# Waveform Design Method for Piezoelectric Print-Head Based on Iterative Learning and Equivalent Circuit Model

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## Abstract

**:**

## 1. Introduction

## 2. Model and Desired Volume Flow Rate

#### 2.1. Equivalent Circuit Model

_{1}represents the compression in the fluid volume at the pressure chamber. The compressibility of the liquid in the right channel is equivalent to capacitance C

_{2}. The compressibility of the liquid in the left channel is ignored because the flow resistance at the exit is very small. The flow resistance and inertial force of the left channel and left half of the pressure chamber are equivalent to resistance R

_{1}and inductance L

_{1}, respectively. Analogously, the flow resistance and inertial force of the right channel and right half of the pressure chamber are equivalent to resistance R

_{2}and inductance L

_{2}, respectively. The flow resistance and inertial force of the nozzle are equivalent to resistance R

_{3}and inductance L

_{3}, respectively. The ambient pressure is equivalent to voltage source Us (typically Us = 1 [atm]) [17]. The voltage source Ud is used to replace a pressure constant derived from the equation of state. Capacitance C

_{3}represents the fluid stored by the meniscus surface tension [18]. The current source i

_{s}is used to simulate the increase or decrease in the fluid inside the pressure chamber.

**A**is the state matrix, x is the state variable,

**B**is the input matrix,

**u**is the system input,

**C**is the output matrix, and

**y**is the system output.

#### 2.2. Parameter Estimation

_{T}is the total length of the channel, r is the inner radius of the channel, and R

_{f}and L

_{f}are the equivalent resistance and inductance of the estimated portion of the channel, respectively.

^{2}dρ) under adiabatic conditions. The state equation of the pressure chamber can be described as

_{0}) is the initial density of the liquid in the pressure chamber, q(t) is the volume flow rate of the fluid in the pressure chamber, and V(t) is the volume of the pressure chamber. As the volume change in the pressure chamber is very small, the change in liquid density caused by the diameter change in the pressure chamber can be simulated using the fluid source q

_{s}(t). Then, Equation (4) can be written as follows:

_{0}) is the initial volume of the pressure chamber. To establish the relationship between V(t) and q

_{s}(t), let Equations (4) and (5) have the same pressure increment. Therefore, the relationship between V(t) and q

_{s}(t) can be described as follows:

_{0}) is the initial pressure in the pressure chamber. In the equivalent circuit illustrated in Figure 2, voltage u

_{1}represents the pressure change in the pressure chamber. According to the Kirchhoff Voltage Law (KVL), u

_{1}can be written as

_{c}

_{1}is the current through capacitance C

_{1}, i

_{s}is the current from the current source, and Q

_{1}(t

_{0}) is the initial charge on capacitor C

_{1}. This current source releases the charge to capacitance C

_{1}, causing voltage u

_{c}

_{1}to increase. The current source absorbs the charge from capacitance C

_{1}, causing voltage u

_{c}

_{1}to decrease. This process is very similar to that causing the pressure changes in the pressure chamber. Therefore, the parameter estimation method for capacitance and the voltage source Ud in the equivalent circuit can be respectively written as

_{f}is the equivalent capacitance of the liquid in the pressure chamber. A capacitor C

_{n}is equivalent to the Laplace pressure caused by surface tension. To estimate the expression of the capacitor parameter, the average values of the flow-out/-in volumes at the nozzle are roof-shaped [17]. The equivalent capacitance of the surface tension can be described as

_{men}is the roof-shaped volume of the fluid with nozzle radius r

_{n}, p

_{men}is the Laplace pressure, σ is the coefficient of surface tension, and C

_{n}is the equivalent capacitance of the meniscus. Based on the above equation, the estimation method of component parameters in the equivalent circuit illustrated in Figure 2 is listed in Table 1. A more detailed method of parameter estimation can be found in Reference [15].

## 3. Iterative Learning Process

#### 3.1. Iterative Learning Method

**A**,

**B**, and

**C**are the discrete system state-transition matrix, input matrix, and output matrix, respectively. For a print-head system, the goal of the iterative learning waveform design method is to find system input

**u**within a given time interval [0, N] so that the system output is consistent with the reference output (desired volume flow rate). According to Equation (12), the error vector

**e**(k) corresponding to iteration k between the system output and reference output can be written as

**u**such that $\underset{k\to \infty}{\mathrm{lim}}E(k)=0$. To rapidly reduce the total error E, the steepest descent method is used to design the iterative learning laws [21,22]. The normalized negative gradient vector of the total error can be written as

**u**is the normalized negative gradient vector of the total error and

**v**is the negative gradient vector without normalization. Therefore, the learning law of the iterative learning system can be written as

#### 3.2. Optimal Iteration Step Length

#### 3.3. Adding to Iterative Learning Constraint

**W**is also a vector defined in n-dimensional real space. This weight function makes iterative learning effective only for a limited amount of time.

**m**= ηΔ

**u**(k)

**w**

^{T}is the weighted iterative learning system input correction, and (

**m**)

_{P}and (

**m**)

_{N}represent the elements of the

**m**vector that are greater than zero and less than zero, respectively. The purpose of Equation (21) is to set the integral of the system input correction ηΔ

**u**(k)

**w**

^{T}calculated in each iteration step to zero, which implies that the starting and ending voltages of the drive waveform obtained by integrating the volume flow rate are zero.

