# Multiparameter Analysis of the Ultrasonic Transducer Transfer Function Using a Genetic Algorithm
## 4 Results of Computational Experiments
The experiments were also performed for other combinations of coefficients \(\alpha_{1}\) and \(\alpha_{2}\). The resultant transfer function curves, however, were very similar to one of the characteristics shown in Figure 5. In this case, it was impossible to obtain \"intermediate\" transfer function characteristics, which was also the case with the transducer operated in air. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Number** & \(\alpha_{1}\) & \(\alpha_{2}\) & \(Z_{c1}\) & \(Z_{c2}\) & \(Z_{c3}\) & \(\Delta\)\(\boldsymbol{\mathit{ff_{0}}}\) \\ \cline{3-8}
**of Layers** & \(\alpha_{1}\) & \(\alpha_{2}\) & [kRay] & [kRay] & [kRay] & [kRay] & \(\Delta\)\(\boldsymbol{\mathit{ff_{0}}}\) \\ \hline  & 1.0 & 0.0 & 14,100 & 17 & - & 0.033 \\  & 2.0 & 1.0 & 3848 & 17 & - & 0.032 \\
2 & 1.0 & 1.0 & 2680 & 17 & - & 0.030 \\  & 1.0 & 1.25 & 1790 & 17 & - & 0.026 \\  & 0.0 & 1.0 & 81 & 80 & - & 0.002 \\ \hline  & 1.0 & 0.0 & 16,200 & 98 & 95 & 0.258 \\  & 1.0 & 4.0 & 16,200 & 211 & 84 & 0.236 \\
3 & 1.0 & 5.0 & 3848 & 1760 & 17 & 0.050 \\  & 1.0 & 7.0 & 203 & 131 & 17 & 0.034 \\  & 0.0 & 1.0 & 2320 & 2310 & 17 & 0.018 \\ \hline \end{tabular}
\end{table} 
Table 1: Acoustic impedances of matching layers obtained with the genetic algorithm (air, \(Z_{T}=0.427\) kRayl). Figure 4: Analysis of the transfer function for the transducer operated in air: (**a**) with 2 matching layers, (**b**) with 3 matching layers. The analysis of the characteristics shown in Figure 5 allows us to conclude that the difference in the transfer function value between the two characteristics for frequency \(\mathrm{f_{0}}\) is approximately 10 dB. Therefore, in the case of a transducer with high bandwidth, the loss of effectiveness may be viewed as insignificant and acceptable. For this reason, the bandwidth was increased by allowing a higher unevenness of the characteristics. Table 3 and Figure 6 show the results for the values of \(\mathrm{\Delta k}\) equal to 3 dB, 6 dB and 10 dB. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \begin{tabular}{c} **Number** \\ **of Layers** \\ \end{tabular} & \(\boldsymbol{\alpha_{1}}\) & \(\boldsymbol{\alpha_{2}}\) & \
\begin{tabular}{c} **Z\({}_{\text{c1}}\)** \\ **[kRayl]** \\ \end{tabular} & \(\boldsymbol{Z_{\text{c2}}}\) &
\begin{tabular}{c} **Z\({}_{\text{c3}}\)** \\ **[kRayl]** \\ \end{tabular} & \(\boldsymbol{\Delta\text{f}/\text{f}_{0}}\) \\ \hline \multirow{2}{*}{2} & 1.0 & 0.0 & 14,100 & 3270 & - & 0.730 \\  & 0.0 & 1.0 & 2420 & 2411 & - & 0.028 \\ \hline \multirow{2}{*}{3} & 1.0 & 0.0 & 16,200 & 4930 & 1950 & 0.800 \\  & 0.0 & 1.0 & 3000 & 2989 & 1745 & 0.038 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Acoustic impedances of the matching layers obtained with the genetic algorithm (water, \(\mathrm{Z_{T}}=1500\) kRayl, the impact of \(\mathrm{\alpha_{1}}\) and \(\mathrm{\alpha_{2}}\) coefficients). Figure 5: Analysis of the transfer function for the transducer operated in water: (**a**) with 2 matching layers, (**b**) with 3 matching layers. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
\begin{tabular}{c} **Number of** \\ **Layers** \\ \end{tabular} & \(\boldsymbol{\Delta k}\) **[dB]** & \(\boldsymbol{Z_{\text{c1}}}\) **[kRayl]** & \(\boldsymbol{Z_{\text{c2}}}\) **[kRayl]** & \(\boldsymbol{Z_{\text{c3}}}\) **[kRayl]** & \(\boldsymbol{\Delta\text{f}/\text{f}_{0}}\) \\ \hline \multirow{2}{*}{2} & 3.0 & 14,100 & 3270 & - & 0.730 \\  & 6.0 & 16,200 & 4130 & - & 0.832 \\  & 10.0 & 16,200 & 5475 & - & 0.904 \\ \hline \multirow{3}{*}{3} & 3.0 & 16,200 & 4930 & 1950 & 0.800 \\  & 6.0 & 16,200 & 7300 & 2858 & 0.956 \\ \cline{1-1}  & 10.0 & 16,200 & 9150 & 3450 & 1.090 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Acoustic impedances of the matching layers obtained with the genetic algorithm (water, \(\mathrm{Z_{T}}=1500\) kRayl, the impact of \(\mathrm{\Delta k}\) coefficient). Subsequently, the analysis focused on the operation of the transducer in the biological medium.