**2. Electrical Breakdown Models**

Generally, a ramp voltage with a constant rising rate is applied to the electrodes on both sides of the polymer nanocomposite to investigate the electrical breakdown properties. Firstly, when the voltage is gradually increased, the electrons and holes in the cathode and anode, respectively, are injected into the nanocomposite. After these electrons and holes enter the nanocomposite, they drift under the driving of the electric field. Since there are deep traps formed by many polar groups inside the nanocomposite, the deep traps capture electrons and holes, thereby accumulating space charges of the same polarity inside the nanocomposite [26]. The space charges cause the electric field inside the nanocomposite to be distorted. When the distorted electric field reaches a certain level, it may lead to the breakdown of the nanocomposite. Nanodoping changes the trap properties inside the nanocomposite, which in turn changes the charge injection, the space charge accumulation, and the electric field concentration properties; these changes ultimately lead to the variation in the breakdown strength with the nanofiller content. This breakdown model is called the electrical breakdown modulated by space charges with a maximum electric field criterion (EBEF).

Secondly, there are spaces inside the polymer nanocomposite that are not occupied by atoms, known as free volumes. When charges are transported inside the nanocomposite, they may enter the free volumes. Charges in the free volumes can be rapidly accelerated by electric fields to obtain high energies. If the energy gained by the charges in the free volumes exceeds the trapping ability of the deep traps, the high-energy charges will result in the breaking of the molecular chain and cause the electric breakdown of the nanocomposite [27]. Nanodoping changes the trap energy, which in turn affects the ability of traps to trap the high-energy charges, and ultimately changes the breakdown strength of the nanocomposite. In this process, the effects of space charge accumulation and electric field concentration on the breakdown strength also need to be considered. This breakdown model is called the electrical breakdown modulated by space charges with a maximum electron energy criterion (EBEG).

Thirdly, since the substance entities trapped inside the polymer nanocomposites are polar groups, when the traps capture charges, the Coulomb force on the trapped charges must be transferred to the molecular chains. The molecular chains will undergo directional displacement under the effect of the Coulomb force, resulting in the expansion of the free volumes. An increase in the size of free volumes allows the charges entering them to accumulate more energies. When the energies accumulated by the charges exceed the trapping ability of the deep traps, it causes the breakdown of the polymer nanocomposite. Nanodoping affects both the molecular chain motion and trap properties in nanocomposites, which in turn affects the energy gain properties of charges and the ability of traps to capture high-energy charges. Finally, the electric breakdown strength of the nanodielectric is changed. This breakdown model is called the EBMD model [13,25].

The three breakdown models of EBEF, EBEG, and EBMD all include charge carrier injection and charge carrier transport processes. The mathematical equations of charge carrier injection and charge carrier transport processes are introduced first. Then, the breakdown criteria of EBEF and EBEG models are given. Finally, the equation of molecular chain displacement under the action of electric force in EBMD model is introduced, and the breakdown criterion is given.
