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A General Network Equation to Unify the Analysis of Normal RLC Circuits and Superconducting Josephson Junction Circuits

Yongliang Wang

Published Nov 15, 2022• 0 citations
Publisher

Abstract

Josephson junction circuits, such as superconducting quantum interference devices (SQUIDs) and single-flux-quantum (SFQ) circuits, have been successfully applied in both analog and digital electronic domains. Their variables of macroscopic quantum phases are distinct from that of the normal resistor-inductor-capacitor (RLC) circuits; their flux-quantization law (FQL) involving magnetic flux couplings are not well supported by the conventional circuit diagrams. This article presents a general network equation to unify the analyses of both normal RLC circuits and superconducting Josephson junction circuits. This network equation uses the flux contributions of noninductive components as variables to unify the definitions of Josephson junctions and normal RLC elements, and unify the closed-loop law for both superconducting and non-superconducting loops. It simplifies the analysis of the electric circuits with various magnetic flux inputs, and is easily understood by electronic engineers who are trained with the conventional circuit theories.

DOI
10.22541/au.166853519.95828508/v1

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Josephson junction circuits, such as superconducting quantum interference devices (SQUIDs) and single-flux-quantum (SFQ) circuits, have been successfully applied in both analog and digital electronic domains. Their variables of macroscopic quantum phases are distinct from that of the normal resistor-inductor-capacitor (RLC) circuits; their flux-quantization law (FQL) involving magnetic flux couplings are not well supported by the conventional circuit diagrams. This article presents a general network equation to unify the analyses of both normal RLC circuits and superconducting Josephson junction circuits. This network equation uses the flux contributions of noninductive components as variables to unify the definitions of Josephson junctions and normal RLC elements, and unify the closed-loop law for both superconducting and non-superconducting loops. It simplifies the analysis of the electric circuits with various magnetic flux inputs, and is easily understood by electronic engineers who are trained with the conventional circuit theories.
Josephson junction circuits, such as superconducting quantum interference devices (SQUIDs) and single-flux-quantum (SFQ) circuits, have been successfully applied in both analog and digital electronic domains. They are considered to be distinct from the normal resistor-inductor-capacitor (RLC) circuits, due to the superconducting physics. To bridge the gap between Josephson junction circuits and conventional RLC circuits, we introduce a magnetic-flux-generator (MFG) model to unify the definitions of Josephson junctions and normal noninductive components, and transform their circuits uniformly into MFG networks. A general network equation for both superconducting and normal MFG networks is derived. It shows that, Josephson junction circuits are the superconducting RLC networks modulated by Josephson currents; their analysis and simulation are unified with that of the normal RLC circuits by solving the general network equation.
Superconducting quantum interference devices (SQUIDs), single flux-quantum (SFQ) logic circuits, and quantum Josephson junction circuits have been developed into a family of superconductor integrated circuit, and are widely applied for subtle magnetic-field measurements, energy-efficient computing, and quantum computing, respectively. They are Josephson junction networks composed of Josephson junctions and normal resistor-inductor-capacitor (RLC) components, working with the fluxoid-quantization principle and Josephson effects to achieve unique flux-modulated dynamics and characteristics; they react to the vector potential of magnetic fields rather than the electric potential. However, the conventional circuit diagrams and nodal analysis methods focus on the electric charges flowing though branches and nodes, ignoring dynamics of the magnetic fluxes flowing from loop to loop. This article introduces a general flux-based circuit theory to unify the analyses of Josephson junction circuits and normal RLC circuits. This theory presents a magnetic-flux-generator (MFG) concept to unify Josephson junctions and normal circuit elements, and abstract both Josephson junction circuits and normal RLC circuits as MFG network; it derives a general network equation to describe dynamics of Josephson junction circuits, and invents a kind of magnetic-flux flow (MFF) diagram to depict the working principles of magnetic-flux flows inside Josephson junction circuits. The flux-based theory is complementary to the conventional circuit theories in the design and analysis of superconductor integrated circuits.
