  We develop a consistent model for a Line Lightning Protection Device and
demonstrate that this model can explain the two modes of current quenching --
impulse quenching and current zero quenching -- observed in such devices. A
dimensional analysis shows that impulse quenching can always be obtained if the
power loss from the electric arcs is large enough as compared to $U_0 I_f$,
where $U_0$ is the grid voltage and $I_f$ is the maximum follow current after a
lightning strike. We further show that the two modes of quenching can be
reproduced in a full 3D arc simulations coupled to the appropriate circuit
model. This means the arc simulations can be used for optimization and
development of future LLPDs.
