# Stability Compensation Design and Analysis of a Piezoelectric Ceramic Driver with an Emitter Follower Stage

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Piezoelectric Ceramic and Driver Circuit

_{f}and R

_{i}. On the non-inverting input, there is a series resistor R

_{c}, the value of which should be about R

_{f}parallel with R

_{i}, to help to cancel out the adverse impact of the bias current of the operational amplifier input stage on the output accuracy.

_{r}. By the function of the current mirror, the bias current flows through R

_{b}; this generates a voltage difference between the base of the output transistors, and consequently determines the quiescent current flowing through the output transistors when no load is driven. The local negative feedback resistors R

_{t}are implemented in series with the emitter of the output transistors, in case the output transistors break down because of the positive correlation between the collector current and the temperature. C

_{p}is the effective capacitance of the driven piezoelectric ceramic.

#### 2.2. Preparation for the Analysis

#### 2.2.1. Determination of the Parameters for the Transistor Model

_{BE}of about 0.65 V, and that the base of the output transistor and the output of the operational amplifier source or sink little current, so the bias current of the transistors involved in the driver can be calculated from Equation (1).

_{T}denotes the thermal voltage, and I

_{s}denotes the saturation current. Together with the above calculated bias current and parameters from the components’ datasheets and SPICE models, the hybrid-π models of the transistors have the following parameters, shown in Table 3.

#### 2.2.2. The Effective Impedance of the Current Mirror

_{m2}, which is far less than r

_{π2}and r

_{o2}; thus, i

_{1}will mainly flow through this effective resistor, and the other resistors parallel to it can be omitted temporarily, in order to facilitate the analysis. Similarly, i

_{2}was considered to mainly flow through r

_{π3}and r

_{o1}, and were temporarily omitted. In this way the DC resistance R

_{0}of the current mirror was found to be about 2.5 MΩ with Equation (2).

_{μ1}, r

_{π1}, r

_{π2}, r

_{π3}, and r

_{μ3}respectively. The calculation of r

_{μ1}is based on modified nodal analysis [29] and is demonstrated in Figure 3, in which the circuit can be described by Equation (3); r

_{μ1}was found to be about 8.5 kΩ with Equation (4). In this way, all of the other effective resistances were calculated and, consequently, all of the time constants were calculated, as listed in Table 4. It was found that τ

_{μ3}dominated, and the overall time constant τ was about 79 μs.

_{0}of 32 pF, in parallel with the DC resistance R

_{0}, which had a corresponding time constant of 79 μs.

#### 2.3. Analysis of the Uncompensated Driver

_{ol}of the operational amplifier and the gain β of the feedback network. The A

_{ol}of the operational amplifier, which has a DC magnitude of about 110 dB, with a dominating pole at about 2 Hz, was measured by the manufacturer, and can be found in the datasheet and the SPICE software model. Thus, the loop gain T is determined mainly by the feedback network gain β, and the following analysis and design in this research were based on the feedback network.

_{0}and C

_{μ3}was in the range of 100 pF, so they functioned only at considerably high frequency, and thus were ignored for an intuitive result. At moderately high frequencies, r

_{o3}conducted little current from node 3 compared with the R

_{t}path, and it was ignored in the analysis; for the same reason, R

_{0}was ignored as well. The circuit of the feedback network can be described by Equation (5), and the transfer function of β can be calculated by Equations (6)–(8).

_{ol}. This usually means a decrease in the phase margin of T and a poor transient response, or even oscillation, which will be verified in Section 3. This is the reason why compensation for the driver was necessary.

#### 2.4. Design of the Compensated Driver

_{s}was inserted between the piezoelectric ceramic and the output of the driver, and a new feedback path was established by C

_{k}and R

_{k}. In this case, the AC equivalent circuit of the top half driver is shown in Figure 6a. It can be seen that a delta topology is formed between node4, node5, and node6, because of the added components. A delta-y transformation was implemented to facilitate the analysis, and the schematic is shown in Figure 6b, in which the impedances, denoted as Z

_{1}, Z

_{2}and Z

_{3}, were calculated with Equation (9).

