# A Combined Electro-Thermal Breakdown Model for Oil-Impregnated Paper

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Electro-Thermal Breakdown Test

^{−6}m

^{2}/s, which ranks tenth and its acid value is 0.012 mg KOH/g. The amount of oil dissolved as alkane gases was 1.24 μL/L. Both paper and oil were strictly processed before the breakdown experiments. Details regarding how samples were processed could be found in [15]. At least 15 samples were used for the breakdown voltage measurement at each temperature.

#### 2.2. Thermal Gravimetric Analysis and Conducitivity Measurement

## 3. Results

#### 3.1. Breakdown Experiments under Different Temperatures and Thermal Gravimetric Test

#### 3.2. Thermal Breakdown Simulation

_{c1}, electric breakdown occurs. The typical phenomenon is that the breakdown voltage is not affected by temperature in general, but sometimes it increases with the increase of temperature. When T

_{c1}< T < T

_{c2}, thermal breakdown occurs, which is usually caused by steady or impulsive heating. This is most often seen in experiments. Once T > T

_{c2}, electro-mechanical breakdown takes place, which results from polymer deformation due to the polymer softening, and electrostatic force between the electrodes.

_{v}is specific heat at constant volume, κ is thermal conductivity of the dielectric, σ is conductivity, T is temperature, E is electric field, t is time.

_{a}is activation energy, a is hopping distance, J

_{0}, σ

_{k}, a

_{k}and b

_{k}are constants, and k is Boltzmann constant. The thermal equilibrium Equation (1) was numerically solved by the Richardson algorithm [19].

#### 3.3. Electro-Thermal Breakdown Simulation

_{0}is the vacuum permittivity, ε

_{r}is relative permittivity of the dielectric, e is electric charge, μ is mobility, n is carrier density, and S is the source term. Details of the model can be found in [21]. Although this model is usually used for charge transport simulations in polymers such as polyethylene, and as oil-impregnated paper is a kind of complex material, it can still apply to charge transport description of oil-impregnated paper [20].

## 4. Discussions

#### 4.1.Thermal Breakdown Simulation

_{c1}≈ 0 °C, T

_{c2}≈ 90 °C. In particular, the breakdown of oil-impregnated paper between 0 °C and 90 °C was due to thermal or electro-thermal breakdown.

_{b}), thermal breakdown would occur, neglecting the dynamic process of breakdown. For simplicity, if the highest local temperature reached T

_{b}, the whole oil-impregnated paper sample would be seen as having failed. Calculated breakdown voltages are shown in Figure 5 and temperature variation within the sample is shown in Figure 6. Because the heat generated by the current was hard to disperse, the temperature within the sample rose sharply and was much higher than the ambient temperature, which finally led to thermal breakdown.

#### 4.2. Electro-Thermal Breakdown Simulation

_{0}is phonon frequency, and it is 4.2 × 10

^{14}Hz here. To accurately determine mobility, we measured the temperature-dependent conductivity of oil-impregnated paper with a 1-kV/mm electric field applied, as shown in Table 3. The temperature-dependent conductivity of oil-impregnated paper under low electric fields can be expressed as:

_{0}is a constant.

_{a}= 0.84 eV, which was consistent with the literature. The calculated breakdown voltages are illustrated in Figure 7 (denoted by the temperature criterion). It could be seen that the calculated results were larger than the experimental results except that at 90 °C. This was because carrier injections from electrodes and carrier mobility were slow under lower temperatures. As a result, the current density was small, and consequently the heat generation was slow. Therefore, the calculated breakdown voltages were higher.

_{b}was about 230 kV/mm. E

_{b}could be treated as an intrinsic feature of oil-impregnated paper; namely, once a local electric field within the sample reached E

_{b}, breakdown happened. With the combination of the temperature criterion and electric field criterion, we obtained a combined electro-thermal criterion. The simulated breakdown voltages are presented in Figure 7 (denoted by electro-thermal criterion). The calculated breakdown voltages were very close to experimental results except the one at 60 °C.

