# Complex Dielectric Permittivity Spectra of Rapeseed in the 20 MHz–3 GHz Frequency Range

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Preparation of the Material under Test

#### 2.2. Dielectric Spectra Measurement

#### 2.3. Dielectric Spectra Modeling and Analysis

- Linear function:$$y=ax+b,$$
- Quadratic function:$$y=a{x}^{2}+bx+c,$$
- Segmented model consisting of two linear segments (introduced in [24]):$$\left(\right)open="\{"\; close>\begin{array}{cc}y={b}_{s1}+{a}_{s1}x\hfill & x\le {x}_{c}\hfill \\ y={b}_{s1}+({a}_{s1}-{a}_{s2}){x}_{c}+{a}_{s2}x\hfill & x\ge {x}_{c}\hfill \end{array}$$

## 3. Results and Discussion

#### 3.1. Dielectric Spectra

#### 3.2. Dielectric Modeling and Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Spectra of the real (

**left**) and imaginary (

**right**) parts of the complex dielectric permittivity of all measured rapeseed samples of moisture content given in the legend (in g ${\mathrm{g}}^{-1})$.

**Figure 2.**Relations between the square root of the real part of dielectric permittivity $\sqrt{{\epsilon}^{\prime}}$ at selected frequencies and moisture content of rapeseed.

**Figure 3.**Slope a and intercept b (blue dots) of the $\sqrt{{\epsilon}^{\prime}}\left(\theta \right)$ relation (Equation (2)) as functions of frequency. The frequency dependencies of the a and b parameters were modeled by a fourth-order polynomial (red lines) with respect to the natural logarithm of the frequency f expressed in GHz, according to Equation (3).

**Figure 4.**Models 3D (solid black lines) and 2D-ABC (dashed black lines) fitted to the spectra (dots of various colors) of ${\epsilon}^{\prime}$ (

**left**graph) and ${\epsilon}^{\u2033}$ (

**right**graph) of rapeseed samples of moisture content given in the legend (in g ${\mathrm{g}}^{-1}$). For clarity, for each moisture content, only one spectrum is shown.

**Figure 5.**Parameters of the fitted models (Equations (4) and (6)) with respect to the rapeseed moisture content. In the case of the 2D-ABC model, parameters ${\epsilon}_{\infty}$, $\Delta {\epsilon}_{3}$ and ${f}_{3}$ were calculated with the use of Equation (7). The parameters’ dependencies on moisture content were modeled by linear, quadratic or segmented functions, as appropriate.

**Table 1.**Slope a, intercept b, coefficient of determination ${R}^{2}$ and root-mean-squared error $RMSE$ of fitting Equation (2), which modeled $\sqrt{{\epsilon}^{\prime}}\left(\theta \right)$ relations at given frequencies f.

f, GHz | a | b | R^{2} | RMSE |
---|---|---|---|---|

0.02 | 18.2 | 0.268 | 0.9788 | 0.076 |

0.05 | 14.5 | 0.560 | 0.9852 | 0.051 |

0.10 | 12.9 | 0.678 | 0.9876 | 0.041 |

0.20 | 11.8 | 0.734 | 0.9885 | 0.036 |

0.50 | 11.0 | 0.760 | 0.9886 | 0.034 |

1.00 | 10.4 | 0.773 | 0.9864 | 0.035 |

2.00 | 9.79 | 0.776 | 0.9846 | 0.035 |

3.00 | 9.22 | 0.801 | 0.9827 | 0.035 |

Parameter | Coefficient | Estimate | $\mathit{SE}$ | t-Statistic | p-Value |
---|---|---|---|---|---|

a | ${c}_{0}$ | 10.3864 | 0.0022 | 4715.20 | 0 |

${c}_{1}$ | −0.8566 | 0.0032 | −263.795 | 0 | |

${c}_{2}$ | −0.0969 | 0.0029 | −33.5964 | 2 $\times {10}^{-125}$ | |

${c}_{3}$ | −0.0616 | 0.0025 | −24.9570 | 2.6 $\times {10}^{-87}$ | |

${c}_{4}$ | 0.0098 | 0.0005 | 20.4419 | 2.1 $\times {10}^{-66}$ | |

b | ${c}_{0}$ | 0.7671 | 0.0002 | 3487.80 | 0 |

${c}_{1}$ | 0.0121 | 0.0003 | 37.4090 | 5 $\times {10}^{-141}$ | |

${c}_{2}$ | 0.0094 | 0.0003 | 32.4952 | 9 $\times {10}^{-121}$ | |

${c}_{3}$ | 0.0079 | 0.0002 | 31.8873 | 4 $\times {10}^{-118}$ | |

${c}_{4}$ | −5.4 $\times {10}^{-4}$ | 5 $\times {10}^{-5}$ | −11.2541 | 4.3 $\times {10}^{-26}$ |

Parameter | Model | Linear Function (Equation (8)) | Quadratic Function (Equation (9)) | |||||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{a}$ | $\mathbf{b}$ | ${\mathit{R}}^{\mathbf{2}}$ | $\mathit{RMSE}$ | $\mathbf{a}$ | $\mathbf{b}$ | $\mathbf{c}$ | ${\mathit{R}}^{\mathbf{2}}$ | $\mathit{RMSE}$ | ||

