# A Dual Stage Low Power Converter Driving for Piezoelectric Actuator Applied in Micro Mobile Robot
## 2 Driving Requirements and Strategies of Piezoelectric Actuator  
### Equivalent Model of a Piezoelectric Actuator  
A bending actuator, called bimorph, with low cost and ultra-high energy density can be fabricated using generic piezo ceramics and high-performance composite materials coupled with the intelligent use of geometry and novel driving techniques. A good understanding of the electrical characteristicsof each layer of a piezoelectric bimorph actuator is essential for designing and analyzing a drive circuit with good performance.  
As a capacitive material, a piezoelectric element exhibits specific natural frequencies in different vibration modes. Many micro bio-inspired robots apply resonant body dynamics to obtain efficient locomotion. When a drive signal is applied to the resonant frequency of a piezoelectric element, the resulting vibration amplitude is considerably higher than those at other frequencies. For example, a piezoelectric bimorph is driven at the lowest resonant frequency in Harvard's microrobot [12]. To design an efficient drive circuit, a dynamic electrical model of the load (i.e., a piezoelectric bimorph driver) must be established. In the electrical domain, a piezoelectric element can be represented by an equivalent model, in which impedance is linked to the mechanical properties of the actuator and its load. The circuit model of Guan is adopted in this study, as shown in Figure 1[26].  
The circuit model includes a static non-resonant part and a series of dynamic mechanical LCR (resistor R, inductor L, and capacitor C) branches. The static part represents the complex capacitive characteristics and loss mechanisms of piezoelectric actuators, which are made up of a parallel connection of series \(C_{0}R_{s}\), and \(R_{p}\). Different mechanical resonance branches correspond to various mechanical resonance modes, where inductance \(L_{n}\), capacitance \(C_{n}\), and resistance \(R_{n}\) are related to the equivalent mass, compliance, and damping coefficient of each resonance mode, respectively. The impedance of the non-resonant part, every resonant shunt circuit, and total impedance are denoted as \(Z_{c}(s)\), \(Z_{n}(s)\), and \(Z_{t}(s)\), respectively, and their corresponding formulas are as follows:  
\[Z_{c}(s)=R_{p}\times[R_{s}+1/(sC_{0})]/\big{[}R_{p}+R_{s}+1/(sC_{0})\big{]}, \tag{1}\]  
\[Z_{n}(s)=R_{n}+sL_{n}+1/(sC_{n}),n=1,2,3,\ldots, \tag{2}\]  
\[Z_{t}(s)=1/[1/Z_{c}(s)+1/Z_{1}(s)+1/Z_{2}(s)+\ldots+1/Z_{n}(s)],n=1,2,3,\ldots. \tag{3}\]  
The piezoelectric bimorph ceramic material (QDTE52-7.0-0.82-4, PANT, Suzhou, Jiangsu, China) used to test dynamic performance has the following parameters: 190 V highest driving voltage, 52 mm long, 7 mm wide, 0.82 mm thick, and 1.2 mm bidirectional displacement. The first resonant frequency of each bimorph layer is approximately 100 Hz close to the driving frequency of an insect-type robot, which makes the research practical.  
The impedance characteristics of the material were measured using a 4294A impedance analyzer (Keysight Technologies, Inc., Santa Rosa, CA, USA). We can fit the experimental curve by choosing the values of the electronic components. Figure 2 shows the frequency response of one of the bimorph PZT layers and its equivalent model, in which the theoretical frequency response is calculated with MATLAB using the transfer function of the circuit model.  
Figure 1: Electrical equivalent model of piezoelectric element.  
Four prominent resonance modes are between 30 Hz and 30 kHz. Hence, the corresponding circuit model has four LCR branches. The parameters of the equivalent model are listed in Table 1.