## 3 Simulation  
In this chapter, MATLAB simulation was performed to analyze the relationship between the entire system resonance frequency of the ultrasonic system output stage and the mechanical resonance frequency of the piezo element. Usually, the capacitor \(C_{f}\) value of the \(LC\) filter is selected to be about 10 times larger than the parasitic capacitor \(C_{0}\) value of the piezo element. The reason is to optimize the performance of the filter and minimize the influence of parasitic capacitance. If \(C_{f}\) is 10 times \(C_{1}\), the value of \(C_{2}\) is about 1.1 times \(C_{f}\). Therefore, the influence of the change in \(C_{1}\) on \(C_{2}\) is reduced to less than 10%; that is, when \(C_{2}\) is about 1.1 times \(C_{f}\), the change in the resonance frequency is about \(\sqrt{1.1}\), which is about 5%. Based on this, the value of \(C_{f}\) was selected as 33 nF and the simulation was performed, and the filter was designed with the corresponding value in the ultrasonic system tested in this study. In an ultrasonic system, matching the resonant frequency of the inverter output stage and the resonant frequency of the piezo element as closely as possible is an important part of the system design. When the resonant frequency of the piezo element and the resonant frequency of the inverter output match, the energy transfer efficiency is maximized. Impedance matching is also an important part, because the impedance of the piezo element is minimal at the resonant frequency, which matches well with the output impedance of the inverter. Therefore, since the \(C_{f}\) value was determined above, the inductor \(L\)f value that matches the corresponding value was selected as 867 uH and the simulation was conducted, and the \(LC\) filter was finally designed with the corresponding values above. The MATLAB simulation conditions that considered the design details of the ultrasonic system as above are as shown in Table 1 below. Although \(L_{1}\) and \(C_{1}\) of the piezo element are values that change depending on the pressure, the simulation was conducted by setting the parameter values for each in order to express them in the graph. However, in the graph of Figure 8, \(L_{1}\) and \(C_{1}\) change in a monotonically increasing pattern. Figure 8 is an s-plane graph according to the pressure fluctuation of the piezo element. The result is that the characteristic root is plotted by assigning different colors to each \(C_{1}\) value and maintaining the same color for the \(L_{1}\) value. The \(LC\) filter value is a fixed value determined during the design of the ultrasonic system, and the graph shows the case where the values fluctuate from \(L_{1}\) 190 to 240 mH and \(C_{1}\) 120 to 140 pF in the equivalent model of the piezo element. As the value of inductor \(L_{1}\) increases, the real part value of the x-axis moves toward 0. As the value of capacitor \(C_{1}\) increases, the imaginary part value of the y-axis moves toward 0. In the graph, blue is when the capacitor value is 120 pF, red is 130 pF, and yellow is 140 pF. And the inductor value at this time ranges from 190 to 240 mH. The range is the range of the minimum and maximum values measured at \(L_{1}\) and \(C_{1}\) when pressure is applied to the piezo element. As shown in Figure 8, it can be separated into the dominant and insignificant poles, and the insignificant pole showed a distribution diagram \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameters** & **Parameter Values** & **Conditions** \\ \hline \(R_{1}\) & 220 \(\Omega\) & - \\ \(L_{1}\) & 190 to 240 mH & Increase by 1 mH \\ \(C_{1}\) & 120 to 140 pF & Increase by 1 pF \\ \(C_{0}\) & 3.25 nF & Fixed value \\ \(L_{f}\) & 870 uH & - \\ \(C_{f}\) & 33 nF & - \\ \(LC\) filter resonant frequency & 28.3 kHz & Filter capacitor \(C_{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MATLAB simulation conditions. like the graph in Figure 8 according to the parameter value fluctuation of the piezo element.