# A Virtual Instrument for Measuring the Piezoelectric Coefficients of a Thin Disc in Radial Resonant Mode

^{1}

^{2}

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^{*}

## Abstract

**:**

^{®}.

## 1. Introduction

^{®}(1994), JAVA

^{®}(2003), and LabVIEW 2008

^{®}. In all previous versions, the number of samples acquired and the iterative method were the same.

- Improve the automatic procedure for the calculation of piezoelectric coefficients in resonant mode.
- Increase the speed of processing and the calculation of constants.
- Increase the accuracy of the results.
- Improve the user interface by making it friendly and intuitive.
- Provide the system with tools for the exploration and export of the acquired and processed data.
- Make the Virtual Instrument software easily readable, expandable, and scalable.

^{®}can perform operations with extended length floating-point numbers of 128 bits (real and imaginary), and it allows substantially to improve the speed and processing capacity of the designed tool.

## 2. Virtual Instrument Description

- The measured admittance of the sample, which is placed in a 16065A Ext Voltage Bias Fixture HP sample holder connected to an HP4194A impedance analyser. The sample holder has to meet the two conditions of allowing the sample to vibrate freely, by not using rigid terminals that will clamp the sample, and avoiding the addition of admittance components that could modify the measurement of the complex admittance of the sample. Figure 4 shows how the sample is connected:
- The start frequency (Hz), that is, the initial excitation frequency.
- The stop frequency (Hz), that is, the final excitation frequency.
- The sample radius a (mm) and sample thickness t (mm).

- ${c}_{11}^{p}$, the elastic stiffness coefficient (N/m
^{2}) in the radial plane:$${c}_{11}^{p}=-\frac{{s}_{11}^{E}}{{\left({s}_{11}^{E}\right)}^{2}-{\left({s}_{12}^{E}\right)}^{2}\text{}}$$ - ${\sigma}^{p}$, the planar radial Poisson’s ratio:$${\sigma}^{p}=-\left(\frac{{s}_{12}^{E}}{{s}_{11}^{E}}\right).$$
- ${\epsilon}_{33}^{T}$, the relative dielectric permittivity when an electric field is applied in the direction of polarisation (i.e., direction 3) at a constant mechanical stress (T = 0, “free” permittivity). Therefore, ${\epsilon}_{33}^{T}={\u03f5}_{0}\xb7{\u03f5}_{33}^{T}$, where ${\u03f5}_{0}=8.8541878176\xb7{10}^{-12}{\text{}\mathrm{C}\text{}}^{2}/\left(\mathrm{N}\xb7{\mathrm{m}}^{2}\right)$.
- d
_{31}, the proportion between the dimensional variation (Δr) of the piezoelectric material (in meters) and the difference of potential applied in volts in axis 3, as well as between the generation of electric charges and the force applied in the material. - k
_{p}, planar radial piezoelectric coupling:$${k}_{p}^{2}=\frac{2\xb7{d}_{31}^{2}}{{\u03f5}_{0}\xb7{\u03f5}_{33}^{T}\xb7\left({s}_{11}^{E}+{s}_{12}^{E}\right)}.$$ - Measurement graphs: The Virtual Instrument provides several graphs from the measured data of electrical admittance, Y = G + iB, as well as utilities for data exploration.
- Logged data file: The Virtual Instrument records the measured and calculated data in files that can be exported to other applications. Moreover, all the intermediate steps of the calculations performed in the iterations are recorded.

_{31}, and k

_{p}while taking into account the energy losses. The numerical iterative method has been described in detail in [18].

#### 2.1. Virtual Instrument Measurements

- f
_{1s}and f_{2s}, the frequencies where G is maximal in the fundamental tone and in the overtone, respectively. The values of these frequencies are used as initial values in the iterative method. - f
_{p}, the frequency where R is maximal. - Δf
_{s}= |f_{Bmax}− f_{Bmin}|, where f_{Bmax}and f_{Bmin}are the frequencies at which B becomes maximal and minimal, respectively, in the fundamental tone. The values of these frequencies are used as initial values in the iterative method.

_{1s}, f

_{2s}, f

_{p}, f

_{Bmax}, and f

_{Bmin}. These values are the initial values to start the implemented iterative method. To do so, there is a previous step of calculation of real and imaginary parts of the measured complex admittance (Y = G + iB) and, also, of its inverse, the complex impedance (Z = R + iX).

#### 2.2. Virtual Instrument Software

^{®}, which is the most widely used language for Virtual Instrumentation. The programs developed with LabVIEW 2019

^{®}consist of two main windows, the front panel (user interface), and the block diagram (program code). The programming is modular, where each module is built with one or more files called SubVIs [38].

^{®}allows the user to implement a state machine with a simple design pattern, which allows for the development of complex programs, making VI scalable, maintainable, and readable. Figure 6 shows the used LabVIEW 2019

^{®}template.

_{1s}, f

_{2s}, f

_{Bmax}, and f

_{Bmin}).

_{p,initial}, f

_{1,initial}, f

_{2,initial}, η

_{1}, and η

_{2}, where:

**(σ**

^{p}_{p}

_{, initial}) and ${\eta}_{1}$ are obtained by solving the system of equations of the Equation (10). The system is solved by the iterative method as described in [18]:

**${\eta}_{1}$, and $\Delta {f}_{s}=\left|{f}_{Bmax}-{f}_{Bmin}\right|$ allows to calculate the initial value of ${c}_{11,initial}^{p}$ as described in [8].**

^{p}## 3. Experimental Results and Discussion

^{TM}Piezoceramics [39], with a density ρ = 7700 kg/m

^{3}, radius = 15 mm, and thickness = 1 mm. Figure 9 shows the front panel of a Virtual Instrument.

