# New Perspective on Planar Inductive Sensors: Radio-Frequency Refractometry for Highly Sensitive Quantification of Magnetic Nanoparticles

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Fundamentals

## 3. Simulation and Experimental Procedures

#### 3.1. Simulation Setup

#### 3.2. Experimental Setup

#### 3.3. Nanoparticles for Testing

## 4. Results and Discussion

#### 4.1. Simulation Results

#### 4.2. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SRF | Self-resonant frequency |

MNPs | Magnetic nanoparticles |

NPs | Nanoparticles |

## References

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**Figure 3.**(

**Left**) schematic representation of the sample (NPs and their embedding matrix) on top of the planar coil (here, the surrounding electronics and materials are not depicted) as used in experimental measurements and simulations; (

**right**) scheme of the uniform particle distribution surrounding the coil as used in the mathematical model, with the same NPs and matrix amounts.

**Figure 4.**Adapter and planar coil on the printed circuit board used for the experimental measurements.

**Figure 5.**(

**Left**) some magnetite NP samples; (

**right**) some gold NPs samples. In all cases, NPs were deposited on blotting paper and placed on the sensing coil in the experimental setup.

**Figure 7.**Dependence of the $\mathrm{ln}\left(\mathrm{SRF}\right)$ on the ratio ${m}_{\mathrm{NP}}/{m}_{\mathrm{max}}$ for the simulated NPs.

**Figure 8.**Dependence of the $\mathrm{ln}\left(\mathrm{SRF}\right)$ on the ratio ${m}_{\mathrm{NP}}/{m}_{\mathrm{max}}$ for different lossy NPs.

**Figure 9.**Dependence of the $\mathrm{ln}\left(\mathrm{SRF}\right)$ on the ratio ${m}_{\mathrm{NP}}/{m}_{\mathrm{max}}$ for several environment refraction indexes. The solid lines are their fitness to our model (12).

**Table 1.**Fitting to Equation (12) of the simulation results for several refraction indexes in the environment.

${\mathit{A}}_{1}$ | ${\mathit{A}}_{0}$ | ${\mathit{R}}^{2}$ | |
---|---|---|---|

${n}_{\mathrm{env}}=1$ | $-1.7313\times {10}^{-1}$ | $-20.9629$ | $0.9776$ |

${n}_{\mathrm{env}}=1.732$ | $-8.7557\times {10}^{-2}$ | $-20.4109$ | $0.9981$ |

${n}_{\mathrm{env}}=2$ | $-7.6000\times {10}^{-2}$ | $-20.2684$ | $0.9940$ |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{0}$ | ${\mathit{R}}^{2}$ | |
---|---|---|---|

Gold NPs | $-2.0871\times {10}^{-6}$ | $-20.0358$ | $0.5815$ |

Magnetite NPs | $-1.0493\times {10}^{-5}$ | $-20.0360$ | $0.9651$ |

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## Share and Cite

**MDPI and ACS Style**

Marqués-Fernández, J.L.; Salvador, M.; Martínez-García, J.C.; Fernández-Miaja, P.; García-Arribas, A.; Rivas, M.
New Perspective on Planar Inductive Sensors: Radio-Frequency Refractometry for Highly Sensitive Quantification of Magnetic Nanoparticles. *Sensors* **2023**, *23*, 2372.
https://doi.org/10.3390/s23052372

**AMA Style**

Marqués-Fernández JL, Salvador M, Martínez-García JC, Fernández-Miaja P, García-Arribas A, Rivas M.
New Perspective on Planar Inductive Sensors: Radio-Frequency Refractometry for Highly Sensitive Quantification of Magnetic Nanoparticles. *Sensors*. 2023; 23(5):2372.
https://doi.org/10.3390/s23052372

**Chicago/Turabian Style**

Marqués-Fernández, José Luis, María Salvador, José Carlos Martínez-García, Pablo Fernández-Miaja, Alfredo García-Arribas, and Montserrat Rivas.
2023. "New Perspective on Planar Inductive Sensors: Radio-Frequency Refractometry for Highly Sensitive Quantification of Magnetic Nanoparticles" *Sensors* 23, no. 5: 2372.
https://doi.org/10.3390/s23052372