  The cavities in superconducting linacs tend to operate with narrow bandwidths
that are comparable to microphonic induced detuning from external forces.
Proportional-Integral (PI) feedback loops are used to stabilize the cavity
voltage in response to the varying cavity resonant frequency. Without feedback,
the cavity voltage variation with detuning is non-linear, but in this paper we
show that in the regime where the detuning amplitude is smaller than the
frequency of the detuning, which is typically the case, the response can be
linearized so as to provide simple analytical characterizations of PI loop
performance as a function of the various parameters involved (integral and
proportional gains, noise and low pass filter bandwidth).
