  We describe a novel scheme for low-noise phase-insensitive linear
amplification at microwave frequencies based on the Superconducting
Low-inductance Undulatory Galvanometer (SLUG). Direct integration of the
junction equations of motion provides access to the full scattering matrix of
the SLUG. We discuss the optimization of SLUG amplifiers and calculate
amplifier gain and noise temperature in both the thermal and quantum regimes.
Loading of the SLUG element by the finite input admittance is taken into
account, and strategies for decoupling the SLUG from the higher-order modes of
the input circuit are discussed. The microwave SLUG amplifier is expected to
achieve noise performance approaching the standard quantum limit in the
frequency range from 5-10 GHz, with gain around 15 dB for a single-stage device
and instantaneous bandwidths of order 1 GHz.
