## 2 Analysis of Piezo Element
### Equivalent Circuit Analysis of the Ultrasonic System Output Stage
The characteristic roots determine the response characteristics of the circuit according to the values of the damping frequency and resonant frequency. In general, the characteristics of piezo elements exhibit under-damping characteristics due to the characteristic that the damping frequency is lower than the resonant frequency, and, in this study, the resonant frequency was analyzed through this characteristic. The resonant frequency at this time means the entire system resonant frequency of the ultrasonic system output terminal. The characteristic equation of the equivalent circuit as shown in Figure 4 is as follows. \[H_{(fi\text{g4})}(s)=\frac{C_{0}C_{1}L_{0}L_{1}(C_{1}L_{1}s^{2}+C_{1}R_{1}s+1) }{C_{0}C_{1}L_{0}L_{1}s^{4}+C_{0}C_{1}L_{0}R_{1}s^{3}+(C_{0}L_{0}+C_{1}L_{0}+C_ {1}L_{1})s^{2}+C_{1}R_{1}s+1} \tag{7}\]  
Figure 5: Equivalent circuit of the ultrasonic entire system impedance separation 2. The characteristic equation of a fourth-order system, as described above, is very complex and difficult to solve for characteristic roots. Therefore, the response characteristics of the circuit can be analyzed using the s-plane, as shown in Figure 6.  
As shown in Figure 6 above, there are poles with different real part sizes and frequency characteristics in the s-plane due to the transfer function of the fourth-order system [31; 32]. At this time, if the real part (\(\alpha 2\)) of the pole located farther from the imaginary part among the two poles is 5 to 10 times larger than the real part (\(\alpha 1\)) of the pole closer to the imaginary part, it can be distinguished as a dominant pole and an insignificant pole [33]. In general, the transient response characteristics of the insignificant pole show fast damping while the dominant pole has a characteristic of relatively slow damping, and the characteristics of this are shown in Figure 7. Using these characteristics, we tried to detect the entire system resonance frequency of the ultrasonic system and find the mechanical resonance frequency of the piezo element through this. The proposed resonance frequency tracking method is summarized as follows. When voltage is applied to the ultrasonic system, a transient phenomenon in the form of under-braking occurs due to the exchange of the magnetic energy and electrostatic energy of the inductor and capacitor [30]. At this time, the transient phenomenon of the fourth-order system applied in this system appears with the damping characteristics and resonance frequency due to the dominant and insignificant poles. After the damping of the insignificant pole with relatively fast damping characteristics is completely achieved, only the damping characteristics and resonance frequency due to the dominant pole exist. At this time, the resonance frequency refers to the resonance frequency due to the inductor and capacitor components included in the entire ultrasonic system. In addition, the resonance frequency of the entire system according to the parameter value change of the piezo element was simulated and verified, and the mechanical resonance frequency of the piezo element was found through this. The resonance frequency of the entire system of the ultrasonic system output stage and the mechanical resonance frequency of the piezo element can be analyzed through a mathematical formula, Equation (6) mentioned above, and the characteristic equation calculation. This process consists of calculating the characteristic equation, calculating the characteristic root, calculating the damping period and the vibration period, and calculating the resonant frequency. The following are the formulas for the ultrasonic system output terminal and the piezo element. Figure 6: Fourth-order system in the s-plane.