# Improved Determination of Q Quality Factor and Resonance Frequency in Sensors Based on the Magnetoelastic Resonance Through the Fitting to Analytical Expressions

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Magnetoelastic Material of the Sensor

#### 2.2. Precipitation Reaction Measurements

#### 2.3. Numerial Fitting of the Sensor Resonance Curves

^{©}software. The parameters ${\omega}_{r}$, ${\omega}_{a}$, and ${\chi}_{0}$, were allowed to vary around their experimental corresponding values (extracted from the experimental data), until the value of $Q$ that minimizes the norm between the fit and the experimental data was found. The norm (or residual) is defined as:

^{®}software and a nonlinear least-squares fitting. The parameters ${\omega}_{r}$, ${\omega}_{a}$, and A to be fitted are initially taken as their values obtained from the experimental curve. All parameters are fitted until those that best fit the experimental curve are found, in this case those that correspond to a local minimum of a function that is a sum of squared residuals (being the residual the difference between the experimental value of the dependent variable and the value predicted by the fitting model).

## 3. Results and Discussion

#### 3.1. Numerical Fitting of the Resonance Curves and Resonance Frequency

#### 3.2. Comparison of the Results Obtained with Both Numerical Fittings

#### 3.3. Determination of the Quality Factor

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Two magnetoelastic resonance curves from the same resonator with different $Q$ values, caused by different amount of mass loading. The peak corresponding to the higher $Q$ (blue) is sharper and narrower, the resonance curve with a lower $Q$ is wider (red).

**Figure 2.**(

**a**) Temporal evolution of the magnetoelastic resonance frequency of the sensor during the reaction of precipitation for different concentrations (30 mM, 50 mM and 100 mM) and a control curve (sensor in a vial with distilled water). The value of the resonance frequency, ${f}_{max}$, is obtained as the frequency at the maximum amplitude of measured resonance curves (Figure 2b); (

**b**) Measured magnetoelastic resonance curve signals of the sensor at different times during the precipitation process for the concentration 30 mM. Data taken from Ref. [20].

**Figure 3.**Fit to Equation (6). (

**a**) Measured magnetoelastic resonance curves of the sensor at different times during the precipitation process for reactants of concentration 30 mM, and corresponding numerical fittings (in dashed lines) to Equation (6); (

**b**) Measured magnetoelastic resonance curve, fitting to Equation (6) (in dashed line), and corresponding error (in red) of the sensor at time t = 125 s for reactants of concentration 30 mM.

**Figure 4.**Fit to Equation (9). (

**a**) Measured magnetoelastic resonance curves of the sensor at different times during the precipitation process for reactants of concentration 30 mM, and corresponding numerical fittings (in dashed lines) to Equation (9); (

**b**) Measured magnetoelastic resonance curve, fitting to Equation (9) (in dashed line), and corresponding error (in red) of the sensor at time t = 125 s for reactants of concentration 30 mM.

**Figure 5.**Evolution of the damping parameters obtained through the numerical fitting to Equation (9) of the resonance curves during the precipitation time for different concentration of reagents (30, 50, and 100 mM). (

**a**) Damping parameter ${\delta}_{r}$; (

**b**) Damping parameter ${\delta}_{a}$.

**Figure 6.**Evolution of the quality factor $Q$ of the resonator during the precipitation process for different reagents concentrations (30, 50, and 100 mM) obtained through the numerical fittings to Equation (6) of the resonance curves.

**Figure 7.**(

**a**) Evolution of the frequency corresponding to the maximum of the resonance curves resulting from the fitting to Equation (9) of the resonance curves measured during the precipitation, when the concentration of the reactants is 50 mM, compared to experimental data (${f}_{max}$); (

**b**) Evolution of the ${f}_{r}$ obtained in the numerical fitting to Equations (6) and (9) of the measured resonance curves, compared to experimental data (${f}_{max}$ ).

**Figure 8.**(

**a**) Frequency response of the system for resonance and anti-resonance behavior, and the total response affected by the damping. Corresponding resonance and anti-resonance frequencies are indicated; (

**b**) Evolution of the difference between the experimental resonance frequency ${f}_{max}$ and the resonance frequency obtained through the fitting to Equation (9) as the damping parameter ${\delta}_{r}$ increases, for the case of concentration 50 mM. The shift in the resonance frequency becomes greater as the damping increases.

**Figure 9.**Relation between the resonance frequency ${f}_{max}$ obtained experimentally and the quality factor $Q$ obtained through the fitting to Equation (9) at different times during the precipitation reaction. The resonance curves corresponding to times t = 6 s (high $Q$ ) and t = 258 s (low $Q$ ) are shown.

**Figure 10.**Comparison of the damping parameter ${\delta}_{r}$ obtained with both numerical fittings of the resonance curves (for reactants of concentration 50 mM) using the equivalences found between ${\delta}_{r}$ and $Q$, Equation (13).

