Phase-locking in series arrays of Josephson junctions shunted by a complex load

The phase-locking in a large series array of Josephson junctions with a finite McCumber parameter shunted by a complex load is studied in the frame of the Stewart–McCumber model. Both the coherent and the incoherent mode are discussed. The influence of weakly time-dependent perturbations on the stability of the phase-locking as well as the optimum operating region are derived in a simple analytical way. The coherent oscillations are nearly sinusoidal and most stable for a bias current something greater than twice the junction critical current and a McCumber parameter of about 0.5; the corresponding oscillation frequency is about twice the junction characteristic frequency. Because of the finite McCumber parameter a complex load is required for the maximum power output given by an eighth of the sum of the single junction characteristic powers. In presence of thermal noise the array shows a second-order phase transition where the coherent mode can be considered as the higher-order phase. The critical temperature is proportional to the junction critical current.