# Model-Guided Manufacturing of Transducer Arrays Based on Single-Fibre Piezocomposites
## 2 Materials and Methods
### Initial Model Fit  
To compare measured and modelled transducer responses, the EMIs of unprocessed PZT fibre discs were measured with an impedance analyser (HP4191A, Hewlett Packard). For quantitative comparison, four properties were obtained from the measurements. * The series resonance \(f_{\text{s}}\) of the thickness resonator resonance. * The EMI magnitude \(Z_{\text{min}}\) at \(f_{\text{s}}\). * The approximated parallel capacitance \(C_{\text{P}}=1/\omega X_{\text{C}}\), calculated at 500 kHz. With phase angles close to \(-90^{\circ}\), the transducers behave almost purely capacitive at this frequency. * The maximum phase angle \(\varphi_{\text{t}}\) of the thickness resonator resonance. Figure 8: Equivalent circuit model of our transducer, based on the KLM model. The fibre and the conductive electrode form the basic model. This is then extended by adding transmission lines for each new layer. As model optimisation metric, the PZT fibre parameters were adjusted until \(f_{\text{s}}\) and \(C_{\text{p}}\) were within 5% of the measured average (see Table 1 for the quantitative comparison at the initial step). Figure 9a shows magnitude and phase of 68 transducers (four PZT fibre discs). This sample size was chosen to be sufficient for estimating average responses. Bold curves are measured data, while the dashed blue curves are the model predictions. The grey areas indicate all measured EMI variations. There, the centre curve is not the measured average but one single transducer, chosen for its good accordance with the initial average data. Using the parameters from the datasheet, the model predicts \(f_{\text{s}}\) at 2.46 MHz. This differs by 0.54-MHz from the measured average. To investigate this difference, \(f_{\text{s}}\) was calculated using the thickness frequency constant \(N_{\text{t}}\). This results according to Equation (1) in 2.44 MHz for \(f_{\text{s}}\). As the calculated value also differs from the measured average, a change in material properties caused by the PZT fibre fabrication process may have occurred. \[f_{\text{s}}=\frac{N_{\text{t}}}{d_{0}} \tag{1}\]  
To improve the prediction accuracy, at first, the conductive adhesive electrode was added to the model. Then, the supply line capacitances \(C_{\text{par}}\) was set to 0.6 pF, accounting for the electrode on the PZT fibre disc. As shown in Figure 9b, adding these parameters reduced the EMI offset, but the difference in \(f_{\text{s}}\) remained almost constant. The series resonance \(f_{\text{s}}\) is predominately affected by the elastic stiffness \(c_{33}^{\text{D}}\). When changing this parameter, it is important to consider the dependent character of piezoelectric material properties. By using the parameter set from the materials datasheet in strain-charge form, \(c_{33}^{\text{D}}\) can be derived \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Assembly** & \(f_{\text{s}}\)**(MHz)** & \multicolumn{3}{c}{\(C_{\text{p}}\)**(pF)**} & \multicolumn{3}{c}{\(Z_{\text{min}}\)**(k\(\Omega\))**} & \multicolumn{3}{c}{\(\varphi_{\text{t}}\)**(\({}^{\circ}\))**} \\ \hline
**Step** & **meas.** & **mod.** & **meas.** & **mod.** & **meas.** & **mod.** & **meas.** & **mod.** \\ \hline (1) Initial & \(1.84\pm 0.04\) & \(1.86\) & \(3.35\pm 0.23\) & \(3.23\) & \(14.18\pm 1.45\) & \(13.38\) & \(-3.12\pm 9.08\) & \(2.61\) \\ (2) PCB & \(1.83\pm 0.04\) & \(1.86\) & \(5.42\pm 0.92\) & \(4.96\) & \(11.19\pm 1.34\) & \(11.27\) & \(-31.56\pm 10.26\) & \(-28.68\) \\ (3) Matching L. & \(1.94\pm 0.05\) & \(2.04\) & \(5.25\pm 1.06\) & \(4.99\) & \(13.42\pm 2.06\) & \(12.71\) & \(-45.54\pm 14.13\) & \(-45.56\) \\ (4) Final & \(1.94\pm 0.05\) & \(2.04\) & \(6.87\pm 0.89\) & \(6.26\) & \(11.74\pm 1.40\) & \(12.80\) & \(-69.27\pm 4.83\) & \(-69.22\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: EMI properties (average and standard deviation) of four PZT fibre discs (68 transducers) initially and after each assembly step. In addition, data from the model responses are listed.