# Model-Guided Manufacturing of Transducer Arrays Based on Single-Fibre Piezocomposites
## Appendix A KLM Model  
The implemented model can be reduced to a 2 \(\times\) 2 transfer matrix, with electrical power (\(P_{\text{in}}=U_{\text{in}}\cdot I_{\text{in}}\)) at the input and mechanical (radiation) power (\(P_{\text{out}}=F_{\text{out}}\cdot v_{\text{out}}\)) at the output. It connects the input and output magnitudes using voltage-to-force conversion. By using the two-port formalism, the model can be divided into arbitrary sub-matrices, which are subsequently concatenated. Figure 11 shows a scheme of the used sub-matrices, where each block correlates to a transferred impedance. As indicated, the backward radiation is connected in parallel with the forward radiation. The blocking capacitance \(C_{0}\) is calculated according to Equation (11), where \(\epsilon_{33}^{\text{S}}\) is the permittivity at constant strain, \(A_{0}=r_{0}^{2}\pi\) the electrode surface area and \(d_{0}\) the thickness of the fibre. Dielectric losses are considered by adding the imaginary part \(j\text{tan}(\delta)\) to the permittivity, which includes the loss factor \(\delta\) of the material. \[C_{0}=e_{33}^{\text{S}}(1-j\text{tan}(\delta))\cdot\frac{A_{0}}{d_{0}} \tag{12}\]  
The acoustic impedance \(Z_{0}\) of the fibre material is determined using Equation (12), where \(\rho\) is the density and \(v_{0}\) the materials speed of sound. Due to the anisotropic behaviour of the material, the speed of sound can only be approximated in thickness direction according to Equation (13). There, \(c_{33}^{\text{D}}\) is the elastic stiffness at constant dielectric displacement. \[Z_{0}=\rho\cdot v_{0} \tag{13}\]  
\[v_{0}=\sqrt{\frac{c_{33}^{\text{D}}}{\rho}} \tag{14}\]  
The fibre material is modelled with a transformer coupled to the centre of a transmission line (see Figure 8). It converts electrical input power with a frequency dependent turns ratio \(\Phi(\omega)\) to a mechanical wave. A frequency dependent capacitance \(C^{\prime}(\omega)\) acts as reactance, transformed to the primary side of the transformer. The used analytical expressions for the turns ratio and the reactance are given in Equation (14) and Equation (15). More detailed descriptions of this modelling approach can be found in [30]. \[\Phi(\omega)=k_{\text{t}}\cdot\sqrt{\frac{d_{0}}{v_{0}C_{0}Z_{0}}}\cdot\text{ sinc}(\omega/2\omega_{0}) \tag{15}\]  
\[C^{\prime}(\omega)=\frac{-C_{0}}{k_{\text{t}}^{2}\cdot\text{sinc}(\omega/ \omega_{0})}\quad\text{with}\quad\omega_{0}=\pi v_{0}/d_{0} \tag{16}\]  
To account for backward radiation, the impedance of the back-side medium \(Z_{\text{B}}\) must be transferred to the centre of the PZT fibre. This is done in two steps. First, \(Z_{\text{B}}\) is transferred to the input of the backing layer \(Z_{3}\) according to Equation (16). Second, the resulting impedance \(Z_{\text{B1}}\) is transferred tothe centre of the PZT fibre using Equation (A7). Elastic losses are modelled by adding an attenuation coefficient \(\alpha\) to the wave number \(\beta\), both individually calculated for each transmission line. \[Z_{\text{B1}}(\omega)=Z_{3}A_{0}\cdot\frac{Z_{\text{B}}+j(Z_{3}\cdot\tan(\beta_{ 3}d_{3}))}{Z_{3}+j(Z_{\text{B}}\cdot\tan(\beta_{3}d_{3}))}\quad\text{with}\quad \beta_{3}=\frac{\omega}{v_{3}}(1-j\alpha_{3}d_{3})\] (A6)  
\[Z_{\text{PB}}(\omega)=Z_{0}A_{0}\cdot\frac{Z_{\text{B1}}+j(Z_{0}\cdot\tan( \beta_{0}d_{0}/2))}{Z_{0}+j(Z_{\text{B1}}\cdot\tan(\beta_{0}d_{0}/2))}\quad \text{with}\quad\beta_{0}=\frac{\omega}{v_{0}}(1-j\alpha_{0}d_{0}/2)\] (A7)  
The overall transmission matrix of the transducer is then calculated according to Equation (A8), where each 2 \(\times\) 2 matrix corresponds to one of the two-ports shown in Figure A1. The wave numbers \(\beta_{1}\) and \(\beta_{2}\) are calculated according to Equation (A6), only with their respective thickness, velocity and attenuation coefficient. \[