Performance Analysis of Digital Flux-locked Loop Circuit with Different SQUID $V$-$\phi$ Transfer Curves for TES Readout System

A superconducting quantum interference device (SQUID), functioning as a nonlinear response device, typically requires the incorporation of a flux-locked loop (FLL) circuit to facilitate linear amplification of the current signal transmitted through a superconducting transition-edge sensor (TES) across a large dynamic range.This work presents a reasonable model of the SQUID-FLL readout system, based on a digital proportional-integral-differential (PID) flux negative feedback algorithm.This work investigates the effect of $V$-$\phi$ shape on the performance of digital FLL circuits.Such as the impact factors of bandwidth, design limits of slew rate of the system and the influence of the shapes of SQUID $V$-$\phi$ curve.Furthermore, the dynamic response of the system to X-ray pulse signals with rise time ranging from $4.4{\sim}281$ $\mathrm{{\mu}s}$ and amplitudes ranging from $6{\sim}8$ $\mathrm{\phi_0}$ was simulated.All the simulation results were found to be consistent with the existing mature theories, thereby validating the accuracy of the model.The results also provide a reliable modelling reference for the design of digital PID flux negative feedback and multiplexing SQUID readout electronic systems.