- The one original zero stemming from C
_{π}stays unchanged and the two original poles stemming from C_{p}and C_{π}change slightly; - The compensated circuit introduces two zeros stemming from C
_{p}and C_{k}and one pole stemming from C_{k}; - The zeros and poles stemming from C
_{p}and C_{k}makes the phase shift of β to be 0 degrees at high frequency under proper design; - The first pole lagging the phase of β in the compensated driver stems from C
_{π}, and it is at a relatively high frequency. In contrast, the first pole lagging the phase of β in the uncompensated driver stems from C_{p}, and it is at a relatively low frequency. Thus, the phase margin of the loop gain is increased.

_{s}and R

_{k}were designed to be 5.6 Ω and 10 kΩ, respectively, and C

_{k}was designed to be 2 nF. As Equation (13) shows, the three poles in β of the compensated driver are located at 816 Hz, 3.9 kHz, and 122 kHz, respectively, and the three zeros in β are located at about 769 Hz, 9.3 kHz, and 2.3 MHz, respectively.

## 3. Results

#### 3.1. Simulation Results

#### 3.1.1. Frequency Domain Simulation Results

#### 3.1.2. Time-Domain Simulation Results

#### 3.2. Experiment Results

#### 3.2.1. Setup of the Experiment

#### 3.2.2. Results of the Experiment

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Mohith, S.; Muralidhara, R.; Navin, K.P.; Kulkarni, S.M.; Adithya, R.U. Development and assessment of large stroke piezo-hydraulic actuator for micro positioning applications. Precis. Eng.
**2021**, 67, 324–338. [Google Scholar] [CrossRef] - Zhou, S.; Yan, P. Design and Analysis of a Hybrid Displacement Amplifier Supporting a High-Performance Piezo Jet Dispenser. Micromachines
**2023**, 14, 322. [Google Scholar] [CrossRef] - Yasinov, R.; Peled, G.; Feinstein, A.; Karasikov, N. Novel Piezo Motor for a High Precision Motion Axis; SPIE: San Francisco, CA, USA, 2020. [Google Scholar] [CrossRef]
- Jin, H.; Gao, X.; Ren, K.; Liu, J.; Qiao, L.; Liu, M.; Chen, W.; He, Y.; Dong, S.; Xu, Z.; et al. Review on Piezoelectric Actuators Based on High-Performance Piezoelectric Materials. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2022**, 69, 3057–3069. [Google Scholar] [CrossRef] - Jansen, B.; Butler, H.; Filippo, R.D. Active Damping of Dynamical Structures Using Piezo Self Sensing. IFAC-Pap.
**2019**, 52, 543–548. [Google Scholar] [CrossRef] - Zhang, Y.; Xu, Q. Adaptive Sliding Mode Control With Parameter Estimation and Kalman Filter for Precision Motion Control of a Piezo-Driven Microgripper. IEEE Trans. Control Syst. Technol.
**2017**, 25, 728–735. [Google Scholar] [CrossRef] - Zheng, S.; Wang, J.; Meng, W.; Zhang, J.; Feng, Q.; Wang, Z.; Hou, Y.; Lu, Q.; Lu, Y. A planar piezoelectric motor of two dimensional XY motions driven by one cross-shaped piezoelectric unit: A new principle. Rev. Sci. Instrum.
**2022**, 93, 043710. [Google Scholar] [CrossRef] [PubMed] - Xu, R.; Zhou, M. Sliding mode control with sigmoid function for the motion tracking control of the piezo-actuated stages. Electron. Lett.
**2017**, 53, 75–77. [Google Scholar] [CrossRef] - Zhong, J.; Nishida, R.; Shinshi, T. A digital charge control strategy for reducing the hysteresis in piezoelectric actuators: Analysis, design, and implementation. Precis. Eng.
**2021**, 67, 370–382. [Google Scholar] [CrossRef] - Li, P.; Wang, X.; Zhao, L.; Zhang, D.; Guo, K. Dynamic linear modeling, identification and precise control of a walking piezo-actuated stage. Mech. Syst. Signal Process.
**2019**, 128, 141–152. [Google Scholar] [CrossRef] - Jalili, H.; Goudarzi, H.; Salarieh, H.; Vossoughi, G. Modeling a multilayer piezo-electric transducer by equivalent electro-mechanical admittance matrix. Sens. Actuators A Phys.