Superconducting Josephson junctions and quantum phase slip (QPS) junctions create various hybrid circuits combined with nonsuperconducting elements. Those circuits require a common analysis method to bridge the gap between superconducting and nonsuperconducting electronics. This article presents an electromagnetic-flux-distribution model to unify the analyses of superconducting and non-superconducting circuits. This model uses electric and magnetic fluxes as the variables to unify the definitions of circuit elements and circuit laws; it derives general circuit equations of the circuits viewed from both electric and magnetic fields, to depict their dynamics and duality principles. This method is demonstrated in the analyses of a Josephson junction circuit and its dual QPS junction circuit. It shows that Josephson junction circuits are the magnetic-flux distribution network tuned by the Josephson current, while QPS junction circuits are the electric-flux distribution network modulated by the QPS voltage.
Josephson junction circuits, such as superconducting quantum interference devices (SQUIDs) and single-flux-quantum circuits, have been successfully applied in both the analog and digital domains. Due to phase-based working principles, their analyses are not well supported by charge-based analysis methods and conventional circuit diagrams. This article introduces a type of magnetic-flux-flow (MFF) diagrams to simplify the design and analysis of Josephson junction circuits. MFF diagrams are the graphical expression of circuit equations derived by loop analysis methods. They vividly depict the functions of Josephson junctions and superconducting loops and visualize how Josephson junctions transfer flux quanta among superconducting loops. The applications of MFF diagrams in the design and analysis of typical Josephson junction circuits are demonstrated case by case; it is shown that MFF diagrams are complementary to conventional circuit diagrams for both Josephson junction circuits and normal electric circuits.
A dc superconducting quantum interference device (SQUID) consisting of a symmetric ring circuit with four Josephson junctions and two independent external currents I1{I}_{1}I1​ and I2{I}_{2}I2​, which enter the circuit at different points, has a phase boundary between the fully superconducting and the normal states that is different from that of a conventional SQUID. The control current I2{I}_{2}I2​ modifies the magnetic flux (\ensuremath{\varphi}) periodicity and the value of the maximum measuring current I1m{I}_{1m}I1m​(\ensuremath{\varphi}). For values of I2{I}_{2}I2​>Ic{I}_{c}Ic​ (critical current of Josephson junction) the fully resistive state exists over flux intervals centered around (n+(1/2)${\ensuremath{\varphi}}_{0}$, where n is an integer and ${\ensuremath{\varphi}}_{0}$ is the fluxoid quantum. A superconducting circuit without Josephson junctions is suggested which should have a phase boundary similar to that found here. Minimization procedures which are usually employed in analyzing SQUID's are discussed.
We adopt superconductor/ferromagnet/insulator/superconductor (SFIS) Josephson junctions (JJs) as both switching JJs and intrinsic π phase shifters in superconductor digital circuits. The critical current density (Jc) and characteristic voltage (Vc) of the SFIS junctions are about 22 A/cm2 and 22 μV, respectively. The intrinsic π phase shift is confirmed by measuring the suppressed nominal critical current Icn and half-period-shifted modulation pattern of a π–π–π superconducting quantum interference device (SQUID) that contains three π-JJs in a superconducting loop. A single-flux-quantum (SFQ) circuit composed of a DC/SFQ, Josephson transmission line (JTL), and SFQ/DC converter based on SFIS JJs alone is demonstrated at 4.2 K. The energy dissipation of the SFQ/DC converter decreases by 80% because some JJs are self-biased by the π phase shifter. The intrinsic circulating currents induced by the π phase shifters lead to a narrow bias margin (±5%) and even error function, which can be solved by parameters optimization or circuit initialization in the future. The half-modulation period (Φ0/2) of a half-flux-quantum (HFQ) SQUID (a partial HFQ JTL) exhibits propagation of HFQ between π–π–π SQUIDs, indicating that more complex HFQ circuits can be developed with π-JJs alone in the future.