# Dielectric Response Model for Transformer Insulation Using Frequency Domain Spectroscopy and Vector Fitting

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## Abstract

**:**

## 1. Introduction

## 2. Overview on Dielectric Response

## 3. Classical and Extended Debye Models

#### 3.1. Classical Debye Model

#### 3.2. Extdended Debye Model

## 4. Calculation of EDM Parameters

## 5. Frequency Domain Spectroscopy

## 6. Vector Fitting Basic Equations

## 7. Modified VF and FDS Algorithm for EDM Identification

## 8. Validation

## 9. Discussion

## 10. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Classical Debye model. (

**b**) Equivalent circuit for conduction. (

**c**) Equivalent circuit for conduction and relaxation.

XFMR | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

MVA | 10 | 12.5 | 2.5 | 2 | 3.75 | 7.5 |

KV Class | 27.6 | 34.5 | 38 | 46 | 69 | 115 |

Fluid Type | FR3 | FR3 | Mineral | FR3 | Mineral | Mineral |

%DF at 60 Hz and 20 °C | 0.35 | 0.37 | 0.32 | 0.41 | 0.32 | 0.29 |

C at 60 Hz (pF) | 7457 | 9301 | 4185 | 4427 | 2686 | 2719 |

Moisture (%) | 1.4 | 1.3 | 0.5 | 1.9 | 1.1 | 1.4 |

Oil Cond (pS/m) | 9.98 | 7.67 | 0.872 | 13.2 | 2.05 | 0.844 |

XFMR #1 | XFMR #2 | XFMR #3 | |||||||

Branch | ${\mathit{R}}_{\mathit{n}}$(GΩ) | ${\mathit{C}}_{\mathit{n}}$(nF) | ${\mathit{\tau}}_{\mathit{n}}$(s) | ${\mathit{R}}_{\mathit{n}}$(GΩ) | ${\mathit{C}}_{\mathit{n}}$(nF) | ${\mathit{\tau}}_{\mathit{n}}$(s) | ${\mathit{R}}_{\mathit{n}}$(GΩ) | ${\mathit{C}}_{\mathit{n}}$(nF) | ${\mathit{\tau}}_{\mathit{n}}$(s) |

0 | 9.251 | 7.387 | 68.336 | 11.69 | 9.192 | 107.50 | 810.2 | 4.129 | 3345.3 |

1 | 0.003 | 0.051 | 0.0002 | 0.002 | 0.080 | 0.0001 | 0.002 | 0.040 | 0.0001 |

2 | 0.066 | 0.030 | 0.0020 | 0.040 | 0.042 | 0.0017 | 0.050 | 0.014 | 0.0007 |

3 | 0.705 | 0.029 | 0.0202 | 0.400 | 0.030 | 0.0121 | 0.375 | 0.015 | 0.0057 |

4 | 1.980 | 0.109 | 0.2154 | 2.474 | 0.039 | 0.0957 | 4.322 | 0.010 | 0.0449 |

5 | 1.318 | 5.015 | 6.6085 | 2.662 | 0.733 | 1.9509 | 24.41 | 0.021 | 0.5200 |

6 | 0.857 | 35.24 | 30.199 | 0.870 | 31.98 | 27.813 | 29.83 | 1.389 | 41.424 |

7 | 2.781 | 37.46 | 104.20 | 2.165 | 46.41 | 100.50 | 24.06 | 8.831 | 212.55 |

8 | - | - | 12.54 | 45.44 | 570.10 | 109.9 | 4.146 | 455.77 | |

9 | - | - | - | - | 218.7 | 8.688 | 1900.5 | ||

XFMR #4 | XFMR #5 | XFMR #6 | |||||||

Branch# | ${\mathit{R}}_{\mathit{n}}$(GΩ) | ${\mathit{C}}_{\mathit{n}}$(nF) | ${\mathit{\tau}}_{\mathit{n}}$(s) | ${\mathit{R}}_{\mathit{n}}$(GΩ) | ${\mathit{C}}_{\mathit{n}}$(nF) | ${\mathit{\tau}}_{\mathit{n}}$(s) | ${\mathit{R}}_{\mathit{n}}$(GΩ) | ${\mathit{C}}_{\mathit{n}}$(nF) | ${\mathit{\tau}}_{\mathit{n}}$(s) |

0 | 29.05 | 4.360 | 126.69 | 71.94 | 2.664 | 191.63 | 88.45 | 2.696 | 238.42 |

1 | 0.002 | 0.049 | 0.0001 | 0.013 | 0.017 | 0.0002 | 0.006 | 0.018 | 0.0001 |

2 | 0.049 | 0.019 | 0.0009 | 3.993 | 0.001 | 0.0020 | 0.169 | 0.007 | 0.0012 |

3 | 0.424 | 0.021 | 0.0088 | 0.265 | 0.010 | 0.0027 | 0.971 | 0.012 | 0.0113 |

4 | 4.260 | 0.026 | 0.1090 | 2.931 | 0.007 | 0.0199 | 7.732 | 0.014 | 0.1098 |

5 | 3.673 | 1.460 | 5.3635 | 7.758 | 0.017 | 0.1288 | 17.27 | 0.066 | 1.1317 |

6 | 2.483 | 11.41 | 28.330 | 25.75 | 0.030 | 0.7649 | 53.13 | 1.093 | 58.081 |

7 | 17.45 | 9.275 | 161.89 | 16.13 | 1.306 | 21.064 | 834.0 | 0.171 | 142.86 |

8 | 104.2 | 9.995 | 1041.9 | 12.64 | 7.703 | 97.432 | 29.00 | 9.096 | 263.79 |

9 | - | - | - | 262.7 | 1.244 | 326.73 | 71.72 | 20.80 | 1492.4 |

10 | - | - | - | 71.39 | 10.80 | 771.11 | - | - | - |

XFMR | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Best Order | 7 | 8 | 9 | 8 | 10 | 9 |

Best Iteration | 2 | 3 | 5 | 3 | 3 | 5 |

rms error | 0.29 | 0.02 | 1.13 | 0.48 | 2.22 | 0.69 |

$\mathrm{AF}\text{}\left|\mathit{C}\right|$ (%) | 99.94 | 99.80 | 99.74 | 99.83 | 99.71 | 99.74 |

$\mathrm{AF}\text{}{\mathit{C}}^{\prime}$(%) | 99.84 | 99.70 | 99.78 | 99.84 | 99.72 | 99.87 |

$\mathrm{AF}\text{}{\mathit{C}}^{\u2033}$(%) | 99.37 | 99.30 | 98.97 | 98.58 | 98.25 | 97.63 |

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**MDPI and ACS Style**

Hernandez, G.; Ramirez, A.
Dielectric Response Model for Transformer Insulation Using Frequency Domain Spectroscopy and Vector Fitting. *Energies* **2022**, *15*, 2655.
https://doi.org/10.3390/en15072655

**AMA Style**

Hernandez G, Ramirez A.
Dielectric Response Model for Transformer Insulation Using Frequency Domain Spectroscopy and Vector Fitting. *Energies*. 2022; 15(7):2655.
https://doi.org/10.3390/en15072655

**Chicago/Turabian Style**

Hernandez, Giovanni, and Abner Ramirez.
2022. "Dielectric Response Model for Transformer Insulation Using Frequency Domain Spectroscopy and Vector Fitting" *Energies* 15, no. 7: 2655.
https://doi.org/10.3390/en15072655