Analytical derivation of DC SQUID response
  We consider voltage and current responses formation in DC SQUID with overdamped Josephson junctions in resistive and superconducting state in the frame of resistively shunted junction (RSJ) model. For simplicity we neglect the junction capacitance and the noise effect. Explicit expressions for the responses in resistive state were obtained for a SQUID which is symmetrical with respect to bias current injection point. Normalized SQUID inductance $l = 2 e I_c L/\hbar$ (where $I_c$ is the critical current of Josephson junction, $L$ is the SQUID inductance, $e$ is the electron charge and $\hbar$ is the Planck constant) was assumed to be within the range $l \leq 1$, subsequently expanded up to $l \approx 7$ using two fitting parameters. SQUID current response in superconducting state was considered for arbitrary value of the inductance. Impact of small technological spread of parameters relevant for low-temperature superconductor (LTS) technology was studied with generalization of the developed analytical approach for a case of small difference of critical currents and shunt resistances of the Josephson junctions, and inequality of SQUID inductive shoulders for both resistive and superconducting states. Comparison with numerical calculation results shows that developed analytical expressions can be used in practical LTS SQUIDs and SQUID-based circuits design, e.g. large serial SQIF, drastically decreasing the time of simulation. 
I. I. Soloviev, N. V. Klenov, A. E. Schegolev, S. V. Bakurskiy and M.   Yu. Kupriyanov
cond-mat.supr-con