Abstract
By postulating that the electrical tree growth rate ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textit {dl} / \textit {dt}$ </tex-math></inline-formula> ) is proportional to the rate of destruction due to partial discharges (PDs), a mathematical model for conductive electrical tree growth in dielectrics is put forward. Three types of solutions are presented for this mathematical model, which predict three cases of electrical tree growth in dielectrics; namely, the development case, the stagnation case, and the transition case. The physical nature of the three cases is presented, which is the comparison of the local field ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${E}_{\text {loc}}$ </tex-math></inline-formula> ) and the critical field ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${E}_{c}$ </tex-math></inline-formula> ) caused by the space charges due to a PD. When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${E}_{\text {loc}} > {E}_{c}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textit {dl}/\textit {dt} > {0}$ </tex-math></inline-formula> and the electrical tree develops. When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${E}_{\text {loc}} < {E}_{c}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textit {dl}/\textit {dt} < {0}$ </tex-math></inline-formula> and the electrical tree stagnates. When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${E}_{\text {loc}} = {E}_{c}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textit {dl}/\textit {dt} = {0}$ </tex-math></inline-formula> and the electrical tree stays at a transition state.
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