**2018**, 277, 92–101. [Google Scholar] [CrossRef] - Ghenna, S.; Bernard, Y.; Daniel, L. Design and experimental analysis of a high force piezoelectric linear motor. Mechatronics
**2023**, 89, 102928. [Google Scholar] [CrossRef] - Wei, F.; Wang, X.; Dong, J.; Guo, K.; Sui, Y. Development of a three-degree-of-freedom piezoelectric actuator. Rev. Sci. Instrum.
**2023**, 94, 025001. [Google Scholar] [CrossRef] - Massavie, V.; Despesse, G.; Carcouet, S.; Maynard, X. Comparison between Piezoelectric Filter and Passive LC filter in a Class L−Piezo Inverter. Electronics
**2022**, 11, 3983. [Google Scholar] [CrossRef] - Li, H.; Chen, W.; Feng, Y.; Deng, J.; Liu, Y. Development of a novel high bandwidth piezo-hydraulic actuator for a miniature variable swept wing. Int. J. Mech. Sci.
**2023**, 240, 107926. [Google Scholar] [CrossRef] - Degefa, T.G.; Wróbel, A.; Płaczek, M. Modelling and Study of the Effect of Geometrical Parameters of Piezoelectric Plate and Stack. Appl. Sci.
**2021**, 11, 11872. [Google Scholar] [CrossRef] - Zhou, C.; Yuan, M.; Feng, C.; Ang, W.T. A Modified Prandtl–Ishlinskii Hysteresis Model for Modeling and Compensating Asymmetric Hysteresis of Piezo-Actuated Flexure-Based Systems. Sensors
**2022**, 22, 8763. [Google Scholar] [CrossRef] - Yu, Z.; Wu, Y.; Fang, Z.; Sun, H. Modeling and compensation of hysteresis in piezoelectric actuators. Heliyon
**2020**, 6, e03999. [Google Scholar] [CrossRef] - Yeh, Y.-L.; Pan, H.-W.; Shen, Y.-H. Model-Free Output-Feedback Sliding-Mode Control Design for Piezo-Actuated Stage. Machines
**2023**, 11, 152. [Google Scholar] [CrossRef] - Zhang, Y.; Sun, M.; Song, Y.; Zhang, C.; Zhou, M. Hybrid Adaptive Controller Design with Hysteresis Compensator for a Piezo-Actuated Stage. Appl. Sci.
**2023**, 13, 402. [Google Scholar] [CrossRef] - Roshandel, E.; Mahmoudi, A.; Kahourzade, S.; Davazdah-Emami, H. DC-DC High-Step-Up Quasi-Resonant Converter to Drive Acoustic Transmitters. Energies
**2022**, 15, 5745. [Google Scholar] [CrossRef] - Pai, F.-S.; Hu, H.-L. Driving Circuit Design for Piezo Ceramics Considering Transformer Leakage Inductance. Processes
**2023**, 11, 247. [Google Scholar] [CrossRef] - Kobayashi, D.; Kawakatsu, H. High slew rate circuit for high rigidity friction-drive. Jpn. J. Appl. Phys.
**2020**, 59, SN1008. [Google Scholar] [CrossRef] - Xu, L.; Li, H.; Li, P.; Ge, C. A High-Voltage and Low-Noise Power Amplifier for Driving Piezoelectric Stack Actuators. Sensors
**2020**, 20, 6528. [Google Scholar] [CrossRef] [PubMed] - Yang, C.; Li, C.; Xia, F.; Zhu, Y.; Zhao, J.; Youcef-Toumi, K. Charge Controller With Decoupled and Self-Compensating Configurations for Linear Operation of Piezoelectric Actuators in a Wide Bandwidth. IEEE Trans. Ind. Electron.
**2019**, 66, 5392–5402. [Google Scholar] [CrossRef] - Bazghaleh, M.; Grainger, S.; Cazzolato, B.; Lu, T.; Oskouei, R. Implementation and analysis of an innovative digital charge amplifier for hysteresis reduction in piezoelectric stack actuators. Rev. Sci. Instrum.
**2014**, 85, 45005. [Google Scholar] [CrossRef] [PubMed] - Jin, T.; Peng, Y.; Xing, Z.; LEI, L. A Charge Controller for Synchronous Linear Operation of Multiple Piezoelectric Actuators. IEEE Access
**2019**, 7, 90741–90749. [Google Scholar] [CrossRef] - Gray, P.R.; Hurst, P.J.; Lewis, S.H.; Meyer, R.G. Analysis and Design of Analog Integrated Circuits, 4th ed.; John Wiley & Sons, Inc.: New York, NY, USA, 2001; pp. 517–522. [Google Scholar]
- DeCarlo, R.A.; Lin, P.M. Linear Circuit Analysis: Time Domain, Phasor, and Laplace Transform Approached, 2nd ed.; Oxford University Press: New York, NY, USA, 2001; pp. 111–116. [Google Scholar]
- Franco, S. Design with Operational Amplifiers and Analog Integrated Circuits, 3rd ed.; McGraw-Hill Higher Education: New York, NY, USA, 2002; pp. 348–355. [Google Scholar]

