The Josephson junction circuit family: network theory

Wave digital models are known to be most suitable for the digital emulation of passive physical systems. Starting from a system description in the form of a suited partial differential equation (PDE), a reference circuit having a one-to-one corresponding wave digital structure has to be synthesized. This synthesis transfers the principle of action at proximity and the passivity of the physical system into algorithms, which offer full parallelism, passivity, and robustness against signal and coefficient quantizations. Finding a reference circuit to a given PDE is an open problem, which has only partially been solved. In this paper, we present a schematic reference circuit synthesis for a special class of hyperbolic PDEs. In order to cut down the complexity of the synthesis, we introduce additional design parameters and decompose the PDE into an ordinary differential equation (ODE) and hyperbolic PDE subsystems. The parameters can be exploited to optimize the reference circuit and its corresponding wave digital structure. The method is applied to the Sine-Gordon equation from the soliton theory and verified by simulation. Figure 1. Steps in wave digital emulation.In the first step, the physical system has to be represented by a mathematical model in the form of passive hyperbolic partial differential equations (PDEs), which reflect its essential characteristics, passivity, and principle of action at proximity. These properties are transferred into a multidimensional (MD) electrical reference circuit, representing the PDEs (e.g. in its loop or node equations). Replacing every circuit element with its wave digital correspondence, one obtains a wave digital structure free of delay-free directed loops. An implementation of this structure yields the resulting wave digital algorithm. During the whole synthesis, the passivity and exclusively local nature of the underlying physical phenomena are retained and finally transferred to the algorithm, which is passive and robust against signal and coefficient quantizations and allows for massive parallelism.The crux of this procedure is the synthesis of a reference circuit, which demands profound background knowledge of the wave digital concept, as well as some essential network theory from the developer. Hence, a schematic approach that allows for computer-aided circuit syntheses is desirable. A reference circuit synthesis based on admittance matrices has been presented in [3]. However, this approach is restricted to linear constant PDEs and steady-state solutions only. In contrast, the syntheses of electrical circuits presented in [4, 5] consider ordinary differential equations (ODEs) only, but allow for the modeling of certain nonlinearities [4]. A first extension of this approach to PDEs is derived in [6], but the resulting circuits are usually unsuited as reference circuits for wave digital modeling and an elaborate post-processing is necessary. For some established examples, this post-processing has manually been performed in [7]. Under further restrictions to the PDEs, fi...

Section: Variable Inductance Matricesmentioning

confidence: 93%

Section: Realization Of the Pde Subsystemsmentioning

confidence: 99%

Section: The 'Perturbed' Sine-gordon Equationmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

See 2 more Smart Citations

On systematic wave digital modeling of passive hyperbolic partial differential equations

Leuer

Ochs

2011

“…Demand for fast switching digital circuits has led the researchers to explore new devices based on superconductor technology 1–3 . In this direction, the Josephson junction (JJ) based on the single flux quantum (SFQ) phenomenon and quantum phase slip (QPS) phenomenon in superconductor nanowires has attracted much attention 3,4 . In a JJ device, the charge tunneling across the insulator barrier between the two sides of the junction induces a quantum flux in the corresponding loop 3,5 .…”

Section: Introductionmentioning

confidence: 99%

Increasing fan‐in and fan‐out of the quantum phase slip junction‐based logic gates

Omranpour

Hashemi

2022

Fan‐in and fan‐out of the quantum phase slip junction (QPSJ)‐based two‐input AND and OR gates are increased for the first time. For increasing their fan‐in, critical voltage, normal and sub‐gap resistances, and kinetic inductance of their constituent QPSJs are changed, and only input and buffer QPSJs are added to the gate. The number of elements in the proposed three‐ and four‐input OR and three‐ and four‐input AND gates is 13, 16, 16, and 20, respectively, which are fewer than those in the serial connection method. For increasing the fan‐out to two, only the mentioned parameters of the constituent QPSJs are changed and no extra element is added to the gate circuits. Comparing with the existing method which adds extra Josephson Junctions and QPSJs to the gate output, the number of elements in the proposed method is that of the basic two‐input gates. For the increased fan‐in/fan‐out OR gates, the critical margins of the external parameters are 10% and 15% of the input voltage, and the critical junction margins are 30% and 26% respectively, while for the increased fan‐in/fan‐out AND gates, they are 9% and 12% of the bias voltage and 26% and 23% for the critical junction margins, respectively.

On stochastic modelling of linear circuits

Rawat

Parthasarathy

2008

SUMMARYIn this paper, the deterministic modelling of linear circuits is replaced by stochastic modelling by including variance in the parameters (resistance, inductance and capacitance). Our method is based on results from the theory of stochastic differential equations. This method is general in the following sense. Any electrical circuit that consists of resistances, inductances and capacitances can be modelled by ordinary differential equations, in which the parameters of the differential operators are the functions of circuit elements. The deterministic ordinary differential equation can be converted into a stochastic differential equation by adding noise to the input potential source and to the circuit elements. The noise added in the potential source is assumed to be a white noise and that added in the parameters is assumed to be a correlated process because these parameters change very slowly with time and hence must be modelled as a correlated process. In this paper, we model a series RLC circuit by using the proposed method. The stochastic differential equation that describes the concentration of charge in the capacitor of a series RLC circuit is solved. Numerical simulations in MATLAB are obtained using the Euler-Maruyama method.