## 2 Analysis of Piezo Element  
### Equivalent Circuit Model Analysis and Resonance Characteristics of Piezo Element  
The piezo element can be expressed as an equivalent model consisting of the \(R_{1}L_{1}C_{1}\) series circuit part that affects the mechanical output and the parasitic capacitor \(C_{0}\), as shown in Figure 1. The first arm is an electrical arm that includes the parasitic capacitor \(C_{0}\), and the other arm is a mechanical arm consisting of the dynamic capacitor \(C_{1}\), the dynamic inductor \(L_{1}\), and the dynamic resistor \(R_{1}\). As can be seen from the equivalent model above, the piezo element has a structure with two resonance points, as shown in the equation below as (1), (2) [21, 22].  
\[f_{r}=\frac{1}{2\pi\sqrt{L_{1}C_{1}}} \tag{1}\]  
\[f_{a}=\frac{1}{2\pi\sqrt{L_{1}\Big{(}\frac{C_{1}C_{0}}{C_{1}+C_{0}}\Big{)}}} \tag{2}\]The resonant frequency (\(f_{r}\)) of the \(R_{1}L_{1}C_{1}\) series circuit of the mechanical arm is organized as in (1), and this becomes the mechanical resonant frequency that delivers the maximum power to the piezo element. And, the total resonant frequency (\(f_{a}\)) of the piezo element including the parasitic capacitor \(C_{0}\) is organized as in (2). In other words, the piezo element exists in a form where both the resonant frequency and the anti-resonant frequency exist due to the influence of the parasitic capacitor \(C_{0}\). Therefore, if the system is operated with an incorrect resonant frequency, the output may not be properly delivered to the piezo element due to the influence of the anti-resonant frequency. Therefore, in order to accurately deliver the output to the piezo element, the system must be operated with the mechanical resonant frequency (\(f_{r}\)), such as (1), that can deliver the maximum power [23; 24; 25].  
The following is the change characteristics of piezo element parameters according to pressure fluctuation. In general, piezo elements have the characteristic that internal parameter values fluctuate according to changes in external conditions (pressure, temperature, etc.), and Figure 2 shows the piezo element parameter values according to pressure fluctuation. Since each parameter value fluctuates according to pressure fluctuation, the mechanical resonance frequency of the piezo element also fluctuates. However, the parasitic capacitor \(C_{0}\) value is almost constant regardless of pressure fluctuation. This can be interpreted as the effect of the resonance frequency fluctuation due to the parasitic capacitor \(C_{0}\) value being very small compared to the resonance frequency fluctuation due to the series \(R_{1}L_{1}C_{1}\) value. Therefore, if the system is designed so that the effect of the parasitic capacitor \(C_{0}\) can be ignored when designing the \(LC\) filter of the inverter output stage in the ultrasonic system, the resonance frequency fluctuation of the piezo element will be even smaller. As a result, the entire equivalent model of the ultrasonic system can be newly interpreted by separating it into two major impedances: the \(R_{1}L_{1}C_{1}\) series circuit and the \(LC\) filter, which affect the mechanical output from the piezo element.  
Figure 1: Piezo element equivalent circuit model.  
Figure 2: Parameter variation characteristics of the piezo element with pressure changes.