_{a}, if the electric field was higher than a critical value E

_{μ}, u

_{a}= 1.1 eV; otherwise, u

_{a}= 0.84 eV. With this assumption and E

_{μ}= 200 kV/mm, the simulated electro-thermal breakdown voltages are illustrated in Figure 8. These decreased with the increase of temperature and they were close to the experimental results.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Temperature-dependent breakdown voltage of oil-impregnated paper insulation. (

**a**) Weibull distribution of breakdown voltages; (

**b**) Temperature-dependent shape and scale parameters.

**Figure 4.**Temperature-dependent breakdown of insulating polymers [16]. (I) T < T

_{c1}, electric breakdown; (II) T

_{c1}< T < T

_{c2}, thermal or electro-thermal breakdown; (III) T > T

_{c2}, electro-mechanical breakdown.

**Figure 7.**Comparison of electro-thermal breakdown simulations and breakdown experiments. Temperature- and electric field-dependent mobility was considered.

**Figure 8.**Comparison of electro-thermal breakdown simulations and breakdown experiments. Temperature- and electric field-dependent mobility was considered, and so was negative differential mobility.

**Figure 9.**Comparison of calculated hopping and Kelvin conductivity under different temperatures and electric fields.

**Figure 10.**Variations in the highest electric field and temperature before breakdown. (

**a**) Variation in the highest electric field, (

**b**) Variation in the highest temperature.

Symbol | Value | Unit | Symbol | Value | Unit |
---|---|---|---|---|---|

a | 2.1 | nm | C_{v} | 2.5 × 10^{6} | J/(m^{3}·K) |

u_{a} | 1.1 | eV | J_{0} | 1.9 × 10^{13} | A/m^{2} |

κ | 0.25 | W/(m·K) |

Symbol | Meaning | Value | Unit |
---|---|---|---|

N_{et0} | Electron trap density | 100 | C/m^{3} |

N_{ht0} | Hole trap density | 100 | C/m^{3} |

B_{e} | Electron trapping coefficient | 7 × 10^{−3} | s^{−1} |

B_{h} | Hole trapping coefficient | 7 × 10^{−3} | s^{−1} |

S_{eμ,ht} | Recombination coefficient | 4 × 10^{−3} | m^{−3} C^{−1} s^{−1} |

S_{et,hμ} | Recombination coefficient | 4 × 10^{−3} | m^{−3} C^{−1} s^{−1} |

S_{et,ht} | Recombination coefficient | 4 × 10^{−3} | m^{−3} C^{−1} s^{−1} |

ω_{e} | Schottky injection barier | 1.2 | eV |

ω_{h} | Schottky injection barrier | 1.2 | eV |

ε_{r} | Relative permittivity | 4.4 | — |

**Table 3.**Temperature-dependent conductivity of oil-impregnated paper (measured at 1 kV/mm electric field).

Temperature T ( °C) | 40 | 50 | 70 | 90 | 110 |
---|---|---|---|---|---|

Conductivity σ(S/m) | 1.90 × 10^{−13} | 4.59 × 10^{−13} | 2.64 × 10^{−12} | 1.38 × 10^{−11} | 5.26 × 10^{−11} |

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**MDPI and ACS Style**

Huang, M.; Zhou, Y.; Zhou, Z.; Qi, B.
A Combined Electro-Thermal Breakdown Model for Oil-Impregnated Paper. *Energies* **2017**, *10*, 2160.
https://doi.org/10.3390/en10122160

**AMA Style**

Huang M, Zhou Y, Zhou Z, Qi B.
A Combined Electro-Thermal Breakdown Model for Oil-Impregnated Paper. *Energies*. 2017; 10(12):2160.
https://doi.org/10.3390/en10122160

**Chicago/Turabian Style**

Huang, Meng, Yuanxiang Zhou, Zhongliu Zhou, and Bo Qi.
2017. "A Combined Electro-Thermal Breakdown Model for Oil-Impregnated Paper" *Energies* 10, no. 12: 2160.
https://doi.org/10.3390/en10122160