Modelling of charge dynamic and breakdown phenomenon in insulating materials

Abstract

In insulating materials, the collection of space charge is actually the main cause of breakdown of dielectric which relies on their lifespan. It can be said that in this research, a model of bipolar transport charge is presented by us for studying the phenomenon of breakdown in the polyethylene which is low-density for electrical insulation. That is why, the impact of sample thickness if studied by us on the dynamic charge and we are specifically intrigued in the current density which is external in the first evidence of the insulator’s dielectric breakdown. The distribution of net charge and electric file is shown by the model results in the insulator before the phenomenon of breakdown. Actually, such distributions are related strongly to the enhancement of current density which is external and illustrates the phenomenon of breakdown.

Keywords: Model, Insulator, Space charge, Sample thickness, Breakdown, Electrical lifetime.

Nomenclature

1. Introduction

In a broad range of electric apparatus, insulating polymers have been utilized like electrical cables of their efficient specifications. The LDPE or low density polyethylene is a principal polymer that is utilized for the insulation of cable in the electrical carrier or transport [1, 2]. Space charges’ existence in the insulating materials which are polymeric has actually been affirmed to be the source of drastic damages in electrical networks under dc voltages which are high applied [3, 4]. For such issues, several methods for the measurements of space charge were designed for understanding and overcoming their impacts on the properties like the conduction and electrical breakdown [5, 6].

In the recent years, simulation and modeling which are theoretical have been combined or integrated for thorough understand the dynamics of space charge [7-12]. Numerical results are presented in this paper by us using our model which focuses on the impact of thickness of sample on the insulator’s electrical breakdown. These results show that the decrease of the sample thickness intensifies the electrical field’s distortion leading to the insulator’s electrical breakdown. We show also an estimation of the electrical lifetime of the sample which is highly affected by the sample thickness. We note that our model’s details were given in our late work where we highlighted a original result under dc applied voltage which is high regarding the space charge’s apparition on the profiles of conduction current and total charge density [12, 13]. In addition, we provided in a previous study an experimental validation proving that our model reproduces qualitatively and quantitatively external current density’s evolutions under several voltages which are applied [14].

2. Hypothesis and model equations

2.1 Hypothesis

The film of polyethylene in between two electrodes is considered by us under the applied voltage of dc. Electrode-polyethylene contact is actually predicted to be flawless and the model of Schottky is utilized for injected current density. Additionally, the sample is supposed to be a plane infinite just to admit that every physical mechanism acts hugely over the direction of sample thickness. Polyethylene is normally under the condition of uniform temperature. It can be said that carriers are assumed to have an effective mobility which is constant as well. Relying on the work of of Meunier et al [15, 16], it is supposed by us that deep and shallow traps are divided in the polyethylene bulk among electrodes. Furthermore, shallow traps played an important role in the mechanism of transport since their residence time is quite weak. Virtually, the deep traps which have a residence time infinitely to contribute, charge accumulation is facilitated. Lastly, it is assumed by us that space charges’ initial density is included in the sample.

2.2. Theoretical equations

2.2 Model equations

The Poisson’s is regrouped by the physical model, transport and continuity expressions or equations.

The equation of Poisson’s, boundary and initial circumstances are as follow:

The complete density of charge ρ(x,t) is composed locally of trapped and mobile holes and electrons:

It can be said that almost all the figures are achieved under the dc applied voltage of 75 kV. In LDPE sample with thickness up to 160 μm, evolution is shown by the external current density by Figure 1(a). It is noted by us that when high dc voltage is implemented, the current density which is external increase until the steady state is reached. This aspect of evolution takes place when the phenomenon of charge packet is formed [13]. For sure, under the voltage of high dc, charge current’s injection is increased after the starting of charge packets of electron-hole recombination. The distortion of electric field is induced by the phenomenon of charge packet: the field of electricity lowers in the mid of sample and rises at the interface of electrode-insulation [13]. The distortion of electric field is theoretically supposed for satisfying the physical condition in the seventh equation. When the applied voltage is adjusted at the steady state, the total along with local contribution of the displacement current disappears, respectively form 20 and 18 equations. Therefore, the external current which is theoretical at the steady state is equivalent to the local conduction current’s integral over the thickness of polyethylene.

As a decrement in the sample thickness occurs (figure 1(b)), injection’s intensification is more important and the external current’s value at the state which is steady is quite significant. For sure, at the state which is steady and as per the numbers 1(a, b), the current density’s value has raised from 5.7 10^-5 A.m^-2 to 9.2 10^-5 A.m^-2. Due to it, when the thickness of sample decreases, insulator’s conduction process becomes more prominent which can harm their breakdown and to electrical lifetime’s reduction.

