# Multiparameter Analysis of the Ultrasonic Transducer Transfer Function Using a Genetic Algorithm
## 2 Transfer Function of Piezoceramic Transducer  
A number of methods allow the analysis and design of piezoelectric transducers with matching layers intended for operation in various media (e.g., solid, liquid, gaseous). These methods employ various mathematical models that describe the behavior of a system consisting of a piezoceramic plate, a matching layer, and a load [6; 10; 11; 12]. One such model analyzes the operation of the transducer from the perspective of the transmission of electric or acoustic signals in the time domain. The model is based on a system of differential equations describing the relationship between electric and acoustic signals, which, in turn, allows the calculation of the transducer's response to any excitation, as well as to the calculated transducer input impedance and its transfer function. The transfer function provides a considerable amount of important information regarding the properties of the multilayer transducer system (especially in the case of designing broadband transducers), and it can therefore be used in selecting the matching layers. Figure 1 shows a physical model of a piezoceramic transducer with multicomponent matching layers. \(Z_{C}\) is the transducer impedance, symbols \(Z_{C1}\)-\(Z_{Cn}\) are the impedances of individual matching layers, \(Z_{T}\) is the impedance of an unlimited medium which receives the energy from the transducer, and \(Z_{b}\) is the impedance of the back layer of the transducer. Symbols \(\mathrm{l}_{i}\) are the thicknesses of individual transducer layers. A piezoceramic transducer with matching layers can be viewed as a linear system. In such a case, the transducer's transfer function is the Fourier transform of its pulse response. A direct calculation of the Fourier transform for a pulse response is a complex task. Therefore, it is more common for the transfer function to be calculated by applying the Fourier transform to the relevant differential equations which describe how the acoustic velocities of waves propagating in the individual layers of the transducer are related to the changes in the electric charge between the electrodes of the transducer due to its excitation. The following relationship is obtained for a transducer with one matching layer [13; 14]:  
\[\mathrm{K_{C2}(j\omega)}=\frac{1}{j\omega}\;\frac{h}{2Z_{T}}\;(1+\beta_{C1})(1 +\beta_{C2})\;\frac{Q_{C2}(j\omega)}{R_{C2}(j\omega)}, \tag{1}\]  
where  
\[Q_{C2}(j\omega)=\left(1-e^{-j\omega\tau}\right)\;\left(1-\beta_{b}e^{-j\omega \tau}\right)e^{-j\omega\tau_{1}} \tag{2}\]  
\[R_{C2}(j\omega)=1+\beta_{C1}\beta_{C2}e^{-j\omega\tau_{1}}+\beta_{b}\beta_{C1} e^{-j\omega\tau}+\beta_{b}\beta_{C2}e^{-j2\omega(\tau+\tau_{1})} \tag{3}\]  
The symbols \(\beta_{b}\), \(\beta_{C1}\) and \(\beta_{C2}\) used in the above equations represent the reflection coefficients on the border of the ceramic material and the back surface of the transducer, the ceramic material and the matching layer, and the matching layer and the medium in which the transducer operates, respectively. The above coefficients depend on impedances \(Z_{b}\), \(Z_{C}\), \(Z_{C1}\) and \(Z_{T}\). The \(\tau\) and \(\tau_{1}\) symbols represent the delay of the acoustic wave in the piezoceramic material and in the matching layer, respectively. The delay depends on the thickness of a particular layer and on the propagation velocity of the wave in a particular layer. The coefficient \(h\) is the piezoelectric constant. In order to calculate the transducer transfer function, a calculation should be performed of the module of function \(\mathrm{K_{C2}(j\omega)}\), as described by relationship (1). Analogical relationships for the transducer with two and three matching layers are described in publications [4; 5]. The calculations for the above relationships are provided in [13]. Relationship (1) indicates that an appropriate selection of the reflection coefficients \(\beta\) and delays \(\tau\) allows the shaping of the attenuation and phase characteristics of the transducer.