Averaged equations for Josephson junction series arrays

We derive the averaged equations describing a series array of Josephson junctions shunted by a parallel inductor-resistor-capacitor load. We assume that the junctions have negligible capacitance (β=0), and derive averaged equations that turn out to be completely tractable: in particular, the stability of both in-phase and splay states depends on a single parameter, δ. We find an explicit expression for δ in terms of the load parameters and the bias current. We recover (and refine) a common claim found in the technical literature, that the in-phase state is stable for inductive loads and unstable for capacitive loads.