$\Delta {\epsilon}_{1}$ | 3D | — | — | — | — | 864.5 | −160.8 | 7.87 | 0.9730 | 0.26 |

2D-ABC | — | — | — | — | 809.8 | −150.8 | 7.43 | 0.9726 | 0.24 | |

$\Delta {\epsilon}_{2}$ | 3D | 12.99 | −0.74 | 0.9742 | 0.06 | — | — | — | — | — |

2D-ABC | 13.40 | -0.74 | 0.9829 | 0.05 | — | — | — | — | — | |

$\Delta {\epsilon}_{3}$ | 3D | 21.94 | −1.50 | 0.9802 | 0.09 | — | — | — | — | — |

2D-ABC | 22.30 | −1.51 | 0.9764 | 0.10 | — | — | — | — | — | |

${\epsilon}_{\infty}$ | 3D | 21.13 | 0.41 | 0.9428 | 0.15 | 162.7 | −18.44 | 2.69 | 0.9680 | 0.11 |

2D-ABC | 18.44 | 0.54 | 0.9256 | 0.15 | 166.1 | −21.95 | 2.87 | 0.9595 | 0.11 | |

${f}_{1}$, GHz | 3D | −0.5013 | 0.1087 | 0.8336 | 0.0064 | 2.285 | −1.057 | 0.1407 | 0.8414 | 0.0064 |

2D-ABC | −0.6296 | 0.1342 | 0.9654 | 0.0034 | 2.662 | −1.277 | 0.1715 | 0.9732 | 0.0031 | |

${f}_{2}$, GHz | 3D | −2.751 | 0.785 | 0.4861 | 0.081 | −29.14 | 4.334 | 0.376 | 0.5107 | 0.080 |

2D-ABC | −3.059 | 0.991 | 0.8260 | 0.040 | −42.24 | 7.211 | 0.399 | 0.8972 | 0.031 | |

${f}_{3}$, GHz | 3D | −6.39 | 4.95 | 0.0358 | 0.95 | — | — | — | — | — |

2D-ABC | −17.72 | 9.04 | 0.3925 | 0.63 | — | — | — | — | — | |

$\sigma $, mS ${\mathrm{m}}^{-1}$ | 3D | — | — | — | — | 1178 | −216.4 | 10.07 | 0.9678 | 0.40 |

2D-ABC | — | — | — | — | 1207 | −220.5 | 10.23 | 0.9689 | 0.40 |

**Table 4.**Parameters of the segmented model (Equation (10)) fitted to the $\Delta {\epsilon}_{1}\left(\theta \right)$ and $\sigma \left(\theta \right)$ relations, with $\Delta {\epsilon}_{1}$ and $\sigma $ obtained with the use of 3D and 2D-ABC dielectric models.

Parameter | Dielectric Model | ${\mathit{a}}_{\mathit{s}1}$ | ${\mathit{a}}_{\mathit{s}2}$ | ${\mathit{b}}_{\mathit{s}1}$ | ${\mathit{\theta}}_{\mathit{c}}$ | ${\mathit{R}}^{2}$ | $\mathit{RMSE}$ |
---|---|---|---|---|---|---|---|

$\Delta {\epsilon}_{1}$ | 3D | 19.57 | 100.28 | −1.31 | 0.129 | 0.9776 | 0.24 |

2D-ABC | 18.81 | 94.48 | −1.22 | 0.130 | 0.9780 | 0.22 | |

$\sigma $, mS ${\mathrm{m}}^{-1}$ | 3D | 29.09 | 139.02 | −2.41 | 0.129 | 0.9717 | 0.38 |

2D-ABC | 29.96 | 142.65 | −2.46 | 0.128 | 0.9724 | 0.39 |

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**MDPI and ACS Style**

Kafarski, M.; Szypłowska, A.; Majcher, J.; Wilczek, A.; Lewandowski, A.; Hlaváčová, Z.; Skierucha, W.
Complex Dielectric Permittivity Spectra of Rapeseed in the 20 MHz–3 GHz Frequency Range. *Materials* **2022**, *15*, 4844.
https://doi.org/10.3390/ma15144844

**AMA Style**

Kafarski M, Szypłowska A, Majcher J, Wilczek A, Lewandowski A, Hlaváčová Z, Skierucha W.
Complex Dielectric Permittivity Spectra of Rapeseed in the 20 MHz–3 GHz Frequency Range. *Materials*. 2022; 15(14):4844.
https://doi.org/10.3390/ma15144844

**Chicago/Turabian Style**

Kafarski, Marcin, Agnieszka Szypłowska, Jacek Majcher, Andrzej Wilczek, Arkadiusz Lewandowski, Zuzana Hlaváčová, and Wojciech Skierucha.
2022. "Complex Dielectric Permittivity Spectra of Rapeseed in the 20 MHz–3 GHz Frequency Range" *Materials* 15, no. 14: 4844.
https://doi.org/10.3390/ma15144844