I. Introduction
The invention of Josephson junction has brought the rapid development of the superconductor electronics, such as the superconducting quantum interference device (SQUID) for analog flux-to-voltage conversion circuits [1] and the single flux quantum device for the digital integrated circuits [2]. With the development of SQUID-based magnetic field measurement systems, practical SQUIDs have been developed into the multi-loop hybrid electric circuits with both superconducting Josephson junctions and normal components such as resistor (R), inductor (L), and capacitor (C), as shown in Figs. 1–3. Those hybrid circuits involve not only the current-voltage relations of normal elements, but also the nonlinear current-phase relation in Josephson junctions and the flux quantization theorem inside superconducting loops. It is inconvenient to implement the conventional circuit analysis methods based on the Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL) in SQUID circuit analysis and simulation [3]. For instance, in a practical radio-frequency (rf) SQUID magnetically driven by a tank circuit as shown Fig. 1(a), and a practical direct current (dc) SQUID tightly coupled with the input coil as shown in Fig. 1(b), studies of the influences of the tank circuit on rf SQUID [4], [5] or the resonances between SQUID and input coils [6], will apply the Josephson equations for Josephson junctions and the second-order differential equations for normal RLC network respectively. The relaxation oscillation SQUID (ROS) [7], [8] and the double ROS (DROS) [9], [10] are another type of hybrid circuit with undamped dc SQUID shunted by a resistor as shown in Fig. 2. To apply the Kirchhoff's laws, the SQUID in those circuits is usually treated as the nonlinear component with static I–V curves [11]. The SQUID additional positive feedback (APF) scheme [12], [13] introduces a normal flux feedback circuit to implement both the electric and magnetic couplings to SQUID loop as shown in Fig. 3; their hybrid circuit analyses also concern both the flux quantization principle and Kirchhoff's laws [14], [15].
(a) Radio-frequency SQUID magnetically coupled with the tank circuit driven by a rf current source. (b) DC SQUID tightly coupled with an input coil. The cross in the circuits denotes the practical resistively shunted Josephson junction.
(a) Relaxation oscillation SQUID circuit. (b) Double relaxation oscillation SQUID circuit. The cross in the circuits denotes the undamped Josephson junction.
(a) DC SQUID in paralleled with an APF circuit, where, La is magnetically coupled to the dc SQUID with mutual inductance. (b) DC SQUID bootstrap circuit (SBC), where both La and Lb are magnetically coupled to the dc SQUID with mutual inductance.