**Figure 11.**Comparison of the damping parameter ${\delta}_{r}$ obtained by the fitting to Equation (9) without taking into account the background parameters ($a$ and $b$ ), and the damping parameter ${\delta}_{r}$ obtained through the fitting to Equation (6).

**Figure 12.**Comparison of the quality factor $Q$ obtained by the fitting to Equations (6) and (9), and the one obtained through Equation (4).

**Figure 13.**Magnetoelastic resonance curve measured at time t = 203 s during the precipitation reaction with 50 mM reactants (in dashed red line), and corresponding fitting to Equation (9) (in dashed black line) for a reduced range of frequencies (in blue). The value of the quality factor obtained with the reduced fitting is $Q$ = 17.4.

**Table 1.**Magnetic, magnetoelastic, and corrosion behaviour parameters of the ${\mathrm{Fe}}_{73}{\mathrm{Cr}}_{5}{\mathrm{Si}}_{10}{\mathrm{B}}_{12}$ sample [21]. ${M}_{s}$ is the spontaneous magnetization, ${\lambda}_{s}$ the saturation magnetostriction, $\mathsf{\Delta}E$ the change in the Young’s modulus with the applied magnetic field (or $\mathsf{\Delta}E$ effect), $k$ the magnetoelastic coupling coefficient and ${E}_{corr}$ the corrosion potential.

Composition | ${\mathit{\mu}}_{0}{\mathit{M}}_{\mathit{s}}\left(\mathit{T}\right)$ | ${\mathit{\lambda}}_{\mathit{s}}$ (ppm) | $\mathbf{\Delta}\mathit{E}(\mathit{\%})$ | $\mathit{k}$ | ${\mathit{E}}_{\mathit{c}\mathit{o}\mathit{r}\mathit{r}}$ (mV) | Corrosion Rate (μm/Year) |
---|---|---|---|---|---|---|

Fe_{73}Cr_{5}Si_{10}B_{12} | 1.12 | 14 | 17 | 0.41 | 47 | 0.035 |

**Table 2.**Comparison of the $Q$ values obtained through Equation (4) and the fitting to Equations (6) and (9) for each concentration of reactants and different times of reaction; and relative error between $Q$ obtained through Equation (4) and Equation (9).

Concentration (mM) | Time (s) | Q (Equation (4)) | Q (Equation (6)) | Q (Equation (9)) | Relative Error * (%) |
---|---|---|---|---|---|

30 | 10.2 | 47.8 | 50.4 | 52.4 | 8.7 |

43 | 45.8 | 47.9 | 50.2 | 8.7 | |

70 | 40.5 | 43.5 | 45.6 | 11.2 | |

125 | 31.5 | 35.2 | 37.0 | 14.8 | |

317 | 16.8 | 22.5 | 24.1 | 30.3 | |

426 | 12.6 | 19.3 | 20.9 | 39.7 | |

50 | 6.3 | 45.6 | 47.7 | 50.0 | 8.8 |

28 | 38.9 | 42.1 | 44.0 | 11.6 | |

72 | 24.6 | 29.8 | 31.3 | 21.4 | |

121 | 15.5 | 21.8 | 23.4 | 33.7 | |

312 | - | 12.9 | 15.0 | - | |

477 | - | 12.1 | 14.2 | - | |

100 | 6 | 38.9 | 41.7 | 44.2 | 12 |

39 | 14.6 | 21.1 | 22.9 | 36.2 | |

99 | 3.6 | 11.2 | 13.4 | 73.1 | |

154 | - | 8.9 | 11.3 | - | |

368 | - | 4.9 | 7.7 | - | |

478 | - | 4.1 | 6.9 | - |

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**MDPI and ACS Style**

Sisniega, B.; Gutiérrez, J.; Muto, V.; García-Arribas, A. Improved Determination of Q Quality Factor and Resonance Frequency in Sensors Based on the Magnetoelastic Resonance Through the Fitting to Analytical Expressions. *Materials* **2020**, *13*, 4708.
https://doi.org/10.3390/ma13214708

**AMA Style**

Sisniega B, Gutiérrez J, Muto V, García-Arribas A. Improved Determination of Q Quality Factor and Resonance Frequency in Sensors Based on the Magnetoelastic Resonance Through the Fitting to Analytical Expressions. *Materials*. 2020; 13(21):4708.
https://doi.org/10.3390/ma13214708

**Chicago/Turabian Style**

Sisniega, Beatriz, Jon Gutiérrez, Virginia Muto, and Alfredo García-Arribas. 2020. "Improved Determination of Q Quality Factor and Resonance Frequency in Sensors Based on the Magnetoelastic Resonance Through the Fitting to Analytical Expressions" *Materials* 13, no. 21: 4708.
https://doi.org/10.3390/ma13214708