## 2 Analysis of Piezo Element
### Equivalent Circuit Analysis of the Ultrasonic System Output Stage  
Figure 3 is the equivalent circuit of the inverter output stage in the ultrasonic system, and can be interpreted as the equivalent circuit of the entire system. As shown in Figure 3, the parasitic capacitor \(C_{0}\) of the piezo element and the capacitor \(C_{f}\) of the \(LC\) filter are connected in parallel, so they can be considered as one capacitor \(C_{2}\). Since \(L_{f}\) and \(C_{f}\) of the \(LC\) filter are values designed by the user, they are fixed values that do not change, and the parasitic capacitor \(C_{0}\) of the piezo element has very little change in value and is almost constant [13; 26; 27; 28; 29]. Accordingly, if the filter \(C_{f}\) value is designed to be sufficiently high enough to ignore the influence of the parasitic capacitor \(C_{0}\), it can be separated into two parts: the \(R_{1}L_{1}C_{1}\) series circuit impedance that affects the mechanical output of the piezo element and the impedance of the \(LC\) filter, as shown in Figure 4. The parameter value change of the capacitor \(C_{2}\) according to the pressure fluctuation of the piezo element is as shown in Figure 2, and the value fluctuation is very small to the extent that it hardly affects the resonant frequency of the piezo element. As a result, it can be separated into two parts: the \(R_{1}L_{1}C_{1}\) series circuit impedance that affects the mechanical output of the piezo element and the impedance of the \(LC\) filter [30]. To analyze the entire system impedance, each impedance \(Z_{1}\), \(Z_{2}\), and \(Z_{p}\) is represented as in Figure 5, and the formula for this is as follows. \(L_{1}\), \(L_{f}\) reactance and \(C_{1}\), \(C_{2}\) reactance are as follows:  
\[ \begin{split} X_{L1}&=wL_{1},X_{C1}=-\frac{1}{wC_{ 1}}\\ X_{Lf}&=wL_{f},X_{C2}=-\frac{1}{wC_{2}}\end{split}  \tag{3}\]  
The piezo element \(R_{1}+L_{1}+C_{1}\) total \(Z_{p}\) impedance is as follows:  
\[ \begin{split} Z_{p}&=R_{1}+jX_{L1}+jX_{C1}\\ &=R_{1}+j(X_{L1}+X_{C1})\end{split}  \tag{4}\]  
Figure 4: Equivalent circuit of the ultrasonic entire system impedance separation 1. Figure 3: Equivalent circuit of the ultrasonic entire system. The \(Z_{1}\) parallel impedance, excluding the filter \(L_{f}\) impedance, is as follows:  
\[ \begin{split} Z_{1}&=\frac{Z_{p}\times jX_{\text{C2}}}{Z _{p}+jX_{\text{C2}}}\\ &=\frac{(R_{1}+j(X_{L1}+X_{\text{C1}}))\times jX_{\text{C2}}}{(R_ {1}+j(X_{L1}+X_{\text{C1}}))+jX_{\text{C2}}}\\ &=\frac{-X_{L1}X_{\text{C2}}-X_{\text{C1}}X_{\text{C2}}+jRX_{ \text{C2}}}{R+j(X_{L1}+X_{\text{C1}}+X_{\text{C2}})}\end{split}  \tag{5}\]  
\(Z_{p}\) is the series \(R_{1}L_{1}C_{1}\) impedance of the piezo element, and \(Z_{1}\) is the sum capacitor \(C_{2}\) of the parallel capacitor \(C_{0}\) of the piezo element and the capacitor \(C_{f}\) of the \(LC\) filter, so it can be expressed as (5) above. Through this, the entire system impedance \(Z_{2}\), including the inductor \(L_{f}\) of the \(LC\) filter, can be expressed as (6) below. The total equivalent impedance \(Z_{2}\), including the inductor \(L_{f}\) of the \(LC\) filter, is as follows:  
\[Z_{2}=\frac{-X_{L1}X_{\text{C2}}-X_{\text{C1}}X_{\text{C2}}+jRX_{\text{C2}}}{R +j(X_{L1}+X_{\text{C1}}+X_{\text{C2}})}+jX_{Lf} \tag{6}\]  
As shown in (6), once the values of the \(LC\) filter components \(L_{f}\) and \(C_{f}\) are determined along with the series \(R_{1}L_{1}C_{1}\) values of the piezo element, the resonant impedance of the entire system, including both the \(LC\) filter and the piezo element, can be analyzed. When voltage is initially applied to an ultrasonic system, a transient phenomenon occurs due to the exchange of magnetic energy and electrostatic energy of the inductor and capacitor configured in the circuit. This can be derived as a characteristic equation that represents the unique characteristics of the actual system, and the natural response characteristics can be confirmed through the roots derived from the characteristic equation, which are called characteristic roots.