**Figure 6.**Equivalent circuit of the compensated piezoelectric ceramic driver: (

**a**) before the delta-y transformation; (

**b**) after the delta-y transformation.

**Figure 7.**Frequency-domain simulation results of β: (

**a**) Bode plot of β of the uncompensated driver; (

**b**) Bode plot of β of the compensated driver.

**Figure 8.**Frequency-domain simulation results of T: (

**a**) Bode plot of T of the uncompensated driver; (

**b**) Bode plot of T of the compensated driver.

**Figure 9.**Time-domain simulation results of the response to the step input signal: (

**a**) output voltage waveform of the uncompensated driver; (

**b**) output voltage waveform of the compensated driver.

**Figure 10.**The experimental setup: (

**a**) the instruments used in the experiment; (

**b**) the schematic of the experimental setup; (

**c**) the piezoelectric ceramic driven in the experiment; (

**d**) the driving board under testing in the experiment.

**Figure 11.**Experimental results of the response to the step input signal: (

**a**) output voltage waveform of the uncompensated driver; (

**b**) output voltage waveform of the compensated driver.

**Figure 12.**Experimental results of the response to the dc input signal: (

**a**) output voltage waveform of the uncompensated driver; (

**b**) output voltage waveform of the compensated driver.

**Figure 13.**Experimental results of the response to the sinusoidal input signal: (

**a**) output voltage waveform of the uncompensated driver; (

**b**) output voltage waveform of the compensated driver.

Dimensions | Nominal Displacement | Blocking Force | Capacitance | Resonant Frequency |
---|---|---|---|---|

7 mm × 7 mm × 18 mm | 15 μm | 1750 N | 3.1 μF | 70 kHz |

Component | Value |
---|---|

OPAMP | OPA547 |

R_{f} | 102 kΩ |

R_{i} | 11.3 kΩ |

R_{c} | 10 kΩ |

Q1 Q2 Q3 | BC856 |

Q4 Q5 Q6 | BC846 |

R_{r} | 90.9 kΩ |

R_{b} | 1 kΩ |

Q7 | 2SCR586J |

Q8 | 2SAR586J |

R_{t} | 1 Ω |

C_{p} | P-887.51 |

Component | r_{π} | C_{π} | g_{m} | C_{μ} | r_{o} |
---|---|---|---|---|---|

BC856 | 13.0 kΩ | 24 pF | 19.2 mA/V | 14 pF | 27.4 kΩ |

BC846 | 13.0 kΩ | 8 pF | 19.2 mA/V | 4 pF | 20.0 kΩ |

2SCR586J | 13.2 kΩ | 1.3 nF | 18.9 mA/V | 188 pF | 321 kΩ |

2SAR586J | 13.2 kΩ | 1.3 nF | 18.9 mA/V | 337 pF | 63 kΩ |

τ_{μ1} | τ_{π1} | τ_{π2} | τ_{π3} | τ_{μ3} |
---|---|---|---|---|

147 ns | 744 ps | 744 ps | 252 ns | 79 μs |

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## Share and Cite

**MDPI and ACS Style**

Wang, X.; Zheng, N.; Wei, F.; Zhou, Y.; Yang, H.
Stability Compensation Design and Analysis of a Piezoelectric Ceramic Driver with an Emitter Follower Stage. *Micromachines* **2023**, *14*, 914.
https://doi.org/10.3390/mi14050914

**AMA Style**

Wang X, Zheng N, Wei F, Zhou Y, Yang H.
Stability Compensation Design and Analysis of a Piezoelectric Ceramic Driver with an Emitter Follower Stage. *Micromachines*. 2023; 14(5):914.
https://doi.org/10.3390/mi14050914

**Chicago/Turabian Style**

Wang, Xueliang, Nan Zheng, Fenglong Wei, Yue Zhou, and Huaijiang Yang.
2023. "Stability Compensation Design and Analysis of a Piezoelectric Ceramic Driver with an Emitter Follower Stage" *Micromachines* 14, no. 5: 914.
https://doi.org/10.3390/mi14050914