  We theoretically study a superradiant laser, deriving both the steady-state
behaviors and small-amplitude responses of the laser's atomic inversion, atomic
polarization, and light field amplitude. Our minimum model for a three-level
laser includes atomic population accumulating outside of the lasing transition
and dynamics of the atomic population distribution causing cavity frequency
tuning, as can occur in realistic experimental systems. We show that the
population dynamics can act as real-time feedback to stabilize or de-stabilize
the laser's output power, and we derive the cavity frequency tuning for a Raman
laser. We extend the minimal model to describe a cold-atom Raman laser using
$^{87}$Rb, showing that the minimal model qualitatively captures the essential
features of the more complex system. This work informs our understanding of the
stability of proposed millihertz linewidth lasers based on ultranarrow optical
atomic transitions and will guide the design and development of these
next-generation optical frequency references.
