By postulating that the electrical tree growth rate ( $\textit {dl} / \textit {dt}$ ) 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 ( ${E}_{\text {loc}}$ ) and the critical field ( ${E}_{c}$ ) caused by the space charges due to a PD. When ${E}_{\text {loc}} > {E}_{c}$ , $\textit {dl}/\textit {dt} > {0}$ and the electrical tree develops. When ${E}_{\text {loc}} < {E}_{c}$ , $\textit {dl}/\textit {dt} < {0}$ and the electrical tree stagnates. When ${E}_{\text {loc}} = {E}_{c}$ , $\textit {dl}/\textit {dt} = {0}$ and the electrical tree stays at a transition state.

中文翻译：

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电介质中导电树生长的数学模型——第一部分：理论

通过假设电树生长率 ( $\textit {dl} / \textit {dt}$) 与局部放电 (PD) 造成的破坏率成正比，提出了电介质中导电电树生长的数学模型。该数学模型提出了三种类型的解决方案，它们预测了电介质中电树生长的三种情况；即发展案例、停滞案例和过渡案例。呈现了三种情况的物理性质，即局部场的比较（ ${E}_{\text {loc}}$) 和临界场 ( ${E}_{c}$) 由 PD 引起的空间电荷引起。什么时候 ${E}_{\text {loc}} > {E}_{c}$, $\textit {dl}/\textit {dt} > {0}$并且电气树发展。什么时候 ${E}_{\text {loc}} < {E}_{c}$, $\textit {dl}/\textit {dt} < {0}$电气树停滞不前。什么时候 ${E}_{\text {loc}} = {E}_{c}$, $\textit {dl}/\textit {dt} = {0}$电气树处于过渡状态。