# Multiparameter Analysis of the Ultrasonic Transducer Transfer Function Using a Genetic Algorithm
## 4 Results of Computational Experiments  
The main goal of this research is to evaluate the potential for using a genetic algorithm for shaping the transfer function of piezoceramic transducers by appropriately selecting the impedances in the matching layers. The use of a genetic algorithm means that the materials forming the matching layer can be selected from the available materials in such a way that allows the optimization of a selected parameter that describes the operation of a transducer in a particular medium (in this case the expected shape of the transfer function). The analysis was performed for three individual media: air (Z\({}_{\text{T}}\) = 427 Rayl), water (Z\({}_{\text{T}}\) = 1.5 MRayl) and a biological medium (skull bone, Z\({}_{\text{T}}\) = 7.8 MRayl). An assumption was made that the transducer was manufactured from PZT ceramic material that has an impedance of Z\({}_{\text{C}}\) = 33 MRayl. The analyzed transducer had either two or three matching layers, without load applied to the back side (Z\({}_{\text{b}}\) = 0). The analysis focused only on the influence of the acoustic impedance of the matching layers on the shape of the transfer function, and therefore assumed a constant thickness of individual layers equal to \(\lambda/4\). The experiments were performed on a database comprising 96 physically available materials with various acoustic properties. The database was developed with the use of data included in a number of scientific articles concerning the design of ultrasonic transducers, and also data in catalogs from the manufacturers of ultrasonic materials [11; 15; 16; 17]. The acoustic properties of the materials used in the experiments are shown in [7]. The analysis of these properties allows us to conclude that the impedance distribution for the available materials is not uniform; the database contains a relatively small number of materials with low and high acoustic impedances. This fact may unfortunately have a negative impact on the obtained results (on the possibility to shape the characteristics in a manner to meet expectations). The transducer's transfer function was evaluated with the use of the fitness function described with Equation (4). The first experiments were performed for a transducer operated in air. In this case, a significant difference is observed between the acoustic impedances of the ceramic material and the air (33 MRayl i 427 Rayl, respectively). The main goal of the experiment was to determine the influence of the \(\alpha_{1}\), \(\alpha_{2}\) and \(\Delta\)k parameters on the obtained transfer function. The properties of the designed transducer can be influenced by adequately selecting the values of \(\alpha_{1}\) and \(\alpha_{2}\). In the case of the combination of \(\alpha_{1}\) = 1.0 and \(\alpha_{2}\) = 0.0, the transducer should demonstrate a significant bandwidth, and in the case of the combination of \(\alpha_{1}\) = 0.0 and \(\alpha_{2}\) = 1.0, the priority is to obtain a transducer with high effectiveness. A transducer with a wide bandwidth has a lower effectiveness, and,conversely, a transducer with high effectiveness has a limited bandwidth. By changing the values of coefficients \(\alpha_{1}\) and \(\alpha_{2}\), a compromise may be reached between the bandwidth and the effectiveness of the transducer. Table 1 shows the obtained values of the acoustic impedances for individual matching layers with respect to the combination of parameters \(\alpha_{1}\) and \(\alpha_{2}\). Figure 4 shows the corresponding transfer functions. The experiments were performed for the value of coefficient \(\Delta\)k = 3 dB. The subsequent experiments were performed for a transducer operated in water. Table 2 and Figure 5 show the results for the following combinations of coefficients: \(\alpha_{1}=1.0/\)\(\alpha_{2}=0.0\) (bandwidth optimization) and \(\alpha_{1}=0.0/\alpha_{2}=1.0\) (effectiveness optimization) and for the assumed value of coefficient \(\Delta\)k = 3 dB.