## 6 Multi-Gas Analysis  
In the multicomponent system, if the response signal of each component at wavelength \(\lambda\) (corresponding to the laser injection current \(I\)) is linear with its concentration, the photoacoustic signal \(Y(I)\) of the mixture can be expressed as the linear superposition of the signal at this wavelength related to each gas:  
\[S(I)=\sum_{i=1}^{n}A_{i}(I)C_{i}+B(I)=A_{CH_{4}}(I)C_{CH_{4}}+A_{C_{2}H_{4}}(I) C_{C_{2}H_{4}}+A_{C_{2}H_{6}}(I)C_{C_{2}H_{6}}+B(I) \tag{1}\]  
where \(S(I)\) is the DHR signal obtained at the laser injected current \(I\) (in mV), \(A_{i}(I)\) is the responsivity at current \(I\) for gas \(i\) (in mV/ppm), \(C_{i}\) is the concentration of gas \(i\) (in ppm), and \(B(I)\) is the background signal (in mV). We prepared seven different gas mixtures; the mixed gas sample is prepared in two ways. One is to dilute a bottle of high-concentration mixture with nitrogen. The high-concentration mixture contains three gases, and their  
Figure 10: (**a**) First harmonic signals of ethane (2 to 90 ppm) with the laser current in the range of 53 to 100 mA; (**b**) Linear fit of ethane absorption peak at \(I_{3}=88.23\) mA; (**c**) Continuous measurement of 18.7 ppm ethane; (**d**) The Allan deviation analysis. concentrations are close to each other. The first four mixtures were prepared by this method. The other way is to dilute three separate bottles of high-concentration single standard gas with nitrogen so that the concentration of each gas obtained can be changed at will. Three samples were prepared this way and the concentration difference between components is set to 3-5 times. The gas-dispensing, inflation, and measurement operations of each sample were repeated three times, and the test results are taken as the average value. The concentration of each component of the mixture is calculated by linear regression using both the single wavelength data and the multiwavelength data. Table 1 lists the responsivity and signal-concentration linearity (R\({}^{2}\)) at the corresponding absorption peak of the three gases. Table 2 gives the nominal concentration of each component and the concentrations retrieved by the two regression procedures. Ethane has a good absorption peak shape and the highest responsivity in this band. Furthermore, the signal at \(I_{3}=88.33\) mA is basically not interfered with by the methane and ethylene signals (<1% interference). Therefore, even if the concentration of ethane is directly calculated from the working curve in Figure 9b, there will be no obvious error, and the relative error of all seven samples is less than \(\pm\)10%. As mentioned above, in the case of a single component gas of methane, ethylene, or ethane, the absorption at right the peak and at wavelengths near the absorption peak show good linearity. However, when the absorption strength is small, the linear relationship far from the absorption peak worsens \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{2}{c}{\(I_{1}\) (62.35 mA)} & \multicolumn{2}{c}{\(I_{2}\) (73.88 mA)} & \multicolumn{2}{c}{\(I_{3}\) (88.33 mA)} \\ \cline{2-7}  & **Slope** & \(\mathbf{R^{2}}\) & **Slope** & \(\mathbf{R^{2}}\) & **Slope** & \(\mathbf{R^{2}}\) \\ \hline CH\({}_{4}\) & 4.953 & 0.999 & \(-\)0.130 & 0.972 & \(-\)0.00622 & 0.822 \\ C\({}_{2}\)H\({}_{4}\) & 0.232 & 0.999 & \(-\)1.778 & 0.999 & \(-\)0.125 & 0.953 \\ C\({}_{2}\)H\({}_{6}\) & \(-\)0.038 & 0.964 & \(-\)0.355 & 0.999 & 13.780 & 0.999 \\ \hline \end{tabular}
\end{table} 
Table 1: Sensitivities (mV/ppm) and linearity (R\({}^{2}\)) of single wavelength linear regression at the peak wavelengths (injection currents) for methane, ethylene, and ethane.