# Dependence of Piezoelectric Discs Electrical Impedance on Mechanical Loading Condition

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Piezoelectric Materials

#### 1.2. Related Work

## 2. Background

#### 2.1. Existing Electrical Equivalent Models for Piezoelectric Elements

#### 2.2. Fundamental Equations

## 3. Research Methodology

#### 3.1. Proposed Work

#### 3.2. Experimental Details

#### 3.2.1. Piezoelectric Discs

#### 3.2.2. Test Equipment

#### 3.3. Method for Electrical Impedance Measurement

- Without mechanical load: To understand the influence of plate diameters, three PZT discs with diameters of 12 $\mathrm{m}\mathrm{m}$, 20 $\mathrm{m}\mathrm{m}$ and 27 $\mathrm{m}\mathrm{m}$ are selected for measurement in the frequency range 0.1–1000 $\mathrm{kHz}$ with resolution of 20 $\mathrm{Hz}$. PZT discs are suspended in the air as shown in Figure 9, such that they can vibrate freely to obtain the impedance response.
- With mechanical load: To observe the influence of varying mechanical load, the impedance response of three discs are measured over the frequency range 0.1–1000 $\mathrm{kHz}$ with a resolution of 20 $\mathrm{Hz}$ at different load weights. The disc is placed flat on the table and then on top of it, equal loads are added one after another. Each load has a mass of 1.25 kg, which corresponds to force F = 12.25 N. Therefore, the pressure P equivalent to each mechanical load is derived from Equation (11). The definition of loading for the existing models mean that the piezoelectric element is mounted on a host structure, which we describe as mechanically unloaded in this paper. But in our case of mechanical loading, we apply a known mechanical load on top of the element as shown in Figure 10 below. The mechanical load on the arrangement generates normal displacement on the disc, perpendicular to its plane.

#### 3.4. Derivation of Electrical Model Parameters

## 4. Results and Discussion

#### 4.1. Without Mechanical Load

#### 4.2. With Mechanical Load

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 17.**Influence of diameter on (

**a**) resistance, (

**b**) capacitance, and (

**c**) inductance for first resonance.

**Figure 18.**Influence of mechanical load on first resonant frequency of (

**a**) PZT-12, (

**b**) PZT-20, and (

**c**) PZT-27.

**Figure 19.**Influence of mechanical load on third resonant frequency of (

**a**) PZT-12, (

**b**) PZT-20, and (

**c**) PZT-27.

**Figure 21.**Influence of mechanical load on frequency dependence of PZT-27 disc around (

**a**) first resonance and (

**b**) third resonance peaks.

Name | Based on | Condition | Remarks |
---|---|---|---|

Mason | Material constants, field constants, and structure geometry | Both loaded and unloaded condition | Negative capacitance and determination of complex elements for lossy materials |

Van Dyke | Material constants, field constants, and structure geometry | Both loaded and unloaded condition | Inaccurate in nonresonant frequency range and for lossy materials |

Sherrit | Material constants, field constants, and structure geometry | Only unloaded condition | Determination of complex elements |

Park | Impedance response | Only unloaded condition | Below ultrasonic frequency range |

Guan | Impedance response | Both loaded and unloaded condition | Based on visual inspection, energy dissipation |

Easy | Impedance response | Both loaded and unloaded condition | Limited frequency range, Simple |

Banerjee | Impedance response | Only unloaded condition | Limited frequency range, Simple |

Our model | Impedance response | Both loaded and unloaded condition | Wide frequency range, Simple |

Product Name | Plate Diameter (mm) | Structure Thickness (mm) | Ceramic Thickness (mm) | Mass (mg) |
---|---|---|---|---|

7BB-12-9 | 12 | 0.22 | 0.12 | 142 |

7BB-20-6 | 20 | 0.42 | 0.22 | 776 |

7BB-27-4 | 27 | 0.54 | 0.24 | 1968 |

Resistance | Capacitance | Inductor | Quality Factor |
---|---|---|---|

R = e6 $\Omega $ | |||

${R}_{0}$ = 0.4 $\Omega $ | ${C}_{0}$ = 16.85 $\mathrm{n}\mathrm{F}$ | ||

${R}_{1}$ = 30,300 $\Omega $ | ${C}_{1}$ = 95.40 $\mathrm{n}\mathrm{F}$ | ${L}_{1}$ = 12.1 $\mathrm{m}\mathrm{H}$ | ${Q}_{1}$ = 85 |

${R}_{2}$ = 355 $\Omega $ | ${C}_{2}$ = 770.56 $\mathrm{n}\mathrm{F}$ | ${L}_{2}$ = 32.1 $\mathsf{\mu}\mathrm{H}$ | ${Q}_{2}$ = 55 |

${R}_{3}$ = 2525 $\Omega $ | ${C}_{3}$ = 118.34 $\mathrm{n}\mathrm{F}$ | ${L}_{3}$ = 20.9 $\mathsf{\mu}\mathrm{H}$ | ${Q}_{3}$ = 190 |

Load | Resistance | Capacitance | Inductor | |||
---|---|---|---|---|---|---|

No load | R = ${10}^{6}$ $\Omega $ | |||||

${R}_{0}$ = 0.4 $\Omega $ | ${C}_{0}$ = 16.85 $\mathrm{n}\mathrm{F}$ | |||||

${R}_{1}$ = 30,300 $\Omega $ | ${C}_{1}$ = 95.40 $\mathrm{n}\mathrm{F}$ | ${L}_{1}$ = 12.1 $\mathrm{m}\mathrm{H}$ | ||||

${R}_{2}$ = 355 $\Omega $ | ${C}_{2}$ = 770.56 $\mathrm{n}\mathrm{F}$ | ${L}_{2}$ = 32.1 $\mathsf{\mu}\mathrm{H}$ | ||||

