CMOS fixed-$g_m$ bias circuits are, as the name suggests, those that generate a bias current which keeps the transconductance of a MOS transistor equal to a constant (off-chip) conductance. Such circuits are useful in many analog and mixed-signal subsystems like filters and data-converters. This work derives the textbook fixed-$g_m$ bias circuit from first principles, and shows these ideas can be used to generate alternative circuits.