  In Kinetic Inductance Detectors (KIDs) and other similar applications of
superconducting microresonators, both the large and small-signal behaviour of
the device may be affected by electrothermal feedback. Microwave power applied
to read out the device is absorbed by and heats the superconductor
quasiparticles, changing the superconductor conductivity and hence the readout
power absorbed in a positive or negative feedback loop. In this work, we
explore numerically the implications of an extensible theoretical model of a
generic superconducting microresonator device for a typical KID, incorporating
recent work on the power flow between superconductor quasiparticles and
phonons. This model calculates the large-signal (changes in operating point)
and small-signal behaviour of a device, allowing us to determine the effect of
electrothermal feedback on device responsivity and noise characteristics under
various operating conditions. We also investigate how thermally isolating the
device from the bath, for example by designing the device on a membrane only
connected to the bulk substrate by thin legs, affects device performance. We
find that at a typical device operating point, positive electrothermal feedback
reduces the effective thermal conductance from the superconductor
quasiparticles to the bath, and so increases responsivity to signal
(pair-breaking) power, increases noise from temperature fluctuations, and
decreases the Noise Equivalent Power (NEP). Similarly, increasing the thermal
isolation of the device while keeping the quasiparticle temperature constant
decreases the NEP, but also decreases the device response bandwidth.