#### 3.4. Calculated Waveform with the Iterative Learning Method

**y**

_{d}. Secondly, the model output

**y**(k) corresponding to any initial system input

**u**(0) is calculated, and the correction amount of the control input is calculated according to Equations (14)–(21). This process is repeated until the total error E(k) converges to the set threshold. Finally, the corresponding control input is converted into a voltage waveform, which generates the expected injection behavior when driving the PPH and thus optimizes the fluid injection performance.

## 4. Results and Discussion

#### 4.1. Experimental Setup

#### 4.2. Suppressed Residual Vibration

#### 4.3. Production of Smaller Droplets

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Abbreviation | Meaning |

PPH | Piezoelectric print-heads |

STDW | Single trapezoidal drive waveform |

DTDW | Double trapezoidal drive waveform |

DWS | Droplet watch system |

KVL | Kirchhoff’s voltage law |

KCL | Kirchhoff’s current law |

PID | proportional integral derivative |

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**Figure 6.**(

**a**). Desired injection volume flow rate designed to suppress residual vibration. (

**b**). The iterative learning process of the injection volume flow rate that inhibits the residual vibrations (

**c**). The change in total error and optimal step length with iterations. (

**d**). Optimal waveform for suppressing residual pressure vibrations.

**Figure 7.**(

**a**). Meniscus vibration process corresponding to STDW. (

**b**). Meniscus vibration process corresponding to the optimized waveform. (

**c**). At 2 kHz frequency, the droplet ejection process corresponding to STDW is unstable. (

**d**). At 7 kHz frequency, the droplet ejection process corresponding to the optimized waveform is stable.

**Figure 8.**(

**a**). Process of forming two smaller droplets (ethanol is used as the fluid). (

**b**). Desired volume flow rate to produce smaller droplet. (

**c**). Optimized waveform obtained using iterative learning. (

**d**). Process of generating a smaller droplet using the optimized waveform.

Equivalent Circuit Components | Mapping Expressions | Parameters for MJ-AL-80 |
---|---|---|

Capacitance C_{1}, C_{2}, C_{3} | $\frac{\pi {r}^{2}{L}_{2}}{{c}^{2}\rho ({t}_{0})}$$\frac{\pi {r}^{2}{L}_{3}}{{c}^{2}\rho ({t}_{0})}$$,\frac{\pi {r}_{o}^{4}}{3\sigma}$ | C_{1} = 1.4374 × 10^{−18}C _{2} = 8.2565 × 10^{−19}C _{3} = 1.1888 × 10^{−16} |

Voltage source Ud | ${c}^{2}\rho ({t}_{0})$ | Ud = 9.897216 × 10^{8} |

Resistance R_{1}, R_{2}, R_{3} | $\sqrt{\frac{2c\rho \mu {({L}_{1}+{L}_{2}/2)}^{2}}{{l}_{T}\pi {r}^{6}}}$$,\sqrt{\frac{2c\rho \mu {({L}_{3}+{L}_{2}/2)}^{2}}{{l}_{T}\pi {r}^{6}}}$$,\sqrt{\frac{2c\rho \mu {L}_{o}{}^{2}}{{l}_{T}\pi {r}_{e}^{6}}}$ | R_{1} = 2.3018 × 10^{11}R _{2} = 1.5635 × 10^{11}R _{3} = 3.7675 × 10^{11} |

Inductance L_{1}, L_{2}, L_{3} | $\frac{\rho \left({L}_{1}+{L}_{2}/2\right)}{\pi {r}^{2}}$$,\frac{\rho \left({L}_{3}+{L}_{2}/2\right)}{\pi {r}^{2}}$$,\frac{\rho {L}_{o}}{\pi {r}_{e}^{2}}$, | R_{1} = 5.8984 × 10^{7}R _{2} = 4.0065 × 10^{7}R _{3} = 3.4867 × 10^{7} |

Current source i_{s} | ${i}_{s}=\frac{\partial V(t)}{\partial t}$ |

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**MDPI and ACS Style**

Wang, J.; Xiong, C.; Huang, J.; Peng, J.; Zhang, J.; Zhao, P.
Waveform Design Method for Piezoelectric Print-Head Based on Iterative Learning and Equivalent Circuit Model. *Micromachines* **2023**, *14*, 768.
https://doi.org/10.3390/mi14040768

**AMA Style**

Wang J, Xiong C, Huang J, Peng J, Zhang J, Zhao P.
Waveform Design Method for Piezoelectric Print-Head Based on Iterative Learning and Equivalent Circuit Model. *Micromachines*. 2023; 14(4):768.
https://doi.org/10.3390/mi14040768

**Chicago/Turabian Style**

Wang, Jianjun, Chuqing Xiong, Jin Huang, Ju Peng, Jie Zhang, and Pengbing Zhao.
2023. "Waveform Design Method for Piezoelectric Print-Head Based on Iterative Learning and Equivalent Circuit Model" *Micromachines* 14, no. 4: 768.
https://doi.org/10.3390/mi14040768