${R}_{3}$ = 2525 $\Omega $ | ${C}_{3}$ = 118.34 $\mathrm{n}\mathrm{F}$ | ${L}_{3}$ = 20.9 $\mathsf{\mu}\mathrm{H}$ | ||||

21.39 $\mathrm{kPa}$ | ${R}_{p1}$ = 900 $\Omega $ | ${C}_{p1}$ = 14.14 $\mathsf{\mu}\mathrm{F}$ | ${L}_{p1}$ = 0.7162 $\mathrm{H}$ | |||

${R}_{p2}$ = 1400 $\Omega $ | ${C}_{p2}$ = 105.2 $\mathrm{n}\mathrm{F}$ | ${L}_{p2}$ = 3.3 $\mathrm{H}$ | ||||

${R}_{p3}$ = 1180 $\Omega $ | ${C}_{p3}$ = 10.5 $\mathrm{n}\mathrm{F}$ | ${L}_{p3}$ = 230.1 $\mathsf{\mu}\mathrm{H}$ | ||||

${R}_{p4}$ = 470 $\Omega $ | ${C}_{p4}$ = 101.5 $\mathrm{n}\mathrm{F}$ | ${L}_{p4}$ = 11.08 $\mathrm{H}$ | ||||

42.79 $\mathrm{kPa}$ | ${R}_{p1}$ = 400 $\Omega $ | ${C}_{p1}$ = 0.795 $\mathsf{\mu}\mathrm{F}$ | ${L}_{p1}$ = 0.0141 $\mathrm{H}$ | |||

${R}_{p2}$ = 1300 $\Omega $ | ${C}_{p2}$ = 65.9 $\mathrm{n}\mathrm{F}$ | ${L}_{p2}$ = 2.526 $\mathrm{H}$ | ||||

${R}_{p3}$ = 980 $\Omega $ | ${C}_{p3}$ = 9.7 $\mathrm{n}\mathrm{F}$ | ${L}_{p3}$ = 190 $\mathsf{\mu}\mathrm{H}$ | ||||

${R}_{p4}$ = 450 $\Omega $ | ${C}_{p4}$ 47.16 $\mathrm{n}\mathrm{F}$ | ${L}_{p4}$ = 0.9549 $\mathrm{H}$ | ||||

64.19 $\mathrm{kPa}$ | ${R}_{p1}$ = 350 $\Omega $ | ${C}_{p1}$ = 0.641 $\mathsf{\mu}\mathrm{F}$ | ${L}_{p1}$ = 0.0137 $\mathrm{H}$ | |||

${R}_{p2}$ = 1000 $\Omega $ | ${C}_{p2}$ = 48.7 $\mathrm{n}\mathrm{F}$ | ${L}_{p2}$ = 2.16 $\mathrm{H}$ | ||||

${R}_{p3}$ = 920 $\Omega $ | ${C}_{p3}$ = 8.5 $\mathrm{n}\mathrm{F}$ | ${L}_{p3}$ = 181.58 $\mathsf{\mu}\mathrm{H}$ | ||||

${R}_{p4}$ = 370 $\Omega $ | ${C}_{p4}$ = 40.4 $\mathrm{n}\mathrm{F}$ | ${L}_{p4}$ = 0.8660 $\mathrm{H}$ | ||||

85.5 $\mathrm{kPa}$ | ${R}_{p1}$ = 300 $\Omega $ | ${C}_{p1}$ = 0.573 $\mathsf{\mu}\mathrm{F}$ | ${L}_{p1}$ = 0.0129 $\mathrm{H}$ | |||

${R}_{p2}$ = 850 $\Omega $ | ${C}_{p2}$ = 44.25 $\mathrm{n}\mathrm{F}$ | ${L}_{p2}$ = 1.89 $\mathrm{H}$ | ||||

${R}_{p3}$ = 900 $\Omega $ | ${C}_{p3}$ = 8.47 $\mathrm{n}\mathrm{F}$ | ${L}_{p3}$ = 181.51 $\mathsf{\mu}\mathrm{H}$ | ||||

${R}_{p4}$ = 300 $\Omega $ | ${C}_{p4}$ = 40.4 $\mathrm{n}\mathrm{F}$ | ${L}_{p4}$ = 0.5684 $\mathrm{H}$ |

Load | Relative Error | Correlation Coefficient |
---|---|---|

0 | 9.46 % | 0.9985 |

21.39 ($\mathrm{kPa}$) | 11.76 % | 0.9995 |

42.79 ($\mathrm{kPa}$) | 11.83 % | 0.9995 |

64.19 ($\mathrm{kPa}$) | 12.76 % | 0.9994 |

85.59 ($\mathrm{kPa}$) | 14.21 % | 0.9995 |

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**MDPI and ACS Style**

Gogoi, N.; Chen, J.; Kirchner, J.; Fischer, G.
Dependence of Piezoelectric Discs Electrical Impedance on Mechanical Loading Condition. *Sensors* **2022**, *22*, 1710.
https://doi.org/10.3390/s22051710

**AMA Style**

Gogoi N, Chen J, Kirchner J, Fischer G.
Dependence of Piezoelectric Discs Electrical Impedance on Mechanical Loading Condition. *Sensors*. 2022; 22(5):1710.
https://doi.org/10.3390/s22051710

**Chicago/Turabian Style**

Gogoi, Niharika, Jie Chen, Jens Kirchner, and Georg Fischer.
2022. "Dependence of Piezoelectric Discs Electrical Impedance on Mechanical Loading Condition" *Sensors* 22, no. 5: 1710.
https://doi.org/10.3390/s22051710