Figure 1. External current density versus time (a) 160 μm thickness and (b) 152 μm thickness

The evolution of density of external current vs. LDPE with the thickness of 151 μm time is shown by the Figure 2(a) which illustrates breakdown thickness’s threshold value. It can be said that it matches with that of previous studies [17]. For sure, the density of external current generally increases from the starting of application of voltage until a steady state and a sudden increment takes place which highlight the sample’s processes regarding electrical post breakdown. Regarding the figure 2a’s profile, it can be concluded by us that beginning from 400s, every trap is filled and every injected charge contributes to the density of external current making us observe the avalanche current amplification’s concept which gets to 2 10^-4 A.m^-2. It is the interfacial electrical field’s intensification that becomes the cause of this aspect and becomes quite significant and densities of injected charge are critical in leading to the conduction process’s amplification in the sample. What’s more, the injection’s intensification of mobile charges plays an important role in the charges density’s augmentation can become the enhancement of local field in polyethylene’s small volumes. The aspect of post breakdown can be shown by both the enhancement of local field and conduction amplification or can lead to the material’s effective breakdown. Results were mentioned by novels cane be observed on the numbers of 2(b)-(f) showing the external density’s evolutions for thicknesses of 140 10^-6 m, 130 10^-6 m, 120 10^-6 m, 110 10^-6 m and 100 10^-6 m, respectively. In them, the sample’s breakdown can take place after 75 s, 67 s, 62 s, 56 s and 51 s for thicknesses of 140 10^-6 m, 130 10^-6 m, 120 10^-6 m, 110 10^-6 m and 100 10^-6 m, respectively. Our model gave these results from which a logical conclusion can be deduced indicating that as there is a decrement in the sample thickness, the establishment of breakdown phenomenon appears quite soon.

Figure 2. External current density versus time (a) 151 μm thickness, (b) 140 μm thickness

(c) 130 μm thickness, (d) 120 μm thickness

(e) 110 μm thickness, (f) 100 μm thickness

For understanding the sudden increment in the current, figures 3(a)-(c) are used for showing the distributions of electric field, the density of mobile electron, and the conduction current density example in polyethylene. These figures’ obtained profiles are determined for examples 151 10^-6 m, right before and at the event of post breakdown for the instants which are corresponding 400 s and 542 s, respectively. Actually, it is shown by figure 3(a) that at the 542 s, electric field’s absolute value at the cathode increases and reaches the 5.5 108 V.m^-1maximum. It can be seen in 3(b) that the injected density’s value of mobile electron increases to 4000 C.m^-3 from 1800 among 400 and 542 s. Considering the fact that the density of external current relies only on mobile charges, it can be observed by us on the figure 2a’s corresponding profile, the sudden increment which is mainly because of the mentioned increment or jump. For explaining more the sample’s conductive character, the conduction current density is shown in figure 3(c) as it jumps from 0.7 A.m^-2 at 400 s to 1.9 A.m^-2 at 542 s.

Figure 3. Profiles before and at the post breakdown (a) Electric field distribution on the sample thickness, (b) Mobile electron density, (c) Conduction current density

For estimating the sample’s electrical lifetime, in figure 4, we plot the lifespan vs. the thickness of sample and it is concluded by us that when there is an increment in the thickness, the sample’s electrical lifespan increases. It is shown clearly that the thickness’s augmentation permits optimizing the insulator’s electrical lifetime and even evading the electrical breakdown. Therefore, this research allows predicting the insulator’s electrical lifetime.

Figure 4. Electrical lifetime of LDPE plotted versus sample thickness

4. Conclusion

A model for the transport of bipolar charge is presented for investigating the effect of temperature on the polyethylene insulator’s post breakdown and lifetime under applied voltages of dc. Particularly, we are intrigued by the sample thickness’s impact on the dynamic external current which is the sample post breakdown’s evidence. It is shown by our model results when there is a decrement in the thickness, external current’s value at the state which is steady is more critical. That is why, the process of conduction in the insulator is becoming more and more intensified which is proven by an increment in the density of conduction current. With a further decrement in the thickness, the charge injection present in the sample increases drastically while threatening the post breakdown and electrical lifetime’s reduction which is marked usually by a sudden increment in the external current density’s profile. These outcomes in our perception are achieved for the very first time in the modeling of breakdown.

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