Measurements of the dynamic input impedance of a dc SQUID

The impedance of a circuit coupled magnetically via a mutual inductanceMi to a dc SQUID of geometric inductanceL is modified by the dynamic input impedance of the SQUID, which can be characterized by the flux-to-current transfer functionJФ≡∂J/∂Ф;J is the current circulating in the SQUID loop and Ф is the flux applied to the loop. At the same time, the SQUID is modified by the presence of the input circuit: in the lumped circuit approximation, one expects its inductance to be reduced toLr=(1−αe2)L, where αe is an effective coupling coefficient. Calculations of JФ using an analog simulator are described and presented in the form of a dynamic inductance ℒ and a dynamic resistance ℛ versus bias currentI and Ф. Experimental measurements of ℒ and ℛ were made on a planar, thin-film SQUID tightly coupled to a spiral input coil that was connected in series with a capacitorCi to form a resonant circuit. Thus,JФ was determined from the change in the resonant freqency and quality factor of this circuit as a function ofI and Ф. At low bias currents (low Josephson frequencies) the measured values of ℒ were in reasonable agreement with values simulated for the reduced SQUID, while at higher bias currents (higher Josephson frequencies) the measured values were in better agreement with values simulated for the unscreened SQUID. Similar conclusions were reached in the comparison of the experimental and simulated values of the flux-to-voltage transfer functionVФ. The reduction in the screening at the higher Josephson frequencies is believed to result from the parasitic capacitanceCp between the SQUID and the input coil. In contrast to the behavior of the input inductance, the change in the input resistance ΔRi could not be explained in terms of the dynamic impedance of the SQUID reflected into the input circuit. Instead, ΔRi was dominated by capacitive feedback between the output of the SQUID and the input circuit viaCp. The experimental values of ΔRi were satisfactorily explained by a simplified model that predicts ΔRi⋍−MiVФr(Cp/Ci).