## 5 Results and Discussion
### Dynamic Performance of Piezoelectric Actuator System  
The step response of the piezoelectric actuator system is represented by the time \(\mathrm{t_{n}}\) in Table 4 and is graphically shown in Figure 13. Case 1 presents the activation of one PE, Case 2 the activation of two PEs, and Case 3 the activation of three PEs. A detailed analysis shows different response times for these scenarios. The shortest response time, \(\mathrm{t_{n3}}=0.185\) ms, is achieved by activating all three PEs. This can be attributed to the higher stiffness of the active PEs compared to the inactive ones, resulting in a stiffer overall piezoelectric actuator system. Conversely, the longest response time, \(\mathrm{t_{n1}}=0.450\) ms, occurs when only one PE is activated. This longer response time is due to energy losses associated with acting on non-active PEs with the generated force. Activating two PEs results in a step response time of \(\mathrm{t_{n2}}=0.280\) ms, reflecting intermediate conditions between the one-PE and three-PE scenarios. Considering the activation of one PE and using PWM control to achieve different discrete values of the piezoelectric actuator system stroke (Figure 13), the step response time varies from 0.15 ms to 0.45 ms. A detailed view of activating one PE with different PWM signal widths, ranging from 20 us to 1000 us, is shown in Figure 14a. Additionally, the step response for a PWM signal width of 20 us is illustrated in Figure 14b. The step response of the piezoelectric actuator system can be significantly improved by using a boosting control strategy (Figure 15, Case 4), which involves activating all three PEs simultaneously with the maximum electrical voltage \(U=200\) V. Similar techniques were used for on/off switching solenoid valves, where the maximum supply voltage (control signal) was used at the beginning of the activation cycle and followed by a proper low-voltage signal to remain in the active state of the actuator [24,25]. This approach achieves a high step response across different strokes, with a response time of approximately 0.25 ms. Compared to other commercially available piezoelectric stack actuators, our design demonstrates equal or potentially superior dynamic performance. For example, according to their technical specifications, the P-810/30, P-840, and P-841 models from Physik Instrumente exhibit sub-millisecond response times [11]. Similarly, Cedrat Technologies' parallel pre-stressed piezo actuators of type PPA40L show a step response time of around 0.15-0.25 ms, which is comparable to our actuator's performance [26]. \begin{table}
\begin{tabular}{l l} \hline \hline
**Active State** & \(\mathrm{t_{n}}\) **[ms]** \\ \hline Case 1—one active PE & 0.450 \\ Case 2—two active PEs & 0.280 \\ Case 3—three active PEs & 0.185 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The step response of the piezoelectric actuator system depended on the control strategy. Figure 13: The step response of the piezoelectric actuator system depended on active PEs. The desired discrete values of the piezoelectric actuator stroke are achieved by activating the PEs for specific durations. Experimental tests show that a 30 \(\upmu\)m stroke is obtained by activating all three PEs for 60 \(\upmu\)s. Similarly, a 52 \(\upmu\)m stroke is achieved with a 90 \(\upmu\)s activation, and a 71 \(\upmu\)m stroke requires 130 \(\upmu\)s of activation. The use of multiple piezoelectric stack actuators (PEs) in series, rather than a single 90 mm actuator, is crucial. Longer PEs have a higher electrical capacitance (C), which results in longer charging times to achieve the desired displacement. To address this, new piezo electronics have been developed to control each PE independently. Each PE is equipped with its own low-voltage and high-voltage amplifier, as well as a capacitor for energy storage. This setup allows for the simultaneous high-potential activation of the PEs, enabling faster charging due to the low electrical capacitance of the stack actuators.