_{\mathbf{ds}}}\), the impedance takes the following form:  
\[\mathbf{z}_{\mathbf{s}}(\mathbf{s})=\mathbf{z}_{\mathbf{c}}+\mathbf{z}_{\mathbf{d}}(\mathbf{s})=\mathbf{z}_{\mathbf{ c}}+\frac{\mathbf{r}_{\mathbf{ds}}\mathbf{C}_{\mathbf{ds}}\mathbf{s}}{\mathbf{r}_{\mathbf{ds}}\mathbf{C}_{\mathbf{ds}} \mathbf{s}+1}. \tag{7}\]  
If \(\overline{\mathbf{t}}=300\ \mathbf{s}\), we obtain \(\mathbf{z}_{\mathbf{c}}=1754362e^{-74.3106}\), \(\mathbf{r}_{\mathbf{ds}}=18167\ \mathbf{\Omega}\), \(\mathbf{C}_{\mathbf{ds}}=9.37\ \mathrm{pF}\), and the impedance as follows:  
\[\mathbf{z}_{\mathbf{s}}(\mathbf{s})=1754362e^{-74.3106}+\frac{0.0001704s}{0.0001704s+1}. \tag{8}\]  
From Equation (8) and using the impedance \(\mathbf{z}_{\mathbf{d}}(\mathbf{s})\), we obtained the Bode diagram shown in Figure 11a. Comparing this diagram with NiSb\({}_{2}\)O\({}_{6}\)'s electrical response (Figure 10a,b), the results are consistent because \(|\mathbf{Z}|\) decreases when the frequency increases from 0.1 to 1 kHz. The detector's electronic diagram in propane atmospheres is shown in Figure 11b. NiSb\({}_{2}\)O\({}_{6}\) pellets working as detectors were placed in one of the Wheatstone bridge's arms, whose output voltages were \(\mathbf{V}_{\mathbf{a}}=\frac{\mathbf{z}_{\mathbf{s}}}{\mathbf{z}_{\mathbf{s}}+\mathbf{z}_{\mathbf{1}}}\mathbf{V} _{\mathbf{ac}}\) and \(\mathbf{V}_{\mathbf{b}}=\frac{\mathbf{z}_{\mathbf{s}}}{\mathbf{z}_{\mathbf{a}}+\mathbf{z}_{\mathbf{2}}}\mathbf{V} _{\mathbf{ac}}\), where \(\mathbf{z}_{\mathbf{ca}}\) is the variable impedance employed to calibrate the bridge, \(\mathbf{z}_{1}\) and \(\mathbf{z}_{2}\) are impedances, and \(\mathbf{V}_{\mathbf{ac}}\) is the supply voltage. Voltages \(\mathbf{V}_{\mathbf{a}}\) and \(\mathbf{V}_{\mathbf{b}}\) are the input signals of an adder-subtractor circuit, whose output voltage is \(\mathbf{V}_{\mathbf{ab}}=\mathbf{V}_{\mathbf{a}}-\mathbf{V}_{\mathbf{b}}=\frac{\mathbf{z}_{\mathbf{s}}}{\mathbf{z} _{\mathbf{s}}+\mathbf{z}_{\mathbf{1}}}\mathbf{V}_{\mathbf{ac}}-\frac{\mathbf{z}_{\mathbf{s}}}{\mathbf{z}_{ \mathbf{ca}}+\mathbf{z}_{\mathbf{2}}}\mathbf{V}_{\mathbf{ac}}\). This output voltage is the input signal of the comparator circuit, whose inverter's input and supply \(-\mathbf{V}_{\mathbf{c}}\) were connected to a common for avoiding negative voltages. The detector's operating principle is shown in Figure 11c, which is based on the following two basic stages: calibration and detection. In the calibration stage, the sensor is exposed to the air-atmosphere, where the Wheatstone bridge was calibrated through the variable impedance \(\mathbf{z}_{\mathbf{c}}\) until the condition \(\mathbf{z}_{1}=\mathbf{z}_{2}=\mathbf{z}_{\mathbf{c}}=\mathbf{z}_{\mathbf{s}}\) was met, yielding \(\mathbf{V}_{\mathbf{ab}}=0\ \mathbf{V}\) and no alarm signal: \(\mathbf{V}_{\mathbf{A}\mathbf{l}\mathbf{am}}\approx 0\ \mathbf{V}\). In the test gas detection stage, the sensor showed electrical variations in propane atmospheres, whereby the Wheatstone bridge was decalibrated, causing \(\mathbf{V}_{\mathbf{ab}}\
eq 0\), \(\mathbf{V}_{\mathbf{a}}>\mathbf{V}_{\mathbf{b}}\), and, as a result, the comparator circuit to be positively saturated with \(\mathbf{V}_{\mathbf{A}\mathbf{l}\mathbf{am}}\approx\mathbf{V}\mathbf{c}\mathbf{c}\), which was the detector's alarm signal in the test gas. In general, it is well known that the performance of semiconductors such as ours (NiSb\({}_{2}\)O\({}_{6}\)) depends, to a great extent, on the microstructure and the crystal size.