- C_11^p ≡ ${c}_{11}^{p}$: stiffness coefficient (N/m
^{2}) - O^p ≡ σ
^{p}: Planar Poisson’s ratio - d_31 ≡ d
_{31}: Charge piezoelectric constant (C/N or m/V) - E_33^T ≡ ${\u03f5}_{33}^{T}$: Dielectric permittivity (relative)
- Kp (%) ≡ k
_{p}: Planar radial piezoelectric coupling

_{1s}, f

_{2s}, f

_{p}, f

_{Bmax}, and f

_{Bmin}, the iterative method described in Figure 8 begins. The iterative method is executed in the CALCULATION state (see Figure 7).

#### 3.1. Processing/Computation Time

^{®}Profile Performance tool [41]. The computation time estimate in this section is the sum of the execution time of the states “MEASUREMENTS”, “GRAPH”, and “CALCULATION”.

_{characterisation}(ms) = 4440 + 7.8 + 15.9 × 6 + 2.1 = 4540.5.

#### 3.2. Comparation with JAVA (2003) Software

- In the Virtual Instrument program, the value of ${c}_{11}^{p}$ is accepted when the relative value of the difference of two consecutive values of ${c}_{11}^{p}$ is lower than ${10}^{-8}$ GN/m
^{2}while in the JAVA (2003) program, it is accepted when it is lower than ${10}^{-6}$ GN/m^{2}. - The values of f
_{1}and f_{2}are accepted in Virtual Instrument when the relative value of the difference of two consecutive values of f_{2}is lower than 5·10^{−4}Hz while in JAVA (2003), they are accepted when this is lower than 10^{−3}Hz. For this reason, the values of ${\epsilon}_{33}^{T},{d}_{31},{\sigma}^{p},\mathrm{and}{k}_{p}$ were accurately estimated. - The number of points acquired in the search for the initial values of the numerical method is three times greater. This gives place to high accuracy of calculation for all values.
- The frequencies of interest, f
_{1s}, f_{2s}, f_{p}, f_{Bmax}, and f_{Bmin}, are automatically searched for with an adequate resolution. This improved the accuracy of calculation for all values. - The calculation and results’ presentation time was much shorter when using the version of Virtual Instrument implemented in this work.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Schematic of the resonant modes of vibration of ceramic resonators: (

**a**) Thickness extensional of a thickness poled thin plate; (

**b**) Shear of a non-standard thickness poled thin plate; (

**c**) Length extensional of a longitudinally poled cylinder or bar; (

**d**) Length extensional of a thickness poled thin bar; (

**e**) Radial extensional of a thickness poled thin disc.

**Figure 5.**Procedure for searching for the frequencies f

_{1s}, f

_{p}, f

_{Bmax}, and f

_{Bmin}from the fundamental tone and f

_{2s}from the first overtone.

State | #Runs | Average (ms) | Shortest (ms) | Longest (ms) |
---|---|---|---|---|

MEASUREMENTS | 400 points 3 measurements 3.7 ms = 4440 ms | |||

GRAPH | 10 | 7.8 | 7.1 | 8.4 |

CALCULATION (6 iterations) | 10 | 15.9 × 6 | 15.1 × 6 | 17.0 × 6 |

RESULTS | 10 | 2.1 | 1.3 | 2.6 |

JAVA (2003) | Virtual Instrument | |
---|---|---|

${c}_{11}^{p}\left(GN/{m}^{2}\right)$ | 73.837502 + 0.80914118 i | 73.7718 + 0.813253 i |

${\sigma}^{p}$ | 0.33042568 + 3.4927518·10^{−5} i | 0.331937 + 1.56062·10^{−5} i |

${\epsilon}_{33}^{T}$ | 1421.2551 − 26.966124 i | 1423.77 − 26.9318 i |

${d}_{31}\left(pC/N\right)$ | 159.00174 − 2.919189 i | 159.395 − 2.92819 i |

${k}_{p}\left(\%\right)$ | 62.632594 − 0.21235708 i | 62.94022 − 0.213614 i |

${f}_{1}\left(Hz\right)$ | 63,848.44 | 63,827.35 |

${f}_{2}\left(Hz\right)$ | 133,641.36 | 133,835.75 |

T_{characterisation} (ms) | ≈8000 | ≈5000 |

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## Share and Cite

**MDPI and ACS Style**

Jiménez, F.J.; González, A.M.; Pardo, L.; Vázquez-Rodríguez, M.; Ochoa, P.; González, B.
A Virtual Instrument for Measuring the Piezoelectric Coefficients of a Thin Disc in Radial Resonant Mode. *Sensors* **2021**, *21*, 4107.
https://doi.org/10.3390/s21124107

**AMA Style**

Jiménez FJ, González AM, Pardo L, Vázquez-Rodríguez M, Ochoa P, González B.
A Virtual Instrument for Measuring the Piezoelectric Coefficients of a Thin Disc in Radial Resonant Mode. *Sensors*. 2021; 21(12):4107.
https://doi.org/10.3390/s21124107

**Chicago/Turabian Style**

Jiménez, Francisco Javier, Amador M. González, Lorena Pardo, Manuel Vázquez-Rodríguez, Pilar Ochoa, and Bernardino González.
2021. "A Virtual Instrument for Measuring the Piezoelectric Coefficients of a Thin Disc in Radial Resonant Mode" *Sensors* 21, no. 12: 4107.
https://doi.org/10.3390/s21124107